Processing math: 23%

Systems of linear equations


In standard Gaussian elimination with backward substitution, achieving row-echelon form requires that each leading entry in a row be a one (1). However, this approach is suboptimal because:

  1. Students face O(mn) additional opportunities to make errors that do not actually reflect their understanding of the algorithm.
  2. These additional errors make it exceedingly difficult to equitably grade a solution when an insignificant error early on leads to significantly divergent subsequent calculations.
  3. The runtime increases by O(mn).
  4. Extra calculations amplify potential numerical errors.

Linear algebra instructors should consider the advantages of defining row-echelon form without the requirement that each leading non-zero entry be scaled to one. By teaching students that row-echelon form must have leading ones rather than simply leading non-zero entries, we inadvertently encourage algorithmic designs that increase runtime and reduce numerical accuracy. Adopting a more flexible definition can streamline calculations and reduce error risks in both learning and implementation.

Another important consideration is the choice between pivoting and partial pivoting. Conventionally, pivoting is applied only when the current pivot entry is zero. However, this approach is numerically unstable. Teaching partial pivoting from the outset avoids the need to address the special case of a zero pivot and ensures numerical stability throughout the Gaussian elimination process.

Introducing students to a numerically unstable algorithm, only to later revise it with partial pivoting, creates unnecessary confusion. It’s more effective to begin with partial pivoting, which is essential for achieving stable results, especially when working with double-precision floating-point numbers. This approach prevents the frustrating scenario of telling students: “What we taught you initially doesn’t actually work for real-world computations, so here’s an additional fix.”

In order to support this, the following gives a plethora of examples of systems of linear equations that can be solved on the blackboard or during an examination using Gaussian elimination with partial pivoting, and where the solution is “nice”, meaning that the computations can be done without fractions and with only a single digit beyond the decimal point.

In each case,

  1. If rank(A)<m, then two examples with no solutions will be given, and two examples with one or infinitely many solutions will be given.
  2. If rank(A)=m, then two examples of either one or infinitely many solutions will be given.
  3. If rank(A)<n, then a basis for the null space found by solving the system of linear equations using Gaussian elimination with partial pivoting will be given.
2×2 2×3 2×4
3×2 3×3 3×4 3×5
4×2 4×3 4×4 4×5 4×6
5×2 5×3 5×4 5×5 5×6
6×2 6×3 6×4 6×5 6×6

One important consideration is the basis for the null space: we make sure that the coefficients of the null space have at most one or two digits after the decimal point. You may now pick any solution vector you choose, and find

  1. 2×2 with 0 free variables
  2. 2×2 with 1 free variables

2×2

2×2 with no free variables

   
* *
0 *

Example matrix

A=(967.27.8)(9.06.003.0) , null(A)={0}

Example solution

b=(30.6) and u=(11)

Example solution

b=(61.2) and u=(22)

Example matrix

A=(611.25.2)(6.01.005.0) , null(A)={0}

Example solution

b=(54) and u=(11)

Example solution

b=(49.2) and u=(12)

Example matrix

A=(8.57.53.410)(8.57.507.0) , null(A)={0}

Example solution

b=(9.53.2) and u=(21)

Example solution

b=(10.59.8) and u=(32)

Example matrix

A=(0.850.42.1)(0.85.000.4) , null(A)={0}

Example solution

b=(1.80.5) and u=(41)

Example solution

b=(2.60.9) and u=(31)

2×2 with one free variable

  v
* *
0 0

Example matrix

A=(221.41.4)(2200) , null(A)=span{(11)}

Example solution

b=(21.4) and u=(10)+u2(11)

Example solution

b=(42.8) and u=(20)+u2(11)

Example matrix

A=(140.72.8)(1400) , null(A)=span{(41)}

Example solution

b=(10.7) and u=(10)+u2(41)

Example solution

b=(21.4) and u=(20)+u2(41)

Example matrix

A=(10.50.20.1)(10.500) , null(A)=span{(0.51)}

Example solution

b=(10.2) and u=(10)+u2(0.51)

Example solution

b=(20.4) and u=(20)+u2(0.51)

Example matrix

A=(22.611.3)(22.600) , null(A)=span{(1.31)}

Example solution

b=(21) and u=(10)+u2(1.31)

Example solution

b=(42) and u=(20)+u2(1.31)

2×3

  1. 2×3 with 1 free variables
  2. 2×3 with 2 free variables

2×3 with one free variable

    v
* * *
0 * *

Example matrix

A=(2891.85.24.1)(289024) , null(A)=span{(3.521)}

Example solution

b=(02) and u=(410)+u3(3.521)

Example solution

b=(20.2) and u=(310)+u3(3.521)

Example matrix

A=(440212)(440012) , null(A)=span{(221)}

Example solution

b=(40) and u=(120)+u3(221)

Example solution

b=(41) and u=(230)+u3(221)

Example matrix

A=(0.59.530.25.80.8)(0.59.53022) , null(A)=span{(2511)}

Example solution

b=(7.55) and u=(410)+u3(2511)

Example solution

b=(85.2) and u=(310)+u3(2511)

Example matrix

A=(282.413.63)(282.400.41.8) , null(A)=span{(19.24.51)}

Example solution

b=(00.4) and u=(410)+u3(19.24.51)

Example solution

b=(20.6) and u=(310)+u3(19.24.51)

  v  
* * *
0 0 *

Example matrix

A=(249122.5)(2.04.09.0002.0) , null(A)=span{(210)}

Example solution

b=(11.5) and u=(401)+u2(210)

Example solution

b=(30.5) and u=(301)+u2(210)

Example matrix

A=(5173.50.78.9)(5.01.07.0004.0) , null(A)=span{(0.210)}

Example solution

b=(31.9) and u=(201)+u2(0.210)

Example solution

b=(25.4) and u=(101)+u2(0.210)

Example matrix

A=(2.539.50.50.60.9)(2.53.09.5001.0) , null(A)=span{(1.210)}

Example solution

b=(0.51.1) and u=(401)+u2(1.210)

Example solution

b=(20.6) and u=(301)+u2(1.210)

Example matrix

A=(23.42.611.73.9)(2.03.42.6005.2) , null(A)=span{(1.710)}

Example solution

b=(0.64.9) and u=(101)+u2(1.710)

Example solution

b=(1.45.9) and u=(201)+u2(1.710)

2×3 with two free variables

  v v
* * *
0 0 0

Example matrix

A=(1560.10.50.6)(156000) , null(A)=span{(510),(601)}

Example solution

b=(10.1) and u=(100)+u2(510)+u3(601)

Example solution

b=(20.2) and u=(200)+u2(510)+u3(601)

Example matrix

A=(1240.91.83.6)(124000) , null(A)=span{(210),(401)}

Example solution

b=(10.9) and u=(100)+u2(210)+u3(401)

Example solution

b=(21.8) and u=(200)+u2(210)+u3(401)

Example matrix

A=(0.58.57.50.35.14.5)(0.58.57.5000) , null(A)=span{(1710),(1501)}

Example solution

b=(0.50.3) and u=(100)+u2(1710)+u3(1501)

Example solution

b=(10.6) and u=(200)+u2(1710)+u3(1501)

Example matrix

A=(0.21.22.40.10.61.2)(0.21.22.4000) , null(A)=span{(610),(1201)}

Example solution

b=(0.20.1) and u=(100)+u2(610)+u3(1201)

Example solution

b=(0.40.2) and u=(200)+u2(610)+u3(1201)

2×4

  1. 2×4 with 2 free variables
  2. 2×4 with 3 free variables

2×4 with two free variables

    v v
* * * *
0 * * *

Example matrix

A=(15430.830.24.4)(15430132) , null(A)=span{(11310),(13201)}

Example solution

b=(10.2) and u=(4100)+u3(11310)+u4(13201)

Example solution

b=(20.6) and u=(3100)+u3(11310)+u4(13201)

Example matrix

A=(11360.64.410.811.6)(11360598) , null(A)=span{(4.81.810),(7.61.601)}

Example solution

b=(23.8) and u=(1100)+u3(4.81.810)+u4(7.61.601)

Example solution

b=(33.2) and u=(2100)+u3(4.81.810)+u4(7.61.601)

Example matrix

A=(0.58.5950.47.34.78)(0.58.59500.52.54) , null(A)=span{(103510),(126801)}

Example solution

b=(6.55.7) and u=(4100)+u3(103510)+u4(126801)

Example solution

b=(76.1) and u=(3100)+u3(103510)+u4(126801)

Example matrix

A=(0.29.6690.15.20.83.3)(0.29.66900.42.21.2) , null(A)=span{(2945.510),(189301)}

Example solution

b=(8.84.8) and u=(4100)+u3(2945.510)+u4(189301)

Example solution

b=(94.9) and u=(3100)+u3(2945.510)+u4(189301)

  v   v
* * * *
0 0 * *

Example matrix

A=(28860.41.60.47.8)(28860029) , null(A)=span{(4100),(1504.51)}

Example solution

b=(02) and u=(4010)+u2(4100)+u4(1504.51)

Example solution

b=(21.6) and u=(3010)+u2(4100)+u4(1504.51)

Example matrix

A=(23090.20.311.1)(23090012) , null(A)=span{(1.5100),(4.5021)}

Example solution

b=(20.8) and u=(1010)+u2(1.5100)+u4(4.5021)

Example solution

b=(21.8) and u=(1020)+u2(1.5100)+u4(4.5021)

Example matrix

A=(0.58.55.520.46.85.44.6)(0.58.55.520013) , null(A)=span{(17100),(29031)}

Example solution

b=(3.53.8) and u=(4010)+u2(17100)+u4(29031)

Example solution

b=(44.2) and u=(3010)+u2(17100)+u4(29031)

Example matrix

A=(0.22.89.27.80.11.43.65.5)(0.22.89.27.80011.6) , null(A)=span{(14100),(112.601.61)}

Example solution

b=(8.43.2) and u=(4010)+u2(14100)+u4(112.601.61)

Example solution

b=(8.63.3) and u=(3010)+u2(14100)+u4(112.601.61)

  v v  
* * * *
0 0 0 *

Example matrix

A=(26770.41.21.46.6)(26770008) , null(A)=span{(3100),(3.5010)}

Example solution

b=(17.8) and u=(3001)+u2(3100)+u3(3.5010)

Example solution

b=(37.4) and u=(2001)+u2(3100)+u3(3.5010)

Example matrix

A=(597135.44.26.6)(59710006) , null(A)=span{(1.8100),(1.4010)}

Example solution

b=(43.6) and u=(1001)+u2(1.8100)+u3(1.4010)

Example solution

b=(90.6) and u=(2001)+u2(1.8100)+u3(1.4010)

Example matrix

A=(55.54.5933.32.713.4)(55.54.590008) , null(A)=span{(1.1100),(0.9010)}

Example solution

b=(64.4) and u=(3001)+u2(1.1100)+u3(0.9010)

Example solution

b=(17.4) and u=(2001)+u2(1.1100)+u3(0.9010)

Example matrix

A=(21.40.21.410.70.19.1)(21.40.21.40009.8) , null(A)=span{(0.7100),(0.1010)}

Example solution

b=(3.48.1) and u=(1001)+u2(0.7100)+u3(0.1010)

Example solution

b=(5.47.1) and u=(2001)+u2(0.7100)+u3(0.1010)

2×4 with three free variables

  v v v
* * * *
0 0 0 0

Example matrix

A=(17730.96.36.32.7)(17730000) , null(A)=span{(7100),(7010),(3001)}

Example solution

b=(10.9) and u=(1000)+u2(7100)+u3(7010)+u4(3001)

Example solution

b=(21.8) and u=(2000)+u2(7100)+u3(7010)+u4(3001)

Example matrix

A=(29711.88.16.30.9)(29710000) , null(A)=span{(4.5100),(3.5010),(0.5001)}

Example solution

b=(21.8) and u=(1000)+u2(4.5100)+u3(3.5010)+u4(0.5001)

Example solution

b=(43.6) and u=(2000)+u2(4.5100)+u3(3.5010)+u4(0.5001)

Example matrix

A=(0.58.5970.23.43.62.8)(0.58.5970000) , null(A)=span{(17100),(18010),(14001)}

Example solution

b=(0.50.2) and u=(1000)+u2(17100)+u3(18010)+u4(14001)

Example solution

b=(10.4) and u=(2000)+u2(17100)+u3(18010)+u4(14001)

Example matrix

A=(21.40.64.610.70.32.3)(21.40.64.60000) , null(A)=span{(0.7100),(0.3010),(2.3001)}

Example solution

b=(21) and u=(1000)+u2(0.7100)+u3(0.3010)+u4(2.3001)

Example solution

b=(42) and u=(2000)+u2(0.7100)+u3(0.3010)+u4(2.3001)

3×2

  1. 3×2 with 0 free variables
  2. 3×2 with 1 free variables

3×2 with no free variables

   
* *
0 *
0 0

Example matrix

A=(844645.2)(840800) , null(A)={0}

Example solution

b=(4109.2) and u=(11)

Example solution

b=(1686.4) and u=(12)

Example matrix

A=(442.40.622.6)(440300) , null(A)={0}

Example solution

b=(43.63.2) and u=(12)

Example solution

b=(45.41.4) and u=(21)

Example matrix

A=(23.51.24.61.63.3)(23.502.500) , null(A)={0}

Example solution

b=(0.52.20.1) and u=(21)

Example solution

b=(2.511.5) and u=(31)

Example matrix

A=(1.312.12.69.81.38.5)(2.69.807.200) , null(A)={0}

Example solution

b=(6.90.63.3) and u=(41)

Example solution

b=(8.224.6) and u=(31)

3×2 with one free variable

  v
* *
0 0
0 0

Example matrix

A=(1.60.8423.21.6)(420000) , null(A)=span{(0.51)}

Example solution

b=(1.643.2) and u=(10)+u2(0.51)

Example solution

b=(3.286.4) and u=(20)+u2(0.51)

Example matrix

A=(40.8514.50.9)(510000) , null(A)=span{(0.21)}

Example solution

b=(454.5) and u=(10)+u2(0.21)

Example solution

b=(8109) and u=(20)+u2(0.21)

Example matrix

A=(2.55.51.53.312.2)(2.55.50000) , null(A)=span{(2.21)}

Example solution

b=(2.51.51) and u=(10)+u2(2.21)

Example solution

b=(532) and u=(20)+u2(2.21)

Example matrix

A=(224422)(440000) , null(A)=span{(11)}

Example solution

b=(242) and u=(10)+u2(11)

Example solution

b=(484) and u=(20)+u2(11)

Example matrix

A=(1.60.8423.21.6)(420000) , null(A)=span{(0.51)}

Example solution

b=(1.643.2) and u=(10)+u2(0.51)

Example solution

b=(3.286.4) and u=(20)+u2(0.51)

Example matrix

A=(40.8514.50.9)(510000) , null(A)=span{(0.21)}

Example solution

b=(454.5) and u=(10)+u2(0.21)

Example solution

b=(8109) and u=(20)+u2(0.21)

Example matrix

A=(2.55.51.53.312.2)(2.55.50000) , null(A)=span{(2.21)}

Example solution

b=(2.51.51) and u=(10)+u2(2.21)

Example solution

b=(532) and u=(20)+u2(2.21)

Example matrix

A=(224422)(440000) , null(A)=span{(11)}

Example solution

b=(242) and u=(10)+u2(11)

Example solution

b=(484) and u=(20)+u2(11)

3×3

  1. 3×3 with 2 free variables
  2. 3×3 with 1 free variables
  3. 3×3 with 2 free variables

3×3 with no free variables

     
* * *
0 * *
0 0 *

Example matrix

A=(4.25.39.46921.83.22.4)(692018001) , null(A)={0}

Example solution

b=(3.243.2) and u=(421)

Example solution

b=(7.421.4) and u=(321)

Example matrix

A=(6911.29.86.83.611.84)(691087001) , null(A)={0}

Example solution

b=(20.60.6) and u=(211)

Example solution

b=(41.21.2) and u=(422)

Example matrix

A=(1470.20.34.10.21.23.7)(14700.55.5009.5) , null(A)={0}

Example solution

b=(24.15.5) and u=(321)

Example solution

b=(14.35.7) and u=(221)

Example matrix

A=(3.81.22.67.66.84.43.82.35.8)(7.66.84.402.24.8006) , null(A)={0}

Example solution

b=(1.40.86.5) and u=(231)

Example solution

b=(0.27.64.2) and u=(241)

3×3 with one free variable

    v
* * *
0 * *
0 0 0

Example matrix

A=(2860.64.41.20.20.60.9)(286023000) , null(A)=span{(91.51)}

Example solution

b=(020.2) and u=(410)+u3(91.51)

Example solution

b=(22.60) and u=(310)+u3(91.51)

Example matrix

A=(0.820.21590.10.92.3)(159027000) , null(A)=span{(8.53.51)}

Example solution

b=(1.210.5) and u=(410)+u3(8.53.51)

Example solution

b=(0.420.6) and u=(310)+u3(8.53.51)

Example matrix

A=(514.540.27.620.24.2)(514.5014000) , null(A)=span{(1.741)}

Example solution

b=(14.82.8) and u=(140)+u3(1.741)

Example solution

b=(24.62.6) and u=(130)+u3(1.741)

Example matrix

A=(0.2310.30.442.60.21.53.2)(0.442.6019000) , null(A)=span{(83.591)}

Example solution

b=(2.22.40.7) and u=(410)+u3(83.591)

Example solution

b=(2.42.80.9) and u=(310)+u3(83.591)

  v  
* * *
0 0 *
0 0 0

Example matrix

A=(4.55.47.156122.42.8)(561008000) , null(A)=span{(1.210)}

Example solution

b=(11.640.8) and u=(101)+u2(1.210)

Example solution

b=(1.9116.8) and u=(201)+u2(1.210)

Example matrix

A=(2791.86.35.11.65.68.7)(279003000) , null(A)=span{(3.510)}

Example solution

b=(12.12.3) and u=(401)+u2(3.510)

Example solution

b=(30.33.9) and u=(301)+u2(3.510)

Example matrix

A=(21.42.353.59.521.45)(53.59.5001.5000) , null(A)=span{(0.710)}

Example solution

b=(1.70.51) and u=(201)+u2(0.710)

Example solution

b=(3.412) and u=(402)+u2(0.710)

Example matrix

A=(0.41.83.80.20.96.30.20.94.1)(0.41.83.8004.4000) , null(A)=span{(4.510)}

Example solution

b=(2.25.53.3) and u=(401)+u2(4.510)

Example solution

b=(2.65.73.5) and u=(301)+u2(4.510)

3×3 with two free variables

  v v
* * *
0 0 0
0 0 0

Example matrix

A=(2341.42.12.81.62.43.2)(234000000) , null(A)=span{(1.510),(201)}

Example solution

b=(21.41.6) and u=(100)+u2(1.510)+u3(201)

Example solution

b=(42.83.2) and u=(200)+u2(1.510)+u3(201)

Example matrix

A=(2941.46.32.80.20.90.4)(294000000) , null(A)=span{(4.510),(201)}

Example solution

b=(21.40.2) and u=(100)+u2(4.510)+u3(201)

Example solution

b=(42.80.4) and u=(200)+u2(4.510)+u3(201)

Example matrix

A=(21.4153.52.542.82)(53.52.5000000) , null(A)=span{(0.710),(0.501)}

Example solution

b=(254) and u=(100)+u2(0.710)+u3(0.501)

Example solution

b=(4108) and u=(200)+u2(0.710)+u3(0.501)

Example matrix

A=(16.86.60.53.43.30.53.43.3)(16.86.6000000) , null(A)=span{(6.810),(6.601)}

Example solution

b=(10.50.5) and u=(100)+u2(6.810)+u3(6.601)

Example solution

b=(211) and u=(200)+u2(6.810)+u3(6.601)

Example matrix

A=(2341.42.12.81.62.43.2)(234000000) , null(A)=span{(1.510),(201)}

Example solution

b=(21.41.6) and u=(100)+u2(1.510)+u3(201)

Example solution

b=(42.83.2) and u=(200)+u2(1.510)+u3(201)

Example matrix

A=(2941.46.32.80.20.90.4)(294000000) , null(A)=span{(4.510),(201)}

Example solution

b=(21.40.2) and u=(100)+u2(4.510)+u3(201)

Example solution

b=(42.80.4) and u=(200)+u2(4.510)+u3(201)

Example matrix

A=(21.4153.52.542.82)(53.52.5000000) , null(A)=span{(0.710),(0.501)}

Example solution

b=(254) and u=(100)+u2(0.710)+u3(0.501)

Example solution

b=(4108) and u=(200)+u2(0.710)+u3(0.501)

Example matrix

A=(16.86.60.53.43.30.53.43.3)(16.86.6000000) , null(A)=span{(6.810),(6.601)}

Example solution

b=(10.50.5) and u=(100)+u2(6.810)+u3(6.601)

Example solution

b=(211) and u=(200)+u2(6.810)+u3(6.601)

3×4

  1. 3×4 with 1 free variables
  2. 3×4 with 2 free variables
  3. 3×4 with 3 free variables

3×4 with one free variable

      v
* * * *
0 * * *
0 0 * *

Example matrix

A=(0.855.87.640621.24.55.52.6)(406205780014) , null(A)=span{(6.57.241)}

Example solution

b=(101.1) and u=(3320)+u4(6.57.241)

Example solution

b=(2.621.1) and u=(2210)+u4(6.57.241)

Example matrix

A=(0.42.47.4727351.23.352.4)(273501860040) , null(A)=span{(18.5601)}

Example solution

b=(6.623.5) and u=(4110)+u4(18.5601)

Example solution

b=(6.244.7) and u=(3110)+u4(18.5601)

Example matrix

A=(2.57.59.54187.82.61.50.51.54.1)(2.57.59.5405410012.5) , null(A)=span{(4.52.22.51)}

Example solution

b=(1.51.40.5) and u=(1220)+u4(4.52.22.51)

Example solution

b=(12.60) and u=(2330)+u4(4.52.22.51)

Example matrix

A=(57.65.22.42.539.60.62.54.20.79.7)(57.65.22.400.870.6000.28.2) , null(A)=span{(588.6359.5411)}

Example solution

b=(0.45.62.3) and u=(4210)+u4(588.6359.5411)

Example solution

b=(08.64.1) and u=(2210)+u4(588.6359.5411)

    v  
* * * *
0 * * *
0 0 0 *

Example matrix

A=(17720.47.86.20.80.62.27.80.2)(177205900001) , null(A)=span{(19.61.810)}

Example solution

b=(24.20.8) and u=(1104)+u3(19.61.810)

Example solution

b=(14.60.4) and u=(2103)+u3(19.61.810)

Example matrix

A=(26.84.86.457761.50.11.11.8)(577604240002) , null(A)=span{(0.70.510)}

Example solution

b=(402) and u=(1102)+u3(0.70.510)

Example solution

b=(3.613.7) and u=(1203)+u3(0.70.510)

Example matrix

A=(52.578.5107.14.331.91117.9)(52.578.500.58.560008) , null(A)=span{(7.11710)}

Example solution

b=(6.56.38.1) and u=(2201)+u3(7.11710)

Example solution

b=(46.310) and u=(2101)+u3(7.11710)

Example matrix

A=(0.46.489.80.245.613.70.23.64.80.3)(0.46.489.800.81.68.80009.6) , null(A)=span{(12210)}

Example solution

b=(1.46.56.7) and u=(4201)+u3(12210)

Example solution

b=(1.86.36.9) and u=(3201)+u3(12210)

  v    
* * * *
0 0 * *
0 0 0 *

Example matrix

A=(1.41.48.38.122930.80.85.414.6)(229300260008) , null(A)=span{(1100)}

Example solution

b=(2.977) and u=(4021)+u2(1100)

Example solution

b=(4.396.2) and u=(3021)+u2(1100)

Example matrix

A=(43.25.20.854161.51.21.50.4)(541600640003) , null(A)=span{(0.8100)}

Example solution

b=(1.215.3) and u=(2012)+u2(0.8100)

Example solution

b=(2.863.8) and u=(1012)+u2(0.8100)

Example matrix

A=(5309.510.684.442.43.213.1)(5309.50082.50004.5) , null(A)=span{(0.6100)}

Example solution

b=(13.27) and u=(4012)+u2(0.6100)

Example solution

b=(5.59.44.3) and u=(3011)+u2(0.6100)

Example matrix

A=(12.76.114.125.43.49.812.75.64.3)(25.43.49.8007.89.20004.6) , null(A)=span{(2.7100)}

Example solution

b=(2.18.62.9) and u=(4021)+u2(2.7100)

Example solution

b=(1.110.63.9) and u=(3021)+u2(2.7100)

Example matrix

A=(1.41.48.38.122930.80.85.414.6)(229300260008) , null(A)=span{(1100)}

Example solution

b=(2.977) and u=(4021)+u2(1100)

Example solution

b=(4.396.2) and u=(3021)+u2(1100)

Example matrix

A=(43.25.20.854161.51.21.50.4)(541600640003) , null(A)=span{(0.8100)}

Example solution

b=(1.215.3) and u=(2012)+u2(0.8100)

Example solution

b=(2.863.8) and u=(1012)+u2(0.8100)

Example matrix

A=(5309.510.684.442.43.213.1)(5309.50082.50004.5) , null(A)=span{(0.6100)}

Example solution

b=(13.27) and u=(4012)+u2(0.6100)

Example solution

b=(5.59.44.3) and u=(3011)+u2(0.6100)

Example matrix

A=(12.76.114.125.43.49.812.75.64.3)(25.43.49.8007.89.20004.6) , null(A)=span{(2.7100)}

Example solution

b=(2.18.62.9) and u=(4021)+u2(2.7100)

Example solution

b=(1.110.63.9) and u=(3021)+u2(2.7100)

3×4 with two free variables

    v v
* * * *
0 * * *
0 0 0 0

Example matrix

A=(1.43.60.53.128570.81.80.82.8)(285702480000) , null(A)=span{(5.5210),(12.5401)}

Example solution

b=(0.620.6) and u=(3100)+u3(5.5210)+u4(12.5401)

Example solution

b=(201.4) and u=(4100)+u3(5.5210)+u4(12.5401)

Example matrix

A=(0.86.24.42.419220.5421)(192201640000) , null(A)=span{(52610),(34401)}

Example solution

b=(352) and u=(4100)+u3(52610)+u4(34401)

Example solution

b=(3.862.5) and u=(3100)+u3(52610)+u4(34401)

Example matrix

A=(10380.253.40.60.834.87)(103805410000) , null(A)=span{(30.810),(80.201)}

Example solution

b=(25.41.4) and u=(2100)+u3(30.810)+u4(80.201)

Example solution

b=(14.83.8) and u=(1100)+u3(30.810)+u4(80.201)

Example matrix

A=(2.54.26.112.158.89.45.42.54.35.47.4)(58.89.45.400.21.49.40000) , null(A)=span{(14.2710),(83.84701)}

Example solution

b=(0.81.20.7) and u=(2100)+u3(14.2710)+u4(83.84701)

Example solution

b=(0.92.61.1) and u=(3200)+u3(14.2710)+u4(83.84701)

  v   v
* * * *
0 0 * *
0 0 0 0

Example matrix

A=(1.21.81.610.423190.60.90.20.2)(231900150000) , null(A)=span{(1.5100),(2051)}

Example solution

b=(201) and u=(1020)+u2(1.5100)+u4(2051)

Example solution

b=(2.411.8) and u=(2030)+u2(1.5100)+u4(2051)

Example matrix

A=(3.67.210.144890243.91.2)(489000240000) , null(A)=span{(2100),(4.5021)}

Example solution

b=(2.910.1) and u=(2010)+u2(2100)+u4(4.5021)

Example solution

b=(0.732.1) and u=(3010)+u2(2100)+u4(4.5021)

Example matrix

A=(188.57.50.43.23.93.50.86.46.75.9)(188.57.5000.50.50000) , null(A)=span{(8100),(1011)}

Example solution

b=(4.52.33.5) and u=(4010)+u2(8100)+u4(1011)

Example solution

b=(5.52.74.3) and u=(3010)+u2(8100)+u4(1011)

Example matrix

A=(0.25.87.64.40.12.92.87.60.12.94.37.1)(0.25.87.64.40019.80000) , null(A)=span{(29100),(350.409.81)}

Example solution

b=(6.82.43.9) and u=(4010)+u2(29100)+u4(350.409.81)

Example solution

b=(72.54) and u=(3010)+u2(29100)+u4(350.409.81)

  v v  
* * * *
0 0 0 *
0 0 0 0

Example matrix

A=(549732.45.41.232.45.43.9)(549700030000) , null(A)=span{(0.8100),(1.8010)}

Example solution

b=(34.82.1) and u=(2001)+u2(0.8100)+u3(1.8010)

Example solution

b=(16.61.2) and u=(3002)+u2(0.8100)+u3(1.8010)

Example matrix

A=(3.26.44.89.648672433.1)(486700040000) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}2 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 1.5 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 3.2 \\ 1 \\ 0.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}2 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 1.5 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}0 \\ 5 \\ 2.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}2 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 1.5 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrr}- 1.6 & 1.6 & 6.4 & -10 \\ -2 & 2 & 8 & - 2.5 \\ 1.6 & - 1.6 & - 6.4 & 3.6 \end{array}\right) \sim \left(\begin{array}{rrrr}-2 & 2 & 8 & - 2.5 \\ 0 & 0 & 0 & -8 \\ 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}1 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}4 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 5.2 \\ - 3.5 \\ 1.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}1 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}4 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 6.8 \\ - 1.5 \\ - 0.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}1 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}4 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrr} 0.4 & - 1.8 & 6.4 & 0.2 \\ 0.2 & - 0.9 & 3.2 & 2.1 \\ - 0.2 & 0.9 & - 3.2 & - 1.1 \end{array}\right) \sim \left(\begin{array}{rrrr} 0.4 & - 1.8 & 6.4 & 0.2 \\ 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 4.5 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-16 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}1 \\ - 1.5 \\ 0.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r} 4.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-16 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 0.6 \\ - 1.7 \\ 0.7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r} 4.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-16 \\ 0 \\ 1 \\ 0 \end{array}\right)

3 \times 4 with three free variables

  v v v
* * * *
0 0 0 0
0 0 0 0

Example matrix

A = \left(\begin{array}{rrrr}- 0.6 & 4.2 & - 4.2 & 1.2 \\ 1 & -7 & 7 & -2 \\ 0.8 & - 5.6 & 5.6 & - 1.6 \end{array}\right) \sim \left(\begin{array}{rrrr}1 & -7 & 7 & -2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}7 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-7 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.6 \\ 1 \\ 0.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}7 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-7 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.2 \\ 2 \\ 1.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}7 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-7 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrr}- 0.4 & - 1.6 & -2 & - 2.4 \\ 1 & 4 & 5 & 6 \\ 0.6 & 2.4 & 3 & 3.6 \end{array}\right) \sim \left(\begin{array}{rrrr}1 & 4 & 5 & 6 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-4 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-5 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}-6 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.4 \\ 1 \\ 0.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-4 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-5 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-6 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.8 \\ 2 \\ 1.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-4 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-5 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-6 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrr}-2 & 6 & - 5.2 & 5.2 \\ - 2.5 & 7.5 & - 6.5 & 6.5 \\ - 0.5 & 1.5 & - 1.3 & 1.3 \end{array}\right) \sim \left(\begin{array}{rrrr}- 2.5 & 7.5 & - 6.5 & 6.5 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}3 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 2.6 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 2.6 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-2 \\ - 2.5 \\ - 0.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}3 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 2.6 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 2.6 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-4 \\ -5 \\ -1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}3 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 2.6 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 2.6 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrr}- 0.1 & 3.4 & 2.8 & 3.2 \\ 0.2 & - 6.8 & - 5.6 & - 6.4 \\ 0.1 & - 3.4 & - 2.8 & - 3.2 \end{array}\right) \sim \left(\begin{array}{rrrr} 0.2 & - 6.8 & - 5.6 & - 6.4 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}34 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}28 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}32 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.1 \\ 0.2 \\ 0.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}34 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}28 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}32 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.2 \\ 0.4 \\ 0.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}34 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}28 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}32 \\ 0 \\ 0 \\ 1 \end{array}\right)

3 \times 5

  1. 3 \times 5 with 2 free variables
  2. 3 \times 5 with 3 free variables
  3. 3 \times 5 with 4 free variables

3 \times 5 with two free variables

      v v
* * * * *
0 * * * *
0 0 * * *

Example matrix

A = \left(\begin{array}{rrrrc}-2 & -3 & -8 & -7 & -1 \\ 1 & - 3.5 & 6 & 3.5 & 4.5 \\ - 0.2 & 3.7 & - 4.4 & - 9.7 & 4.7 \end{array}\right) \sim \left(\begin{array}{rrrrc}-2 & -3 & -8 & -7 & -1 \\ 0 & -5 & 2 & 0 & 4 \\ 0 & 0 & -2 & -9 & 8 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 17.2 \\ - 1.8 \\ - 4.5 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}- 20.1 \\ 2.4 \\ 4 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}6 \\ 5 \\ - 3.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -2 \\ -1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 17.2 \\ - 1.8 \\ - 4.5 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 20.1 \\ 2.4 \\ 4 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-3 \\ - 5.5 \\ 7.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 1 \\ -1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 17.2 \\ - 1.8 \\ - 4.5 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 20.1 \\ 2.4 \\ 4 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc}-1 & -9 & -4 & -2 & 9 \\ - 0.7 & - 4.3 & - 3.8 & 2.6 & - 1.7 \\ - 0.9 & - 7.5 & - 8.9 & - 3.6 & 11.7 \end{array}\right) \sim \left(\begin{array}{rrrrc}-1 & -9 & -4 & -2 & 9 \\ 0 & 2 & -1 & 4 & -8 \\ 0 & 0 & -5 & -3 & 6 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 21.1 \\ - 2.3 \\ - 0.6 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}- 37.2 \\ 4.6 \\ 1.2 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}3 \\ - 0.9 \\ - 3.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ -1 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 21.1 \\ - 2.3 \\ - 0.6 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 37.2 \\ 4.6 \\ 1.2 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}2 \\ - 1.6 \\ - 4.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ -1 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 21.1 \\ - 2.3 \\ - 0.6 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 37.2 \\ 4.6 \\ 1.2 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc} 2.5 & -4 & 9.5 & -7 & - 6.5 \\ -1 & 2.6 & - 8.3 & - 5.7 & 8.1 \\ 1 & - 0.8 & - 0.3 & - 0.6 & 0.3 \end{array}\right) \sim \left(\begin{array}{rrrrc} 2.5 & -4 & 9.5 & -7 & - 6.5 \\ 0 & 1 & - 4.5 & - 8.5 & 5.5 \\ 0 & 0 & - 0.5 & 9 & - 1.5 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 77.6 \\ 89.5 \\ 18 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}- 16.4 \\ -19 \\ -3 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}1 \\ - 0.9 \\ - 0.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 4 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 77.6 \\ 89.5 \\ 18 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 16.4 \\ -19 \\ -3 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.5 \\ 0.1 \\ - 1.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 4 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 77.6 \\ 89.5 \\ 18 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 16.4 \\ -19 \\ -3 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc} 0.4 & 3.8 & 8.4 & - 6.8 & 9.4 \\ 0.2 & 1.7 & 5.8 & 0.6 & 2.3 \\ - 0.2 & - 1.8 & -9 & - 6.2 & 5.3 \end{array}\right) \sim \left(\begin{array}{rrrrc} 0.4 & 3.8 & 8.4 & - 6.8 & 9.4 \\ 0 & - 0.2 & 1.6 & 4 & - 2.4 \\ 0 & 0 & -4 & - 7.6 & 8.8 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 11.3 \\ 4.8 \\ - 1.9 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}- 122.9 \\ 5.6 \\ 2.2 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 2.2 \\ 1.1 \\ -4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ -3 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 11.3 \\ 4.8 \\ - 1.9 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 122.9 \\ 5.6 \\ 2.2 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.8 \\ 1.3 \\ - 4.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ -3 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 11.3 \\ 4.8 \\ - 1.9 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 122.9 \\ 5.6 \\ 2.2 \\ 0 \\ 1 \end{array}\right)

    v   v
* * * * *
0 * * * *
0 0 0 * *

Example matrix

A = \left(\begin{array}{rrrrc} 0.7 & - 5.2 & - 1.9 & 4.2 & 3.3 \\ -1 & 6 & 7 & -6 & -9 \\ 0.6 & - 4.1 & - 2.7 & 7.6 & 5.9 \end{array}\right) \sim \left(\begin{array}{rrrrc}-1 & 6 & 7 & -6 & -9 \\ 0 & -1 & 3 & 0 & -3 \\ 0 & 0 & 0 & 4 & 2 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}25 \\ 3 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-24 \\ -3 \\ 0 \\ - 0.5 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 3.4 \\ 2 \\ 1.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 2 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}25 \\ 3 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}-24 \\ -3 \\ 0 \\ - 0.5 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 4.1 \\ 3 \\ 1.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 2 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}25 \\ 3 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}-24 \\ -3 \\ 0 \\ - 0.5 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc}-5 & 1 & -6 & -3 & -9 \\ 4 & - 1.8 & 8.8 & - 4.6 & 9.2 \\ 3.5 & - 1.5 & 7.4 & - 5.5 & 2.9 \end{array}\right) \sim \left(\begin{array}{rrrrc}-5 & 1 & -6 & -3 & -9 \\ 0 & -1 & 4 & -7 & 2 \\ 0 & 0 & 0 & -2 & -5 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 0.4 \\ 4 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 3.6 \\ 19.5 \\ 0 \\ - 2.5 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}2 \\ 1.4 \\ 3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 4 \\ 0 \\ -1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 0.4 \\ 4 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 3.6 \\ 19.5 \\ 0 \\ - 2.5 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}1 \\ 3.2 \\ 4.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 3 \\ 0 \\ -1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 0.4 \\ 4 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 3.6 \\ 19.5 \\ 0 \\ - 2.5 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc}-5 & 0 & 7.5 & -6 & -9 \\ -2 & 4 & 9 & - 10.4 & - 5.6 \\ 4 & - 0.8 & - 7.2 & 8.4 & 3.6 \end{array}\right) \sim \left(\begin{array}{rrrrc}-5 & 0 & 7.5 & -6 & -9 \\ 0 & 4 & 6 & -8 & -2 \\ 0 & 0 & 0 & 2 & -4 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 1.5 \\ - 1.5 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 4.2 \\ 4.5 \\ 0 \\ 2 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}1 \\ 0.4 \\ - 2.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ -2 \\ 0 \\ -1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 1.5 \\ - 1.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 4.2 \\ 4.5 \\ 0 \\ 2 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-3 \\ - 1.2 \\ - 1.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ -4 \\ 0 \\ -2 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 1.5 \\ - 1.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 4.2 \\ 4.5 \\ 0 \\ 2 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc}- 0.2 & 3.9 & 4.6 & 2.3 & 10.8 \\ - 0.4 & 8.2 & 0.4 & 0.2 & 8 \\ 0.2 & -4 & - 2.4 & - 0.2 & 1.8 \end{array}\right) \sim \left(\begin{array}{rrrrc}- 0.4 & 8.2 & 0.4 & 0.2 & 8 \\ 0 & - 0.2 & 4.4 & 2.2 & 6.8 \\ 0 & 0 & 0 & 1 & 9.2 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}452 \\ 22 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 1362.2 \\ - 67.2 \\ 0 \\ - 9.2 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.5 \\ 6.2 \\ - 2.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 1 \\ 0 \\ -2 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}452 \\ 22 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 1362.2 \\ - 67.2 \\ 0 \\ - 9.2 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 0.8 \\ 6.4 \\ -3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 1 \\ 0 \\ -1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}452 \\ 22 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 1362.2 \\ - 67.2 \\ 0 \\ - 9.2 \\ 1 \end{array}\right)

    v v  
* * * * *
0 * * * *
0 0 0 0 *

Example matrix

A = \left(\begin{array}{rrrrc}1 & -9 & 0 & -7 & -1 \\ - 0.5 & 5.5 & 5 & - 2.5 & 2.5 \\ - 0.1 & 1.5 & 3 & - 2.9 & - 2.7 \end{array}\right) \sim \left(\begin{array}{rrrrc}1 & -9 & 0 & -7 & -1 \\ 0 & 1 & 5 & -6 & 2 \\ 0 & 0 & 0 & 0 & -4 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-45 \\ -5 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}61 \\ 6 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-4 \\ 1 \\ 3.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 1 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_3 \left(\begin{array}{r}-45 \\ -5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}61 \\ 6 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-5 \\ 1.5 \\ 3.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 1 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_3 \left(\begin{array}{r}-45 \\ -5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}61 \\ 6 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc}2 & -4 & 0 & 7 & -5 \\ - 0.6 & - 0.8 & 5 & - 1.1 & 10.5 \\ 0.8 & -2 & 1 & 3 & 5.8 \end{array}\right) \sim \left(\begin{array}{rrrrc}2 & -4 & 0 & 7 & -5 \\ 0 & -2 & 5 & 1 & 9 \\ 0 & 0 & 0 & 0 & 6 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}5 \\ 2.5 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 2.5 \\ 0.5 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-5 \\ 6.5 \\ 5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 2 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_3 \left(\begin{array}{r}5 \\ 2.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 2.5 \\ 0.5 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-7 \\ 7.1 \\ 4.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 2 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_3 \left(\begin{array}{r}5 \\ 2.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 2.5 \\ 0.5 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc} 2.5 & - 9.5 & 4.5 & - 4.5 & - 3.5 \\ 2 & - 8.1 & 1.1 & - 3.1 & - 10.8 \\ 1 & - 3.9 & 1.3 & - 1.7 & 6.5 \end{array}\right) \sim \left(\begin{array}{rrrrc} 2.5 & - 9.5 & 4.5 & - 4.5 & - 3.5 \\ 0 & - 0.5 & - 2.5 & 0.5 & -8 \\ 0 & 0 & 0 & 0 & 9.5 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 20.8 \\ -5 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 5.6 \\ 1 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-1 \\ 6.7 \\ - 8.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 1 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_3 \left(\begin{array}{r}- 20.8 \\ -5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 5.6 \\ 1 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 1.5 \\ 8.7 \\ - 7.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 1 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_3 \left(\begin{array}{r}- 20.8 \\ -5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 5.6 \\ 1 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc}- 0.1 & - 1.9 & - 8.8 & 9.3 & 4.1 \\ - 0.2 & 0.2 & -6 & 9.8 & 7 \\ 0.1 & 0.9 & 5.9 & - 7.1 & -13 \end{array}\right) \sim \left(\begin{array}{rrrrc}- 0.2 & 0.2 & -6 & 9.8 & 7 \\ 0 & -2 & - 5.8 & 4.4 & 0.6 \\ 0 & 0 & 0 & 0 & - 9.2 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 32.9 \\ - 2.9 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 51.2 \\ 2.2 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-2 \\ 6.8 \\ - 9.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 3 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_3 \left(\begin{array}{r}- 32.9 \\ - 2.9 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 51.2 \\ 2.2 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 3.8 \\ 7.2 \\ - 9.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 4 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_3 \left(\begin{array}{r}- 32.9 \\ - 2.9 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 51.2 \\ 2.2 \\ 0 \\ 1 \\ 0 \end{array}\right)

  v     v
* * * * *
0 0 * * *
0 0 0 * *

Example matrix

A = \left(\begin{array}{rrrrc}2 & 7 & -9 & 3 & -3 \\ - 0.6 & - 2.1 & 6.7 & - 8.9 & - 5.1 \\ 1 & 3.5 & - 6.9 & 4.3 & 7.1 \end{array}\right) \sim \left(\begin{array}{rrrrc}2 & 7 & -9 & 3 & -3 \\ 0 & 0 & 4 & -8 & -6 \\ 0 & 0 & 0 & -2 & 5 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 3.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}27 \\ 0 \\ 6.5 \\ 2.5 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-7 \\ 2.1 \\ - 5.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 2 \\ 1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 3.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}27 \\ 0 \\ 6.5 \\ 2.5 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-9 \\ 2.7 \\ - 6.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 2 \\ 1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 3.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}27 \\ 0 \\ 6.5 \\ 2.5 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc}- 4.5 & - 6.3 & -2 & 5.5 & - 5.4 \\ 5 & 7 & 0 & -5 & -4 \\ 1 & 1.4 & 0.8 & - 2.4 & 11.8 \end{array}\right) \sim \left(\begin{array}{rrrrc}5 & 7 & 0 & -5 & -4 \\ 0 & 0 & -2 & 1 & -9 \\ 0 & 0 & 0 & -1 & 9 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 1.4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 9.8 \\ 0 \\ 0 \\ 9 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 0.5 \\ -5 \\ - 1.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 3 \\ 2 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 1.4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 9.8 \\ 0 \\ 0 \\ 9 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.5 \\ -5 \\ - 0.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 4 \\ 2 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 1.4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 9.8 \\ 0 \\ 0 \\ 9 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc} 2.5 & 5 & 4 & 7.5 & - 8.5 \\ 2 & 4 & 5.2 & 14 & - 9.8 \\ 0.5 & 1 & 0.4 & 0.9 & 1.9 \end{array}\right) \sim \left(\begin{array}{rrrrc} 2.5 & 5 & 4 & 7.5 & - 8.5 \\ 0 & 0 & 2 & 8 & -3 \\ 0 & 0 & 0 & 1 & 3 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 9.2 \\ 0 \\ 13.5 \\ -3 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-1 \\ - 0.8 \\ 0.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ -4 \\ 1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 9.2 \\ 0 \\ 13.5 \\ -3 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 1.5 \\ 1.2 \\ 1.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ -4 \\ 1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 9.2 \\ 0 \\ 13.5 \\ -3 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc}- 0.5 & - 3.5 & - 6.7 & 9.2 & 3.8 \\ 1 & 7 & 5.4 & -6 & - 9.2 \\ 0.5 & 3.5 & 0.7 & - 0.1 & 1.8 \end{array}\right) \sim \left(\begin{array}{rrrrc}1 & 7 & 5.4 & -6 & - 9.2 \\ 0 & 0 & -4 & 6.2 & - 0.8 \\ 0 & 0 & 0 & - 0.2 & 6.8 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-7 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 70.3 \\ 0 \\ 52.5 \\ 34 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.3 \\ - 0.2 \\ 0.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ -3 \\ -2 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-7 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 70.3 \\ 0 \\ 52.5 \\ 34 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 0.2 \\ - 1.2 \\ - 0.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ -3 \\ -2 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-7 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 70.3 \\ 0 \\ 52.5 \\ 34 \\ 1 \end{array}\right)

  v   v  
* * * * *
0 0 * * *
0 0 0 0 *

Example matrix

A = \left(\begin{array}{rrrrc} 1.6 & - 0.8 & 4.6 & -2 & 0.4 \\ -2 & 1 & -7 & 0 & -8 \\ 1.8 & - 0.9 & 6.2 & - 0.2 & - 2.4 \end{array}\right) \sim \left(\begin{array}{rrrrc}-2 & 1 & -7 & 0 & -8 \\ 0 & 0 & -1 & -2 & -6 \\ 0 & 0 & 0 & 0 & -9 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}7 \\ 0 \\ -2 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-1 \\ -5 \\ -5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ -1 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r} 0.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}7 \\ 0 \\ -2 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 0.6 \\ -7 \\ - 3.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ -1 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r} 0.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}7 \\ 0 \\ -2 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc}- 0.4 & - 0.8 & 5.2 & - 2.2 & - 6.4 \\ 2 & 4 & -1 & 1 & -3 \\ - 0.2 & - 0.4 & 0.6 & - 0.3 & - 6.4 \end{array}\right) \sim \left(\begin{array}{rrrrc}2 & 4 & -1 & 1 & -3 \\ 0 & 0 & 5 & -2 & -7 \\ 0 & 0 & 0 & 0 & -6 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 0.3 \\ 0 \\ 0.4 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 2.8 \\ 1 \\ - 5.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 2 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}-2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 0.3 \\ 0 \\ 0.4 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 2.4 \\ 3 \\ -6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 2 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}-2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 0.3 \\ 0 \\ 0.4 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc}- 0.5 & 7 & -4 & -1 & -8 \\ 0.1 & - 1.4 & - 1.7 & 9.2 & 0.1 \\ - 0.4 & 5.6 & - 1.7 & - 6.2 & -12 \end{array}\right) \sim \left(\begin{array}{rrrrc}- 0.5 & 7 & -4 & -1 & -8 \\ 0 & 0 & - 2.5 & 9 & - 1.5 \\ 0 & 0 & 0 & 0 & - 6.5 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}14 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 30.8 \\ 0 \\ 3.6 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-2 \\ - 3.1 \\ 7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 2 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}14 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 30.8 \\ 0 \\ 3.6 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.5 \\ - 3.2 \\ 7.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 2 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}14 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 30.8 \\ 0 \\ 3.6 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc} 0.2 & - 1.6 & - 3.8 & 2.8 & 3.1 \\ 0.4 & - 3.2 & - 9.6 & - 5.6 & 9.8 \\ - 0.2 & 1.6 & 4.3 & 0 & - 0.4 \end{array}\right) \sim \left(\begin{array}{rrrrc} 0.4 & - 3.2 & - 9.6 & - 5.6 & 9.8 \\ 0 & 0 & 1 & 5.6 & - 1.8 \\ 0 & 0 & 0 & 0 & 3.6 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}8 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 120.4 \\ 0 \\ - 5.6 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.6 \\ - 8.4 \\ - 4.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ -1 \\ 0 \\ -2 \end{array}\right) + u_2 \left(\begin{array}{r}8 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 120.4 \\ 0 \\ - 5.6 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.8 \\ - 8.8 \\ - 4.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ -1 \\ 0 \\ -2 \end{array}\right) + u_2 \left(\begin{array}{r}8 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 120.4 \\ 0 \\ - 5.6 \\ 1 \\ 0 \end{array}\right)

  v v    
* * * * *
0 0 0 * *
0 0 0 0 *

Example matrix

A = \left(\begin{array}{rrrrc}2 & -2 & - 1.6 & - 0.2 & - 7.4 \\ 5 & -5 & -4 & 7 & -1 \\ 3.5 & - 3.5 & - 2.8 & 2.5 & - 2.3 \end{array}\right) \sim \left(\begin{array}{rrrrc}5 & -5 & -4 & 7 & -1 \\ 0 & 0 & 0 & -3 & -7 \\ 0 & 0 & 0 & 0 & 4 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}1 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 0.8 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-1 \\ 0 \\ 3.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ -2 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}1 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 0.8 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 3.2 \\ 2 \\ 2.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ -1 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}1 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 0.8 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc} 3.6 & 1.8 & 7.2 & - 1.6 & - 3.7 \\ 4 & 2 & 8 & 6 & -3 \\ 1.6 & 0.8 & 3.2 & 3.8 & -10 \end{array}\right) \sim \left(\begin{array}{rrrrc}4 & 2 & 8 & 6 & -3 \\ 0 & 0 & 0 & -7 & -1 \\ 0 & 0 & 0 & 0 & -9 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 0.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 5.1 \\ -1 \\ - 10.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ -1 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}- 0.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 10.5 \\ -5 \\ 4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 0 \\ -2 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}- 0.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc}- 0.2 & 0.8 & - 1.4 & - 5.9 & 4.6 \\ - 0.5 & 2 & - 3.5 & - 8.5 & 1.5 \\ 0.1 & - 0.4 & 0.7 & 3.2 & - 9.7 \end{array}\right) \sim \left(\begin{array}{rrrrc}- 0.5 & 2 & - 3.5 & - 8.5 & 1.5 \\ 0 & 0 & 0 & - 2.5 & 4 \\ 0 & 0 & 0 & 0 & -7 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-7 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 6.4 \\ 13.5 \\ 3.7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 0 \\ -2 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-7 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 6.6 \\ 14 \\ 3.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ -2 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-7 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc} 0.4 & 1.6 & - 1.4 & 2 & 6.4 \\ - 0.2 & - 0.8 & 0.7 & 3.2 & -1 \\ 0.2 & 0.8 & - 0.7 & 3.1 & - 3.7 \end{array}\right) \sim \left(\begin{array}{rrrrc} 0.4 & 1.6 & - 1.4 & 2 & 6.4 \\ 0 & 0 & 0 & 4.2 & 2.2 \\ 0 & 0 & 0 & 0 & -8 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 3.5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 2.8 \\ 3.4 \\ 7.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 0 \\ 1 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}-4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 3.5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 3.2 \\ 3.6 \\ 7.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 1 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}-4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 3.5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)

3 \times 5 with three free variables

    v v v
* * * * *
0 * * * *
0 0 0 0 0

Example matrix

A = \left(\begin{array}{rrrrc}-2 & 1.4 & - 7.6 & - 10.6 & - 4.8 \\ -5 & 6 & 1 & -4 & -7 \\ -3 & 4.1 & 4.6 & 2.1 & - 3.2 \end{array}\right) \sim \left(\begin{array}{rrrrc}-5 & 6 & 1 & -4 & -7 \\ 0 & -1 & -8 & -9 & -2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 9.4 \\ -8 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 11.6 \\ -9 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}- 3.8 \\ -2 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 3.8 \\ -2 \\ 0.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 3 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 9.4 \\ -8 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 11.6 \\ -9 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 3.8 \\ -2 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 3.2 \\ -3 \\ - 0.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 2 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 9.4 \\ -8 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 11.6 \\ -9 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 3.8 \\ -2 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc}-1 & -5 & -6 & 9 & 1 \\ 0.4 & 7 & - 6.6 & 2.4 & - 7.4 \\ - 0.2 & -4 & 4.2 & - 1.8 & 4.4 \end{array}\right) \sim \left(\begin{array}{rrrrc}-1 & -5 & -6 & 9 & 1 \\ 0 & 5 & -9 & 6 & -7 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-15 \\ 1.8 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}15 \\ - 1.2 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}-6 \\ 1.4 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}1 \\ - 5.4 \\ 3.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-15 \\ 1.8 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}15 \\ - 1.2 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}-6 \\ 1.4 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}2 \\ - 5.8 \\ 3.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ -1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-15 \\ 1.8 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}15 \\ - 1.2 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}-6 \\ 1.4 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc}5 & 0 & - 6.5 & -1 & - 3.5 \\ -4 & - 0.5 & - 4.3 & - 1.2 & 11.8 \\ 4 & 0.2 & - 1.4 & 0 & - 6.4 \end{array}\right) \sim \left(\begin{array}{rrrrc}5 & 0 & - 6.5 & -1 & - 3.5 \\ 0 & - 0.5 & - 9.5 & -2 & 9 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 1.3 \\ -19 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 0.2 \\ -4 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 0.7 \\ 18 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}5 \\ -2 \\ 3.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ -4 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 1.3 \\ -19 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 0.2 \\ -4 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 0.7 \\ 18 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}5 \\ - 2.5 \\ 3.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ -3 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 1.3 \\ -19 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 0.2 \\ -4 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 0.7 \\ 18 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc} 0.2 & - 1.3 & - 4.2 & - 4.8 & - 8.2 \\ 0.4 & - 4.6 & 2.4 & - 6.8 & -4 \\ 0.2 & - 2.8 & 3.9 & - 2.7 & 1.1 \end{array}\right) \sim \left(\begin{array}{rrrrc} 0.4 & - 4.6 & 2.4 & - 6.8 & -4 \\ 0 & 1 & - 5.4 & - 1.4 & - 6.2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 56.1 \\ 5.4 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 33.1 \\ 1.4 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 81.3 \\ 6.2 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.5 \\ -3 \\ -2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 56.1 \\ 5.4 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 33.1 \\ 1.4 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 81.3 \\ 6.2 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.7 \\ - 3.4 \\ - 2.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 56.1 \\ 5.4 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 33.1 \\ 1.4 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 81.3 \\ 6.2 \\ 0 \\ 0 \\ 1 \end{array}\right)

  v   v v
* * * * *
0 0 * * *
0 0 0 0 0

Example matrix

A = \left(\begin{array}{rrrrc}- 4.5 & - 1.8 & 9.1 & 11.7 & 8.8 \\ 5 & 2 & -9 & -3 & -2 \\ 4 & 1.6 & - 7.5 & - 5.1 & - 3.7 \end{array}\right) \sim \left(\begin{array}{rrrrc}5 & 2 & -9 & -3 & -2 \\ 0 & 0 & 1 & 9 & 7 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 0.4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 15.6 \\ 0 \\ -9 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}- 12.2 \\ 0 \\ -7 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 0.1 \\ 1 \\ 0.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 0.4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 15.6 \\ 0 \\ -9 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 12.2 \\ 0 \\ -7 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 0.2 \\ 2 \\ 1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 2 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 0.4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 15.6 \\ 0 \\ -9 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 12.2 \\ 0 \\ -7 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc} 0.3 & 1.8 & - 0.4 & 6.8 & - 9.9 \\ -1 & -6 & -2 & 4 & 3 \\ 0.4 & 2.4 & 1.3 & - 5.6 & 3.3 \end{array}\right) \sim \left(\begin{array}{rrrrc}-1 & -6 & -2 & 4 & 3 \\ 0 & 0 & -1 & 8 & -9 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-6 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-12 \\ 0 \\ 8 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}21 \\ 0 \\ -9 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}1 \\ 0 \\ - 0.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ -1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-6 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-12 \\ 0 \\ 8 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}21 \\ 0 \\ -9 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 0.7 \\ 1 \\ - 0.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ -1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-6 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-12 \\ 0 \\ 8 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}21 \\ 0 \\ -9 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc}- 0.4 & - 1.6 & 2.6 & - 0.1 & 8.5 \\ - 0.5 & -2 & 9.5 & -7 & 7.5 \\ - 0.2 & - 0.8 & 4.8 & - 3.9 & 2.5 \end{array}\right) \sim \left(\begin{array}{rrrrc}- 0.5 & -2 & 9.5 & -7 & 7.5 \\ 0 & 0 & -5 & 5.5 & 2.5 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 6.9 \\ 0 \\ 1.1 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 24.5 \\ 0 \\ 0.5 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}1 \\ 7.5 \\ 4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 6.9 \\ 0 \\ 1.1 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 24.5 \\ 0 \\ 0.5 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 1.4 \\ 8 \\ 4.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 6.9 \\ 0 \\ 1.1 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 24.5 \\ 0 \\ 0.5 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc}- 0.2 & - 6.4 & 6 & 4.2 & - 8.2 \\ 0.1 & 3.2 & - 2.8 & - 9.9 & 1.9 \\ - 0.1 & - 3.2 & 3.1 & - 1.8 & - 5.2 \end{array}\right) \sim \left(\begin{array}{rrrrc}- 0.2 & - 6.4 & 6 & 4.2 & - 8.2 \\ 0 & 0 & 0.2 & - 7.8 & - 2.2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-32 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}1191 \\ 0 \\ 39 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}289 \\ 0 \\ 11 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 5.2 \\ - 2.4 \\ 2.7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-32 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}1191 \\ 0 \\ 39 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}289 \\ 0 \\ 11 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 5.4 \\ - 2.5 \\ 2.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-32 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}1191 \\ 0 \\ 39 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}289 \\ 0 \\ 11 \\ 0 \\ 1 \end{array}\right)

  v v   v
* * * * *
0 0 0 * *
0 0 0 0 0

Example matrix

A = \left(\begin{array}{rrrrc} 0.3 & - 0.6 & 1.5 & 1.1 & - 7.8 \\ 1 & -2 & 5 & -3 & 4 \\ - 0.6 & 1.2 & -3 & 1 & 1.2 \end{array}\right) \sim \left(\begin{array}{rrrrc}1 & -2 & 5 & -3 & 4 \\ 0 & 0 & 0 & 2 & -9 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 9.5 \\ 0 \\ 0 \\ 4.5 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 1.7 \\ -1 \\ - 0.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 9.5 \\ 0 \\ 0 \\ 4.5 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}2 \\ 0 \\ - 0.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 9.5 \\ 0 \\ 0 \\ 4.5 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc}-2 & - 2.4 & - 1.2 & 8.2 & - 3.6 \\ 5 & 6 & 3 & -8 & 9 \\ 2.5 & 3 & 1.5 & -2 & 4.5 \end{array}\right) \sim \left(\begin{array}{rrrrc}5 & 6 & 3 & -8 & 9 \\ 0 & 0 & 0 & 5 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 1.2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 0.6 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 1.8 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 4.2 \\ 2 \\ 3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 1.2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 0.6 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 1.8 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 2.2 \\ 7 \\ 5.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 1.2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 0.6 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 1.8 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc}- 0.6 & - 5.4 & - 2.4 & 3.4 & 4.2 \\ -1 & -9 & -4 & 6.5 & 7 \\ - 0.4 & - 3.6 & - 1.6 & 2.9 & 2.8 \end{array}\right) \sim \left(\begin{array}{rrrrc}-1 & -9 & -4 & 6.5 & 7 \\ 0 & 0 & 0 & - 0.5 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-9 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-4 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}7 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}1 \\ 2.5 \\ 1.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-9 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-4 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}7 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 1.6 \\ 3.5 \\ 1.7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-9 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-4 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}7 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc}-2 & 8.2 & - 7.8 & -6 & - 4.8 \\ -1 & 4.1 & - 3.9 & -1 & 3 \\ 1 & - 4.1 & 3.9 & 2 & - 0.3 \end{array}\right) \sim \left(\begin{array}{rrrrc}-2 & 8.2 & - 7.8 & -6 & - 4.8 \\ 0 & 0 & 0 & 2 & 5.4 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 4.1 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 3.9 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 5.7 \\ 0 \\ 0 \\ - 2.7 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}2 \\ -1 \\ 0 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ -1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 4.1 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 3.9 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 5.7 \\ 0 \\ 0 \\ - 2.7 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}0 \\ -2 \\ 1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ -1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 4.1 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 3.9 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 5.7 \\ 0 \\ 0 \\ - 2.7 \\ 1 \end{array}\right)

  v v v  
* * * * *
0 0 0 0 *
0 0 0 0 0

Example matrix

A = \left(\begin{array}{rrrrc}- 0.9 & - 2.7 & 0.9 & - 4.5 & 9.6 \\ 1 & 3 & -1 & 5 & -4 \\ 0.7 & 2.1 & - 0.7 & 3.5 & 2.6 \end{array}\right) \sim \left(\begin{array}{rrrrc}1 & 3 & -1 & 5 & -4 \\ 0 & 0 & 0 & 0 & 6 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-3 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}1 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}6 \\ 0 \\ 5.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}-3 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}1 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 6.9 \\ -1 \\ 4.7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}-3 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}1 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc}-5 & 9 & 8 & -1 & 2 \\ 2.5 & - 4.5 & -4 & 0.5 & 6 \\ - 4.5 & 8.1 & 7.2 & - 0.9 & - 0.3 \end{array}\right) \sim \left(\begin{array}{rrrrc}-5 & 9 & 8 & -1 & 2 \\ 0 & 0 & 0 & 0 & 7 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 1.8 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 1.6 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 0.2 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-7 \\ - 3.5 \\ - 4.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r} 1.8 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 1.6 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 0.2 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-9 \\ - 9.5 \\ - 3.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 0 \\ 0 \\ -2 \end{array}\right) + u_2 \left(\begin{array}{r} 1.8 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 1.6 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 0.2 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc} 2.5 & - 2.5 & 4.5 & 5 & - 3.5 \\ - 1.5 & 1.5 & - 2.7 & -3 & 2.6 \\ -2 & 2 & - 3.6 & -4 & 3 \end{array}\right) \sim \left(\begin{array}{rrrrc} 2.5 & - 2.5 & 4.5 & 5 & - 3.5 \\ 0 & 0 & 0 & 0 & 0.5 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}1 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 1.8 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-2 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 0.5 \\ 0.7 \\ 0 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 0 \\ 2 \end{array}\right) + u_2 \left(\begin{array}{r}1 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 1.8 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-2 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 1.5 \\ - 0.4 \\ -1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}1 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 1.8 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-2 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc} 0.5 & - 4.8 & 3.6 & 0.6 & - 7.6 \\ 1 & - 9.6 & 7.2 & 1.2 & -6 \\ - 0.5 & 4.8 & - 3.6 & - 0.6 & 5.3 \end{array}\right) \sim \left(\begin{array}{rrrrc}1 & - 9.6 & 7.2 & 1.2 & -6 \\ 0 & 0 & 0 & 0 & - 4.6 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 9.6 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 7.2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 1.2 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 5.6 \\ -2 \\ 3.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r} 9.6 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 7.2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 1.2 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 6.1 \\ -3 \\ 3.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r} 9.6 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 7.2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 1.2 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

3 \times 5 with four free variables

  v v v v
* * * * *
0 0 0 0 0
0 0 0 0 0

Example matrix

A = \left(\begin{array}{rrrrc}5 & 7 & -4 & 4 & -8 \\ 3.5 & 4.9 & - 2.8 & 2.8 & - 5.6 \\ -1 & - 1.4 & 0.8 & - 0.8 & 1.6 \end{array}\right) \sim \left(\begin{array}{rrrrc}5 & 7 & -4 & 4 & -8 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 1.4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 0.8 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 0.8 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 1.6 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}5 \\ 3.5 \\ -1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 1.4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 0.8 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 0.8 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 1.6 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}10 \\ 7 \\ -2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 1.4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 0.8 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 0.8 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 1.6 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc}2 & 1 & 9 & -1 & -9 \\ 0.4 & 0.2 & 1.8 & - 0.2 & - 1.8 \\ 0.6 & 0.3 & 2.7 & - 0.3 & - 2.7 \end{array}\right) \sim \left(\begin{array}{rrrrc}2 & 1 & 9 & -1 & -9 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 0.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 4.5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 0.5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 4.5 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}2 \\ 0.4 \\ 0.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 0.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 4.5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 0.5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 4.5 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}4 \\ 0.8 \\ 1.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 0.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 4.5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 0.5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 4.5 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc}-4 & 3.6 & - 6.4 & - 5.2 & 6 \\ -5 & 4.5 & -8 & - 6.5 & 7.5 \\ -2 & 1.8 & - 3.2 & - 2.6 & 3 \end{array}\right) \sim \left(\begin{array}{rrrrc}-5 & 4.5 & -8 & - 6.5 & 7.5 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.9 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 1.6 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 1.3 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 1.5 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-4 \\ -5 \\ -2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 0.9 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 1.6 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 1.3 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 1.5 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-8 \\ -10 \\ -4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 0.9 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 1.6 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 1.3 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 1.5 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc}- 0.2 & 3 & - 3.7 & - 4.6 & 4.5 \\ 0.4 & -6 & 7.4 & 9.2 & -9 \\ 0.2 & -3 & 3.7 & 4.6 & - 4.5 \end{array}\right) \sim \left(\begin{array}{rrrrc} 0.4 & -6 & 7.4 & 9.2 & -9 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}15 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 18.5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-23 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 22.5 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.2 \\ 0.4 \\ 0.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}15 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 18.5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-23 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 22.5 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.4 \\ 0.8 \\ 0.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}15 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 18.5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-23 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 22.5 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

4 \times 2

  1. 4 \times 2 with 0 free variables
  2. 4 \times 2 with 1 free variables

4 \times 2 with no free variables

   
* *
0 *
0 0
0 0

Example matrix

A = \left(\begin{array}{rr}- 3.6 & 4.2 \\ 4 & 0 \\ 3.2 & 6 \\ 1.6 & 3.6 \end{array}\right) \sim \left(\begin{array}{rr}4 & 0 \\ 0 & 6 \\ 0 & 0 \\ 0 & 0 \end{array}\right) , \textrm{null}(A) = \{\textbf{0}\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 7.8 \\ 4 \\ - 2.8 \\ -2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ -1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 11.4 \\ 8 \\ 0.4 \\ - 0.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ -1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rr} 1.6 & - 2.7 \\ -2 & -8 \\ 4 & 2 \\ - 0.4 & - 2.3 \end{array}\right) \sim \left(\begin{array}{rr}4 & 2 \\ 0 & -7 \\ 0 & 0 \\ 0 & 0 \end{array}\right) , \textrm{null}(A) = \{\textbf{0}\}

Example solution

\textbf{b} = \left(\begin{array}{r} 4.3 \\ 6 \\ 2 \\ 1.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ -1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 5.9 \\ 4 \\ 6 \\ 1.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ -1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rr}- 0.9 & 3.6 \\ 1.8 & - 3.6 \\ - 4.5 & 8 \\ - 2.7 & 6.4 \end{array}\right) \sim \left(\begin{array}{rr}- 4.5 & 8 \\ 0 & 2 \\ 0 & 0 \\ 0 & 0 \end{array}\right) , \textrm{null}(A) = \{\textbf{0}\}

Example solution

\textbf{b} = \left(\begin{array}{r} 1.8 \\ 0 \\ -1 \\ 1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 3.6 \\ 0 \\ -2 \\ 2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 2 \end{array}\right)

Example matrix

A = \left(\begin{array}{rr} 4.8 & 3.9 \\ 4.8 & 7 \\ 9.6 & 1.6 \\ - 4.8 & - 3.9 \end{array}\right) \sim \left(\begin{array}{rr} 9.6 & 1.6 \\ 0 & 6.2 \\ 0 & 0 \\ 0 & 0 \end{array}\right) , \textrm{null}(A) = \{\textbf{0}\}

Example solution

\textbf{b} = \left(\begin{array}{r} 0.9 \\ - 2.2 \\ 8 \\ - 0.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ -1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-3 \\ - 9.2 \\ 6.4 \\ 3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ -2 \end{array}\right)

4 \times 2 with one free variable

  v
* *
0 0
0 0
0 0

Example matrix

A = \left(\begin{array}{rr}-4 & -2 \\ - 0.4 & - 0.2 \\ - 2.4 & - 1.2 \\ 1.6 & 0.8 \end{array}\right) \sim \left(\begin{array}{rr}-4 & -2 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 0.5 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-4 \\ - 0.4 \\ - 2.4 \\ 1.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 0.5 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-8 \\ - 0.8 \\ - 4.8 \\ 3.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 0.5 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rr}2 & - 2.4 \\ - 1.5 & 1.8 \\ -5 & 6 \\ -4 & 4.8 \end{array}\right) \sim \left(\begin{array}{rr}-5 & 6 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 1.2 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}2 \\ - 1.5 \\ -5 \\ -4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 1.2 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}4 \\ -3 \\ -10 \\ -8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 1.2 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rr} 0.5 & - 1.4 \\ -2 & 5.6 \\ 2.5 & -7 \\ - 1.5 & 4.2 \end{array}\right) \sim \left(\begin{array}{rr} 2.5 & -7 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 2.8 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 0.5 \\ -2 \\ 2.5 \\ - 1.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 2.8 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}1 \\ -4 \\ 5 \\ -3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 2.8 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rr} 0.5 & - 3.8 \\ - 0.5 & 3.8 \\ 1 & - 7.6 \\ - 0.5 & 3.8 \end{array}\right) \sim \left(\begin{array}{rr}1 & - 7.6 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 7.6 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 0.5 \\ - 0.5 \\ 1 \\ - 0.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 7.6 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}1 \\ -1 \\ 2 \\ -1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 7.6 \\ 1 \end{array}\right)

4 \times 3

  1. 4 \times 3 with 0 free variables
  2. 4 \times 3 with 2 free variables

4 \times 3 with no free variables

     
* * *
0 * *
0 0 *
0 0 0

Example matrix

A = \left(\begin{array}{rrr}- 8.1 & 2.7 & - 11.6 \\ -9 & -3 & 2 \\ 4.5 & - 4.5 & 5 \\ 5.4 & 4.2 & -6 \end{array}\right) \sim \left(\begin{array}{rrr}-9 & -3 & 2 \\ 0 & -6 & 6 \\ 0 & 0 & -8 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \{\textbf{0}\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.9 \\ -5 \\ 8.5 \\ 3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ -2 \\ -1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}7 \\ -4 \\ 8 \\ 4.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ -3 \\ -2 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrr} 2.5 & - 7.3 & - 6.9 \\ - 3.5 & 10.4 & 1.4 \\ -5 & 2 & -8 \\ 2 & 1.9 & 4.1 \end{array}\right) \sim \left(\begin{array}{rrr}-5 & 2 & -8 \\ 0 & 9 & 7 \\ 0 & 0 & -6 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \{\textbf{0}\}

Example solution

\textbf{b} = \left(\begin{array}{r} 4.6 \\ 2 \\ 0 \\ 1.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 1 \\ -1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 7.1 \\ - 1.5 \\ -5 \\ 3.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 1 \\ -1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrr}- 0.3 & -4 & 2.2 \\ - 0.2 & - 2.8 & -6 \\ 0.5 & 5 & - 9.5 \\ - 0.1 & - 1.4 & - 0.9 \end{array}\right) \sim \left(\begin{array}{rrr} 0.5 & 5 & - 9.5 \\ 0 & -1 & - 3.5 \\ 0 & 0 & -7 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \{\textbf{0}\}

Example solution

\textbf{b} = \left(\begin{array}{r} 4.6 \\ 10.8 \\ 1.5 \\ 3.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -2 \\ -1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 4.9 \\ 11 \\ 1 \\ 3.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ -2 \\ -1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrr} 2.5 & - 1.3 & - 1.6 \\ 2.5 & 10.4 & - 3.7 \\ 5 & 5.2 & - 9.4 \\ - 2.5 & 1.3 & 7 \end{array}\right) \sim \left(\begin{array}{rrr}5 & 5.2 & - 9.4 \\ 0 & 7.8 & 1 \\ 0 & 0 & 3.6 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \{\textbf{0}\}

Example solution

\textbf{b} = \left(\begin{array}{r} 7.2 \\ - 6.6 \\ 0.4 \\ - 1.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ -1 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 4.7 \\ - 9.1 \\ - 4.6 \\ 0.7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ -1 \\ 1 \end{array}\right)

4 \times 3 with one free variable

    v
* * *
0 * *
0 0 0
0 0 0

Example matrix

A = \left(\begin{array}{rrr}-1 & 6.5 & 4 \\ 2 & -3 & 4 \\ 0.2 & 1.2 & 2.2 \\ - 0.4 & - 0.9 & - 2.6 \end{array}\right) \sim \left(\begin{array}{rrr}2 & -3 & 4 \\ 0 & 5 & 6 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 3.8 \\ - 1.2 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 4.5 \\ 1 \\ 1.6 \\ - 1.7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 3.8 \\ - 1.2 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 3.5 \\ 3 \\ 1.8 \\ - 2.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 3.8 \\ - 1.2 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrr} 0.1 & -2 & 2.3 \\ 1 & -8 & -1 \\ - 0.8 & 2.4 & 8.8 \\ 0.9 & - 10.4 & 5.5 \end{array}\right) \sim \left(\begin{array}{rrr}1 & -8 & -1 \\ 0 & -4 & 8 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}17 \\ 2 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.6 \\ -4 \\ - 0.8 \\ - 6.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}17 \\ 2 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.7 \\ -5 \\ 0 \\ - 7.7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}17 \\ 2 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrr} 2.4 & - 1.7 & - 2.7 \\ -4 & 1.5 & 0.5 \\ 1.6 & 0.4 & 2.8 \\ - 1.6 & 0.4 & - 0.4 \end{array}\right) \sim \left(\begin{array}{rrr}-4 & 1.5 & 0.5 \\ 0 & 1 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-1 \\ -3 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-1 \\ -1 \\ 2.4 \\ - 0.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 2 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-1 \\ -3 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 2.7 \\ 0.5 \\ 2.8 \\ - 0.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 3 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-1 \\ -3 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrr}- 0.4 & - 3.1 & - 0.7 \\ 0.4 & 1.6 & -5 \\ - 0.8 & - 4.2 & 6.2 \\ - 0.4 & - 1.6 & 5 \end{array}\right) \sim \left(\begin{array}{rrr}- 0.8 & - 4.2 & 6.2 \\ 0 & -1 & - 3.8 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 27.7 \\ - 3.8 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 1.5 \\ 0 \\ 1 \\ 0 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 27.7 \\ - 3.8 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 1.9 \\ - 0.4 \\ 1.8 \\ 0.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ -1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 27.7 \\ - 3.8 \\ 1 \end{array}\right)

  v  
* * *
0 0 *
0 0 0
0 0 0

Example matrix

A = \left(\begin{array}{rrr}-1 & 4 & 1.4 \\ 2 & -8 & -2 \\ 0.8 & - 3.2 & - 4.8 \\ 0.6 & - 2.4 & -3 \end{array}\right) \sim \left(\begin{array}{rrr}2 & -8 & -2 \\ 0 & 0 & -4 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}4 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.6 \\ 2 \\ - 3.2 \\ - 1.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}4 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.6 \\ 4 \\ - 2.4 \\ - 1.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}4 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrr}- 2.5 & 3 & 5.5 \\ -2 & 2.4 & - 7.5 \\ 5 & -6 & 3 \\ - 0.5 & 0.6 & 1.1 \end{array}\right) \sim \left(\begin{array}{rrr}5 & -6 & 3 \\ 0 & 0 & 7 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 1.2 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-8 \\ 5.5 \\ 2 \\ - 1.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r} 1.2 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 10.5 \\ 3.5 \\ 7 \\ - 2.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r} 1.2 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrr}-1 & - 0.4 & 0.3 \\ -4 & - 1.6 & 3 \\ 5 & 2 & - 7.5 \\ -3 & - 1.2 & 5.7 \end{array}\right) \sim \left(\begin{array}{rrr}5 & 2 & - 7.5 \\ 0 & 0 & -3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 0.4 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.7 \\ -5 \\ 2.5 \\ - 0.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}- 0.4 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 2.4 \\ -6 \\ 0 \\ 2.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 2 \end{array}\right) + u_2 \left(\begin{array}{r}- 0.4 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrr}-4 & 4.8 & 6.6 \\ -2 & 2.4 & 0.4 \\ -2 & 2.4 & 9.1 \\ -2 & 2.4 & 6.2 \end{array}\right) \sim \left(\begin{array}{rrr}-4 & 4.8 & 6.6 \\ 0 & 0 & 5.8 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 1.2 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.4 \\ - 3.6 \\ 5.1 \\ 2.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r} 1.2 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 5.4 \\ - 5.6 \\ 3.1 \\ 0.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r} 1.2 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrr}-1 & 4 & 1.4 \\ 2 & -8 & -2 \\ 0.8 & - 3.2 & - 4.8 \\ 0.6 & - 2.4 & -3 \end{array}\right) \sim \left(\begin{array}{rrr}2 & -8 & -2 \\ 0 & 0 & -4 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}4 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.6 \\ 2 \\ - 3.2 \\ - 1.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}4 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.6 \\ 4 \\ - 2.4 \\ - 1.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}4 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrr}- 2.5 & 3 & 5.5 \\ -2 & 2.4 & - 7.5 \\ 5 & -6 & 3 \\ - 0.5 & 0.6 & 1.1 \end{array}\right) \sim \left(\begin{array}{rrr}5 & -6 & 3 \\ 0 & 0 & 7 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 1.2 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-8 \\ 5.5 \\ 2 \\ - 1.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r} 1.2 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 10.5 \\ 3.5 \\ 7 \\ - 2.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r} 1.2 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrr}-1 & - 0.4 & 0.3 \\ -4 & - 1.6 & 3 \\ 5 & 2 & - 7.5 \\ -3 & - 1.2 & 5.7 \end{array}\right) \sim \left(\begin{array}{rrr}5 & 2 & - 7.5 \\ 0 & 0 & -3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 0.4 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.7 \\ -5 \\ 2.5 \\ - 0.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}- 0.4 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 2.4 \\ -6 \\ 0 \\ 2.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 2 \end{array}\right) + u_2 \left(\begin{array}{r}- 0.4 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrr}-4 & 4.8 & 6.6 \\ -2 & 2.4 & 0.4 \\ -2 & 2.4 & 9.1 \\ -2 & 2.4 & 6.2 \end{array}\right) \sim \left(\begin{array}{rrr}-4 & 4.8 & 6.6 \\ 0 & 0 & 5.8 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 1.2 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.4 \\ - 3.6 \\ 5.1 \\ 2.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r} 1.2 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 5.4 \\ - 5.6 \\ 3.1 \\ 0.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r} 1.2 \\ 1 \\ 0 \end{array}\right)

4 \times 3 with two free variables

  v v
* * *
0 0 0
0 0 0
0 0 0

Example matrix

A = \left(\begin{array}{rrr}- 0.4 & - 1.2 & - 3.6 \\ 1 & 3 & 9 \\ - 0.6 & - 1.8 & - 5.4 \\ - 0.3 & - 0.9 & - 2.7 \end{array}\right) \sim \left(\begin{array}{rrr}1 & 3 & 9 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-3 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}-9 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.4 \\ 1 \\ - 0.6 \\ - 0.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-3 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-9 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.8 \\ 2 \\ - 1.2 \\ - 0.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-3 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-9 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrr}2 & -2 & 3 \\ 0.6 & - 0.6 & 0.9 \\ 0.2 & - 0.2 & 0.3 \\ 0.4 & - 0.4 & 0.6 \end{array}\right) \sim \left(\begin{array}{rrr}2 & -2 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}1 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}- 1.5 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}2 \\ 0.6 \\ 0.2 \\ 0.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}1 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 1.5 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}4 \\ 1.2 \\ 0.4 \\ 0.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}1 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 1.5 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrr}-1 & - 0.5 & 1.6 \\ 5 & 2.5 & -8 \\ -2 & -1 & 3.2 \\ 4 & 2 & - 6.4 \end{array}\right) \sim \left(\begin{array}{rrr}5 & 2.5 & -8 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 0.5 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 1.6 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-1 \\ 5 \\ -2 \\ 4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 0.5 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 1.6 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-2 \\ 10 \\ -4 \\ 8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 0.5 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 1.6 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrr}1 & - 1.1 & 3.8 \\ 2 & - 2.2 & 7.6 \\ -1 & 1.1 & - 3.8 \\ -1 & 1.1 & - 3.8 \end{array}\right) \sim \left(\begin{array}{rrr}2 & - 2.2 & 7.6 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 1.1 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}- 3.8 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}1 \\ 2 \\ -1 \\ -1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 1.1 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 3.8 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}2 \\ 4 \\ -2 \\ -2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 1.1 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 3.8 \\ 0 \\ 1 \end{array}\right)

4 \times 4

  1. 4 \times 4 with 0 free variables
  2. 4 \times 4 with 1 free variables
  3. 4 \times 4 with 2 free variables
  4. 4 \times 4 with 3 free variables

4 \times 4 with no free variables

       
* * * *
0 * * *
0 0 * *
0 0 0 *

Example matrix

A = \left(\begin{array}{rrrr} 1.8 & - 3.8 & - 9.6 & 2.7 \\ 2 & -4 & -6 & 5 \\ 1.2 & - 1.4 & 2.4 & 7 \\ - 0.2 & 0.1 & - 3.6 & - 11.5 \end{array}\right) \sim \left(\begin{array}{rrrr}2 & -4 & -6 & 5 \\ 0 & 1 & 6 & 4 \\ 0 & 0 & -3 & -1 \\ 0 & 0 & 0 & -9 \end{array}\right) , \textrm{null}(A) = \{\textbf{0}\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.1 \\ 1 \\ 0.6 \\ 7.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ -2 \\ 1 \\ -1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 4.9 \\ 1 \\ 4.6 \\ 3.7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ -4 \\ 2 \\ -1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrr}- 4.2 & - 8.6 & 3.6 & - 4.8 \\ 6.3 & - 0.7 & 0.8 & 4.2 \\ 7 & 1 & 4 & 8 \\ - 2.1 & 3.7 & -1 & -5 \end{array}\right) \sim \left(\begin{array}{rrrr}7 & 1 & 4 & 8 \\ 0 & -8 & 6 & 0 \\ 0 & 0 & -4 & -3 \\ 0 & 0 & 0 & -5 \end{array}\right) , \textrm{null}(A) = \{\textbf{0}\}

Example solution

\textbf{b} = \left(\begin{array}{r} 2.8 \\ 7.4 \\ -8 \\ - 3.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ -2 \\ -3 \\ -1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 5.2 \\ 8.8 \\ -8 \\ 3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ -1 \\ -3 \\ -2 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrr} 1.2 & 5.5 & - 9.6 & 3.7 \\ 3 & 9 & - 8.5 & -1 \\ - 2.4 & - 16.7 & 5.3 & 2.8 \\ - 1.2 & - 5.5 & - 0.8 & 0.5 \end{array}\right) \sim \left(\begin{array}{rrrr}3 & 9 & - 8.5 & -1 \\ 0 & - 9.5 & - 1.5 & 2 \\ 0 & 0 & - 6.5 & 4.5 \\ 0 & 0 & 0 & -3 \end{array}\right) , \textrm{null}(A) = \{\textbf{0}\}

Example solution

\textbf{b} = \left(\begin{array}{r} 4.7 \\ -3 \\ - 2.1 \\ - 8.7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 1 \\ 2 \\ 4 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 2.2 \\ 1 \\ - 7.3 \\ - 10.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 1 \\ 2 \\ 3 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrr}-8 & 6.8 & 0 & - 3.4 \\ -4 & 4.7 & 3.8 & - 13.8 \\ 4 & -6 & - 4.8 & 7.1 \\ -4 & 4.7 & 3.1 & - 11.3 \end{array}\right) \sim \left(\begin{array}{rrrr}-8 & 6.8 & 0 & - 3.4 \\ 0 & - 2.6 & - 4.8 & 5.4 \\ 0 & 0 & 1.4 & - 9.4 \\ 0 & 0 & 0 & - 2.2 \end{array}\right) , \textrm{null}(A) = \{\textbf{0}\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 5.8 \\ 2.9 \\ 4.5 \\ 1.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 1 \\ -2 \\ -1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-7 \\ 3.6 \\ 2.5 \\ 2.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 2 \\ -2 \\ -1 \end{array}\right)

4 \times 4 with one free variable

      v
* * * *
0 * * *
0 0 * *
0 0 0 0

Example matrix

A = \left(\begin{array}{rrrr}-1 & -7 & 0 & -9 \\ - 0.4 & - 6.8 & 6 & 4.4 \\ - 0.3 & - 1.3 & - 6.2 & - 10.3 \\ 0.1 & 4.3 & - 8.4 & - 9.9 \end{array}\right) \sim \left(\begin{array}{rrrr}-1 & -7 & 0 & -9 \\ 0 & -4 & 6 & 8 \\ 0 & 0 & -5 & -6 \\ 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 10.4 \\ 0.2 \\ - 1.2 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}10 \\ 6 \\ 7.6 \\ 0.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -2 \\ -1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 10.4 \\ 0.2 \\ - 1.2 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}11 \\ 6.4 \\ 7.9 \\ 0.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ -2 \\ -1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 10.4 \\ 0.2 \\ - 1.2 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrr}- 0.1 & 1.2 & 5.9 & 4 \\ - 0.9 & 2.5 & 7.6 & 10.8 \\ 1 & -2 & -9 & 0 \\ - 0.4 & 1.6 & 7.2 & 4 \end{array}\right) \sim \left(\begin{array}{rrrr}1 & -2 & -9 & 0 \\ 0 & 1 & 5 & 4 \\ 0 & 0 & -4 & 8 \\ 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-10 \\ -14 \\ 2 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.2 \\ 1.5 \\ 2 \\ - 1.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 4 \\ -1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-10 \\ -14 \\ 2 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 2.1 \\ - 1.7 \\ -1 \\ 1.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ -3 \\ 1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-10 \\ -14 \\ 2 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrr}-2 & - 1.9 & - 6.2 & - 6.3 \\ 5 & 6 & 8 & 9.5 \\ 4 & 4.4 & 7.8 & 11.6 \\ 2 & 2.5 & 3.2 & 2.1 \end{array}\right) \sim \left(\begin{array}{rrrr}5 & 6 & 8 & 9.5 \\ 0 & 0.5 & -3 & - 2.5 \\ 0 & 0 & -1 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 25.5 \\ 17 \\ 2 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}2 \\ 0 \\ - 0.6 \\ - 0.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -2 \\ -1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 25.5 \\ 17 \\ 2 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 4.5 \\ 0 \\ 2.6 \\ - 0.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ -3 \\ 1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 25.5 \\ 17 \\ 2 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrr} 0.1 & - 0.4 & - 2.1 & - 1.6 \\ - 0.1 & 1 & - 5.6 & 4.6 \\ 0.2 & - 1.6 & 5.8 & 2.4 \\ - 0.1 & 1 & - 5.3 & - 6.2 \end{array}\right) \sim \left(\begin{array}{rrrr} 0.2 & - 1.6 & 5.8 & 2.4 \\ 0 & 0.4 & -5 & - 2.8 \\ 0 & 0 & - 0.2 & 7.2 \\ 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}2600 \\ 457 \\ 36 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 3.4 \\ - 1.9 \\ 0 \\ - 1.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 4 \\ 1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}2600 \\ 457 \\ 36 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 3.5 \\ - 1.8 \\ - 0.2 \\ - 1.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 4 \\ 1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}2600 \\ 457 \\ 36 \\ 1 \end{array}\right)

    v  
* * * *
0 * * *
0 0 0 *
0 0 0 0

Example matrix

A = \left(\begin{array}{rrrr}4 & 1 & -2 & -3 \\ - 0.8 & 4.8 & - 5.6 & 0.6 \\ -2 & 2.5 & - 2.6 & 3.5 \\ - 1.2 & 0.2 & 0 & 0.7 \end{array}\right) \sim \left(\begin{array}{rrrr}4 & 1 & -2 & -3 \\ 0 & 5 & -6 & 0 \\ 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.2 \\ 1.2 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-3 \\ - 4.4 \\ 2.5 \\ 0 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ -1 \\ 0 \\ 2 \end{array}\right) + u_3 \left(\begin{array}{r} 0.2 \\ 1.2 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-2 \\ - 4.6 \\ 4 \\ - 0.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ -1 \\ 0 \\ 3 \end{array}\right) + u_3 \left(\begin{array}{r} 0.2 \\ 1.2 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrr}1 & 3 & -3 & -4 \\ 0.2 & 2.6 & - 4.2 & -8 \\ 0.4 & - 3.8 & 7.8 & - 3.6 \\ 0.1 & - 1.2 & 2.4 & - 5.8 \end{array}\right) \sim \left(\begin{array}{rrrr}1 & 3 & -3 & -4 \\ 0 & -5 & 9 & -2 \\ 0 & 0 & 0 & -8 \\ 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 2.4 \\ 1.8 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-3 \\ - 9.8 \\ 1.8 \\ - 4.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -1 \\ 0 \\ 1 \end{array}\right) + u_3 \left(\begin{array}{r}- 2.4 \\ 1.8 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-4 \\ -10 \\ 1.4 \\ - 4.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ -1 \\ 0 \\ 1 \end{array}\right) + u_3 \left(\begin{array}{r}- 2.4 \\ 1.8 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrr}-2 & - 9.2 & - 3.2 & - 0.6 \\ - 1.5 & - 1.7 & 5.4 & - 2.9 \\ - 2.5 & - 9.5 & -1 & - 6.5 \\ 0.5 & 2.7 & 1.4 & 3.5 \end{array}\right) \sim \left(\begin{array}{rrrr}- 2.5 & - 9.5 & -1 & - 6.5 \\ 0 & 4 & 6 & 1 \\ 0 & 0 & 0 & 5 \\ 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 5.3 \\ - 1.5 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 4.6 \\ - 4.2 \\ -2 \\ 1.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ -1 \\ 0 \\ 1 \end{array}\right) + u_3 \left(\begin{array}{r} 5.3 \\ - 1.5 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 2.6 \\ - 5.7 \\ - 4.5 \\ 2.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ -1 \\ 0 \\ 1 \end{array}\right) + u_3 \left(\begin{array}{r} 5.3 \\ - 1.5 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrr} 0.8 & - 9.4 & 1.6 & 3.4 \\ - 0.4 & 4.5 & - 0.4 & - 11.5 \\ - 0.4 & 4.6 & - 0.6 & 0.2 \\ - 0.4 & 4.8 & -1 & - 0.2 \end{array}\right) \sim \left(\begin{array}{rrrr} 0.8 & - 9.4 & 1.6 & 3.4 \\ 0 & - 0.2 & 0.4 & - 9.8 \\ 0 & 0 & 0 & 6.8 \\ 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 21.5 \\ 2 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 12.2 \\ - 4.1 \\ 7.8 \\ 7.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 2 \\ 0 \\ 1 \end{array}\right) + u_3 \left(\begin{array}{r} 21.5 \\ 2 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-13 \\ - 3.7 \\ 8.2 \\ 8.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 2 \\ 0 \\ 1 \end{array}\right) + u_3 \left(\begin{array}{r} 21.5 \\ 2 \\ 1 \\ 0 \end{array}\right)

  v    
* * * *
0 0 * *
0 0 0 *
0 0 0 0

Example matrix

A = \left(\begin{array}{rrrr} 0.8 & - 2.4 & - 6.8 & 9.2 \\ - 1.4 & 4.2 & - 2.5 & 10.1 \\ -2 & 6 & 5 & 3 \\ - 1.2 & 3.6 & 4.2 & - 3.4 \end{array}\right) \sim \left(\begin{array}{rrrr}-2 & 6 & 5 & 3 \\ 0 & 0 & -6 & 8 \\ 0 & 0 & 0 & 4 \\ 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}3 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.2 \\ - 0.5 \\ 5 \\ 0.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 2 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}3 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-2 \\ 0.9 \\ 7 \\ 1.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 2 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}3 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrr} 1.4 & - 6.3 & 4.6 & - 0.5 \\ -2 & 9 & -7 & 0 \\ 1 & - 4.5 & 4.5 & -5 \\ 1.6 & - 7.2 & 6.5 & - 4.7 \end{array}\right) \sim \left(\begin{array}{rrrr}-2 & 9 & -7 & 0 \\ 0 & 0 & 1 & -5 \\ 0 & 0 & 0 & -2 \\ 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 4.5 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 3.1 \\ 6 \\ 0 \\ - 1.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ -2 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r} 4.5 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 0.5 \\ -1 \\ - 5.5 \\ - 4.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ -1 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r} 4.5 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrr} 0.5 & - 8.5 & - 6.5 & - 2.5 \\ - 0.3 & 5.1 & - 3.1 & - 6.5 \\ 0.4 & - 6.8 & - 6.6 & 2.4 \\ - 0.2 & 3.4 & 8.2 & 11 \end{array}\right) \sim \left(\begin{array}{rrrr} 0.5 & - 8.5 & - 6.5 & - 2.5 \\ 0 & 0 & -7 & -8 \\ 0 & 0 & 0 & 6 \\ 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}17 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-2 \\ 2.2 \\ - 7.4 \\ - 3.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 1 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}17 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 2.5 \\ 2.5 \\ - 7.8 \\ - 3.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 1 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}17 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrr}-1 & 9.2 & 2.6 & 0.8 \\ 0.5 & - 4.6 & 6.7 & - 5.2 \\ - 0.5 & 4.6 & - 2.7 & - 1.6 \\ 0.5 & - 4.6 & - 5.3 & - 0.2 \end{array}\right) \sim \left(\begin{array}{rrrr}-1 & 9.2 & 2.6 & 0.8 \\ 0 & 0 & 8 & - 4.8 \\ 0 & 0 & 0 & - 4.4 \\ 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 9.2 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 1.2 \\ - 2.2 \\ - 7.4 \\ - 4.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 1 \\ 2 \end{array}\right) + u_2 \left(\begin{array}{r} 9.2 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 0.2 \\ - 1.7 \\ - 7.9 \\ - 3.7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 1 \\ 2 \end{array}\right) + u_2 \left(\begin{array}{r} 9.2 \\ 1 \\ 0 \\ 0 \end{array}\right)

4 \times 4 with two free variables

    v v
* * * *
0 * * *
0 0 0 0
0 0 0 0

Example matrix

A = \left(\begin{array}{rrrr}- 0.9 & 4.3 & 0.2 & - 9.1 \\ 0.1 & - 1.5 & - 3.6 & 0.5 \\ -1 & 7 & 8 & -9 \\ - 0.5 & 3.1 & 2.6 & - 4.7 \end{array}\right) \sim \left(\begin{array}{rrrr}-1 & 7 & 8 & -9 \\ 0 & -2 & -7 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 16.5 \\ - 3.5 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}- 12.5 \\ - 0.5 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 0.7 \\ - 1.1 \\ 3 \\ 1.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 16.5 \\ - 3.5 \\ 1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 12.5 \\ - 0.5 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 1.6 \\ - 1.2 \\ 4 \\ 1.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 16.5 \\ - 3.5 \\ 1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 12.5 \\ - 0.5 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrr}- 3.5 & 4.5 & 8 & 7.4 \\ 5 & -5 & 0 & -2 \\ 2.5 & - 2.6 & - 0.8 & - 1.6 \\ - 4.5 & 4.6 & 0.8 & 2.4 \end{array}\right) \sim \left(\begin{array}{rrrr}5 & -5 & 0 & -2 \\ 0 & 1 & 8 & 6 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-8 \\ -8 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}- 5.6 \\ -6 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.5 \\ 5 \\ 2.2 \\ - 4.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 3 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-8 \\ -8 \\ 1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 5.6 \\ -6 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.5 \\ 5 \\ 2.3 \\ - 4.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 2 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-8 \\ -8 \\ 1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 5.6 \\ -6 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrr}- 0.6 & - 0.5 & - 4.6 & 4.1 \\ 0.8 & - 2.5 & 1.7 & - 11.8 \\ 1 & 0 & 6.5 & - 8.5 \\ - 0.2 & - 1.5 & - 3.4 & - 1.3 \end{array}\right) \sim \left(\begin{array}{rrrr}1 & 0 & 6.5 & - 8.5 \\ 0 & - 2.5 & - 3.5 & -5 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 6.5 \\ - 1.4 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 8.5 \\ -2 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.7 \\ - 0.9 \\ 2 \\ - 1.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 6.5 \\ - 1.4 \\ 1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 8.5 \\ -2 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.1 \\ 3.3 \\ 1 \\ 1.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ -1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 6.5 \\ - 1.4 \\ 1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 8.5 \\ -2 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrr} 0.5 & 1 & 6.1 & - 1.4 \\ 1 & 0 & 9.2 & 5.6 \\ 0.5 & 2 & 7.6 & - 5.6 \\ 0.5 & -1 & 3.1 & 7 \end{array}\right) \sim \left(\begin{array}{rrrr}1 & 0 & 9.2 & 5.6 \\ 0 & 2 & 3 & - 8.4 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 9.2 \\ - 1.5 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}- 5.6 \\ 4.2 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.5 \\ 1 \\ - 1.5 \\ 1.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ -1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 9.2 \\ - 1.5 \\ 1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 5.6 \\ 4.2 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}0 \\ 2 \\ -1 \\ 2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ -1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 9.2 \\ - 1.5 \\ 1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 5.6 \\ 4.2 \\ 0 \\ 1 \end{array}\right)

  v   v
* * * *
0 0 * *
0 0 0 0
0 0 0 0

Example matrix

A = \left(\begin{array}{rrrr}- 0.8 & 0.8 & 3.4 & 10.2 \\ - 3.2 & 3.2 & - 2.9 & 11.1 \\ 4 & -4 & 8 & -6 \\ 3.6 & - 3.6 & 8.2 & - 3.6 \end{array}\right) \sim \left(\begin{array}{rrrr}4 & -4 & 8 & -6 \\ 0 & 0 & 5 & 9 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}1 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 5.1 \\ 0 \\ - 1.8 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-5 \\ - 3.5 \\ 0 \\ -1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ -1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}1 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 5.1 \\ 0 \\ - 1.8 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 4.2 \\ - 0.3 \\ -4 \\ - 4.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ -1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}1 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 5.1 \\ 0 \\ - 1.8 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrr} 0.7 & - 2.8 & 6.6 & - 8.8 \\ 1 & -4 & 3 & -1 \\ 0.4 & - 1.6 & - 3.8 & 8.6 \\ - 0.2 & 0.8 & - 2.1 & 2.9 \end{array}\right) \sim \left(\begin{array}{rrrr}1 & -4 & 3 & -1 \\ 0 & 0 & -5 & 9 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}4 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 4.4 \\ 0 \\ 1.8 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 3.8 \\ 1 \\ 5.4 \\ 1.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ -1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}4 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 4.4 \\ 0 \\ 1.8 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 4.5 \\ 0 \\ 5 \\ 1.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ -1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}4 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 4.4 \\ 0 \\ 1.8 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrr}- 0.3 & 2.1 & - 3.4 & 0.4 \\ - 0.5 & 3.5 & -6 & 1.5 \\ - 0.2 & 1.4 & - 1.4 & - 1.9 \\ 0.3 & - 2.1 & 3.4 & - 0.4 \end{array}\right) \sim \left(\begin{array}{rrrr}- 0.5 & 3.5 & -6 & 1.5 \\ 0 & 0 & 1 & - 2.5 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}7 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-27 \\ 0 \\ 2.5 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 2.2 \\ 4 \\ 0.6 \\ - 2.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ -1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}7 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-27 \\ 0 \\ 2.5 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 2.5 \\ 4.5 \\ 0.8 \\ - 2.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ -1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}7 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-27 \\ 0 \\ 2.5 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrr} 0.4 & - 5.8 & - 2.8 & 2 \\ 0.2 & - 2.9 & 2.6 & - 8.6 \\ 0.2 & - 2.9 & - 3.4 & 5.8 \\ 0.2 & - 2.9 & 0.6 & - 3.8 \end{array}\right) \sim \left(\begin{array}{rrrr} 0.4 & - 5.8 & - 2.8 & 2 \\ 0 & 0 & 4 & - 9.6 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 14.5 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 11.8 \\ 0 \\ 2.4 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.2 \\ 3.4 \\ - 2.6 \\ 1.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 14.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 11.8 \\ 0 \\ 2.4 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.6 \\ 3.2 \\ - 2.8 \\ 1.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 14.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 11.8 \\ 0 \\ 2.4 \\ 1 \end{array}\right)

  v v  
* * * *
0 0 0 *
0 0 0 0
0 0 0 0

Example matrix

A = \left(\begin{array}{rrrr}-2 & 9 & 6 & -5 \\ 1 & - 4.5 & -3 & - 3.5 \\ 0.4 & - 1.8 & - 1.2 & 3.4 \\ 1.8 & - 8.1 & - 5.4 & 0.3 \end{array}\right) \sim \left(\begin{array}{rrrr}-2 & 9 & 6 & -5 \\ 0 & 0 & 0 & -6 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 4.5 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}3 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}3 \\ 4.5 \\ -3 \\ 1.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r} 4.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}3 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}1 \\ 5.5 \\ - 2.6 \\ 3.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r} 4.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}3 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrr} 4.5 & 3.6 & - 0.9 & 7 \\ -3 & - 2.4 & 0.6 & - 5.6 \\ 5 & 4 & -1 & 0 \\ 1.5 & 1.2 & - 0.3 & 0.7 \end{array}\right) \sim \left(\begin{array}{rrrr}5 & 4 & -1 & 0 \\ 0 & 0 & 0 & 7 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 0.8 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 0.2 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 2.5 \\ 2.6 \\ 5 \\ 0.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}- 0.8 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 0.2 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}2 \\ - 0.4 \\ 10 \\ 2.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}- 0.8 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 0.2 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrr}- 0.3 & - 5.4 & - 2.4 & 3 \\ 0.3 & 5.4 & 2.4 & - 3.9 \\ 0.5 & 9 & 4 & 1 \\ 0.2 & 3.6 & 1.6 & 2.2 \end{array}\right) \sim \left(\begin{array}{rrrr} 0.5 & 9 & 4 & 1 \\ 0 & 0 & 0 & - 4.5 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-18 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-8 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 2.4 \\ - 3.3 \\ 2 \\ 2.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}-18 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-8 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 2.1 \\ -3 \\ 2.5 \\ 2.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}-18 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-8 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrr}- 0.2 & 0.5 & - 0.5 & 1.4 \\ - 0.4 & 1 & -1 & 2.6 \\ 0.2 & - 0.5 & 0.5 & - 1.5 \\ 0.2 & - 0.5 & 0.5 & - 1.4 \end{array}\right) \sim \left(\begin{array}{rrrr}- 0.4 & 1 & -1 & 2.6 \\ 0 & 0 & 0 & - 0.2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 2.5 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 2.5 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 0.6 \\ 1 \\ - 0.7 \\ - 0.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r} 2.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 2.5 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 0.8 \\ 1.4 \\ - 0.9 \\ - 0.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r} 2.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 2.5 \\ 0 \\ 1 \\ 0 \end{array}\right)

4 \times 4 with three free variables

  v v v
* * * *
0 0 0 0
0 0 0 0
0 0 0 0

Example matrix

A = \left(\begin{array}{rrrr}4 & 2 & 6 & -4 \\ - 1.6 & - 0.8 & - 2.4 & 1.6 \\ 1.6 & 0.8 & 2.4 & - 1.6 \\ - 0.8 & - 0.4 & - 1.2 & 0.8 \end{array}\right) \sim \left(\begin{array}{rrrr}4 & 2 & 6 & -4 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 0.5 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 1.5 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}1 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}4 \\ - 1.6 \\ 1.6 \\ - 0.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 0.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 1.5 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}1 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}8 \\ - 3.2 \\ 3.2 \\ - 1.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 0.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 1.5 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}1 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrr} 2.4 & - 3.6 & 3.6 & - 1.2 \\ - 2.8 & 4.2 & - 4.2 & 1.4 \\ 4 & -6 & 6 & -2 \\ -2 & 3 & -3 & 1 \end{array}\right) \sim \left(\begin{array}{rrrr}4 & -6 & 6 & -2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 1.5 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 1.5 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 0.5 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 2.4 \\ - 2.8 \\ 4 \\ -2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 1.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 1.5 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 0.5 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 4.8 \\ - 5.6 \\ 8 \\ -4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 1.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 1.5 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 0.5 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrr}- 0.4 & - 2.4 & - 3.2 & 3.6 \\ - 0.5 & -3 & -4 & 4.5 \\ 0.1 & 0.6 & 0.8 & - 0.9 \\ - 0.2 & - 1.2 & - 1.6 & 1.8 \end{array}\right) \sim \left(\begin{array}{rrrr}- 0.5 & -3 & -4 & 4.5 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-6 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-8 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}9 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.4 \\ - 0.5 \\ 0.1 \\ - 0.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-6 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-8 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}9 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.8 \\ -1 \\ 0.2 \\ - 0.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-6 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-8 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}9 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrr}2 & - 4.6 & 3.4 & - 2.4 \\ 1 & - 2.3 & 1.7 & - 1.2 \\ -1 & 2.3 & - 1.7 & 1.2 \\ -1 & 2.3 & - 1.7 & 1.2 \end{array}\right) \sim \left(\begin{array}{rrrr}2 & - 4.6 & 3.4 & - 2.4 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 2.3 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 1.7 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 1.2 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}2 \\ 1 \\ -1 \\ -1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 2.3 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 1.7 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 1.2 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}4 \\ 2 \\ -2 \\ -2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 2.3 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 1.7 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 1.2 \\ 0 \\ 0 \\ 1 \end{array}\right)

4 \times 5

  1. 4 \times 5 with 1 free variables
  2. 4 \times 5 with 2 free variables
  3. 4 \times 5 with 3 free variables
  4. 4 \times 5 with 4 free variables

4 \times 5 with one free variable

        v
* * * * *
0 * * * *
0 0 * * *
0 0 0 * *

Example matrix

A = \left(\begin{array}{rrrrc}-1 & -2 & 9 & 8 & -4 \\ 0.3 & - 4.4 & - 10.7 & - 9.4 & - 7.8 \\ 0.6 & 0.2 & -12 & - 6.2 & 5.6 \\ - 0.7 & - 5.4 & 4.4 & 2 & - 17.5 \end{array}\right) \sim \left(\begin{array}{rrrrc}-1 & -2 & 9 & 8 & -4 \\ 0 & -5 & -8 & -7 & -9 \\ 0 & 0 & -5 & 0 & 5 \\ 0 & 0 & 0 & 2 & -3 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}28 \\ - 5.5 \\ 1 \\ 1.5 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}4 \\ - 1.2 \\ 7.6 \\ - 0.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -1 \\ -2 \\ 3 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}28 \\ - 5.5 \\ 1 \\ 1.5 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}6 \\ - 1.8 \\ 6.4 \\ 1.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ -1 \\ -2 \\ 3 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}28 \\ - 5.5 \\ 1 \\ 1.5 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc}- 0.6 & -2 & - 8.2 & -5 & - 8.8 \\ 0.8 & -1 & 11.2 & - 6.2 & 5.8 \\ 1 & 5 & 9 & -4 & 6 \\ 0.4 & - 1.5 & 7.4 & 2.3 & 12.6 \end{array}\right) \sim \left(\begin{array}{rrrrc}1 & 5 & 9 & -4 & 6 \\ 0 & -5 & 4 & -3 & 1 \\ 0 & 0 & -2 & -8 & -5 \\ 0 & 0 & 0 & 2 & 7 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-181 \\ 11.5 \\ 11.5 \\ - 3.5 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 3.2 \\ - 16.2 \\ 1 \\ - 6.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 2 \\ -1 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}-181 \\ 11.5 \\ 11.5 \\ - 3.5 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 2.6 \\ -17 \\ 0 \\ - 6.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 2 \\ -1 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}-181 \\ 11.5 \\ 11.5 \\ - 3.5 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc}-1 & 0.8 & - 3.8 & - 5.5 & - 5.4 \\ -2 & - 2.8 & - 0.9 & 6.3 & - 9.4 \\ 2.5 & 1 & 5.5 & -6 & 8 \\ 2 & - 0.4 & 6.6 & - 3.3 & 3.8 \end{array}\right) \sim \left(\begin{array}{rrrrc} 2.5 & 1 & 5.5 & -6 & 8 \\ 0 & -2 & 3.5 & 1.5 & -3 \\ 0 & 0 & 0.5 & -7 & -4 \\ 0 & 0 & 0 & 2 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 25.8 \\ 12.5 \\ 8 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 5.9 \\ - 1.3 \\ 1 \\ - 0.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -4 \\ -2 \\ -1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 25.8 \\ 12.5 \\ 8 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 3.1 \\ - 4.4 \\ -2 \\ - 4.7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 2 \\ -1 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 25.8 \\ 12.5 \\ 8 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrc}- 0.4 & - 1.1 & 1.1 & - 4.8 & 11.6 \\ - 0.4 & - 1.6 & - 0.9 & 8.1 & 2.3 \\ 0.8 & 4.2 & 6.6 & - 2.8 & -6 \\ - 0.4 & - 2.6 & - 5.6 & - 0.8 & - 0.8 \end{array}\right) \sim \left(\begin{array}{rrrrc} 0.8 & 4.2 & 6.6 & - 2.8 & -6 \\ 0 & 1 & 4.4 & - 6.2 & 8.6 \\ 0 & 0 & 0.2 & 9.8 & -5 \\ 0 & 0 & 0 & - 0.4 & -2 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 4207.4 \\ - 1227.6 \\ 270 \\ -5 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 7.6 \\ - 5.8 \\ 0 \\ 1.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -3 \\ 1 \\ -1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 4207.4 \\ - 1227.6 \\ 270 \\ -5 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}8 \\ - 5.4 \\ - 0.8 \\ 1.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ -3 \\ 1 \\ -1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 4207.4 \\ - 1227.6 \\ 270 \\ -5 \\ 1 \end{array}\right)

      v  
* * * * *
0 * * * *
0 0 * * *
0 0 0 0 *

Example matrix

A = \left(\begin{array}{rrrrr}-5 & 7 & 5 & 0 & -9 \\ - 1.5 & 0.3 & 1.1 & - 9.2 & - 4.8 \\ 1 & - 3.4 & 3 & -8 & 2.8 \\ 0.5 & - 0.9 & 1.1 & - 0.2 & 10.9 \end{array}\right) \sim \left(\begin{array}{rrrrr}-5 & 7 & 5 & 0 & -9 \\ 0 & -2 & 4 & -8 & 1 \\ 0 & 0 & -4 & -2 & -3 \\ 0 & 0 & 0 & 0 & 9 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 7.5 \\ -5 \\ - 0.5 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}6 \\ 1.3 \\ 3.8 \\ - 7.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 1 \\ 2 \\ 0 \\ -1 \end{array}\right) + u_4 \left(\begin{array}{r}- 7.5 \\ -5 \\ - 0.5 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}8 \\ 0.5 \\ - 2.6 \\ - 9.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 2 \\ 1 \\ 0 \\ -1 \end{array}\right) + u_4 \left(\begin{array}{r}- 7.5 \\ -5 \\ - 0.5 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr}- 1.5 & 1.1 & 1.1 & - 2.2 & 10.4 \\ 5 & -2 & 8 & 9 & -3 \\ -1 & - 0.6 & - 10.6 & 5.2 & - 6.4 \\ 1.5 & - 1.5 & - 5.1 & 6.6 & - 7.8 \end{array}\right) \sim \left(\begin{array}{rrrrr}5 & -2 & 8 & 9 & -3 \\ 0 & -1 & -9 & 7 & -7 \\ 0 & 0 & -1 & 4 & 6 \\ 0 & 0 & 0 & 0 & 3 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 19.8 \\ -29 \\ 4 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 4.1 \\ 3 \\ 3.4 \\ 3.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ -2 \\ -1 \\ 0 \\ 1 \end{array}\right) + u_4 \left(\begin{array}{r}- 19.8 \\ -29 \\ 4 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}7 \\ 0 \\ 0 \\ - 1.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 2 \\ -1 \\ 0 \\ 1 \end{array}\right) + u_4 \left(\begin{array}{r}- 19.8 \\ -29 \\ 4 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr}2 & 9.5 & -9 & 0.5 & - 4.5 \\ - 1.2 & - 5.8 & 6.2 & 5 & 0.4 \\ - 0.4 & - 2.4 & 0.8 & - 8.6 & - 0.6 \\ 1.2 & 6.1 & - 4.2 & 9.9 & 4.2 \end{array}\right) \sim \left(\begin{array}{rrrrr}2 & 9.5 & -9 & 0.5 & - 4.5 \\ 0 & - 0.5 & -1 & - 8.5 & - 1.5 \\ 0 & 0 & 1 & 7 & -2 \\ 0 & 0 & 0 & 0 & 6.5 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 17.5 \\ -3 \\ -7 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}2 \\ 2.6 \\ - 1.4 \\ - 2.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 1 \\ 2 \\ 0 \\ -1 \end{array}\right) + u_4 \left(\begin{array}{r}- 17.5 \\ -3 \\ -7 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}4 \\ 1.4 \\ - 1.8 \\ - 1.7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 1 \\ 2 \\ 0 \\ -1 \end{array}\right) + u_4 \left(\begin{array}{r}- 17.5 \\ -3 \\ -7 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr}- 0.2 & 4.5 & - 4.2 & - 7.7 & - 11.5 \\ 0.2 & - 3.5 & 0.2 & 6.5 & 13.5 \\ - 0.4 & 5 & 3.6 & 0.6 & - 9.4 \\ - 0.2 & 1.5 & 4.3 & 5.7 & - 1.8 \end{array}\right) \sim \left(\begin{array}{rrrrr}- 0.4 & 5 & 3.6 & 0.6 & - 9.4 \\ 0 & 2 & -6 & -8 & - 6.8 \\ 0 & 0 & -1 & 2.8 & 5.4 \\ 0 & 0 & 0 & 0 & - 3.2 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 181.7 \\ 12.4 \\ 2.8 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 2.5 \\ - 5.5 \\ 1.4 \\ 2.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -2 \\ 1 \\ 0 \\ -1 \end{array}\right) + u_4 \left(\begin{array}{r} 181.7 \\ 12.4 \\ 2.8 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 2.3 \\ - 5.7 \\ 1.8 \\ 2.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ -2 \\ 1 \\ 0 \\ -1 \end{array}\right) + u_4 \left(\begin{array}{r} 181.7 \\ 12.4 \\ 2.8 \\ 1 \\ 0 \end{array}\right)

    v    
* * * * *
0 * * * *
0 0 0 * *
0 0 0 0 *

Example matrix

A = \left(\begin{array}{rrrrr}-4 & 0 & 2 & 1 & -2 \\ 0.4 & 5 & - 9.2 & - 7.1 & - 3.8 \\ - 0.8 & - 4.5 & 8.5 & 15.5 & 2.2 \\ - 2.4 & - 3.5 & 7.5 & 11.8 & - 4.1 \end{array}\right) \sim \left(\begin{array}{rrrrr}-4 & 0 & 2 & 1 & -2 \\ 0 & 5 & -9 & -7 & -4 \\ 0 & 0 & 0 & 9 & -1 \\ 0 & 0 & 0 & 0 & -5 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.5 \\ 1.8 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-1 \\ 0.1 \\ - 7.2 \\ 4.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ -3 \\ 0 \\ -1 \\ -2 \end{array}\right) + u_3 \left(\begin{array}{r} 0.5 \\ 1.8 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-5 \\ - 4.5 \\ - 3.5 \\ 5.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ -4 \\ 0 \\ -1 \\ -2 \end{array}\right) + u_3 \left(\begin{array}{r} 0.5 \\ 1.8 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr}- 0.5 & 0.6 & - 3.5 & - 0.9 & 3.5 \\ - 4.5 & - 9.2 & 5 & 8.2 & - 8.4 \\ -5 & -8 & 0 & 8 & -6 \\ 0.5 & - 0.8 & 4 & - 0.2 & 2.6 \end{array}\right) \sim \left(\begin{array}{rrrrr}-5 & -8 & 0 & 8 & -6 \\ 0 & -2 & 5 & 1 & -3 \\ 0 & 0 & 0 & -1 & 2 \\ 0 & 0 & 0 & 0 & 4 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-4 \\ 2.5 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.9 \\ - 0.8 \\ -2 \\ 5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -1 \\ 0 \\ 2 \\ 1 \end{array}\right) + u_3 \left(\begin{array}{r}-4 \\ 2.5 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.4 \\ 3.7 \\ 3 \\ 4.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ -1 \\ 0 \\ 2 \\ 1 \end{array}\right) + u_3 \left(\begin{array}{r}-4 \\ 2.5 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr}- 0.2 & -1 & 1.4 & 9.3 & 1.7 \\ - 0.5 & 1.5 & - 2.5 & -6 & 7 \\ 0.3 & - 2.9 & 4.5 & 12.6 & - 8.7 \\ 0.2 & - 0.2 & 0.4 & 1.5 & 2.1 \end{array}\right) \sim \left(\begin{array}{rrrrr}- 0.5 & 1.5 & - 2.5 & -6 & 7 \\ 0 & -2 & 3 & 9 & - 4.5 \\ 0 & 0 & 0 & 4.5 & 2.5 \\ 0 & 0 & 0 & 0 & 3.5 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 0.5 \\ 1.5 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 4.4 \\ 5 \\ - 8.5 \\ 2.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -4 \\ 0 \\ -1 \\ 1 \end{array}\right) + u_3 \left(\begin{array}{r}- 0.5 \\ 1.5 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 4.2 \\ 5.5 \\ - 8.8 \\ 2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ -4 \\ 0 \\ -1 \\ 1 \end{array}\right) + u_3 \left(\begin{array}{r}- 0.5 \\ 1.5 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr}- 0.4 & 6 & 6.6 & - 3.2 & - 7.2 \\ - 0.2 & 4 & 7.5 & 7.4 & - 11.2 \\ 0.2 & - 2.5 & - 1.2 & 2.3 & 0.4 \\ 0.2 & - 2.5 & - 1.2 & 8 & - 1.9 \end{array}\right) \sim \left(\begin{array}{rrrrr}- 0.4 & 6 & 6.6 & - 3.2 & - 7.2 \\ 0 & 1 & 4.2 & 9 & - 7.6 \\ 0 & 0 & 0 & - 3.8 & 0.6 \\ 0 & 0 & 0 & 0 & - 1.4 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 46.5 \\ - 4.2 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 2.4 \\ 2 \\ 5 \\ 0.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -4 \\ 0 \\ -2 \\ -3 \end{array}\right) + u_3 \left(\begin{array}{r}- 46.5 \\ - 4.2 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.2 \\ - 3.8 \\ - 3.6 \\ - 2.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 3 \\ 0 \\ 1 \\ 2 \end{array}\right) + u_3 \left(\begin{array}{r}- 46.5 \\ - 4.2 \\ 1 \\ 0 \\ 0 \end{array}\right)

  v      
* * * * *
0 0 * * *
0 0 0 * *
0 0 0 0 *

Example matrix

A = \left(\begin{array}{rrrrr} 2.5 & -2 & -7 & 4.3 & - 6.8 \\ 3.5 & - 2.8 & - 0.6 & 1.9 & 12.2 \\ 5 & -4 & -8 & 7 & 6 \\ 1.5 & - 1.2 & - 0.4 & 0.1 & 2 \end{array}\right) \sim \left(\begin{array}{rrrrr}5 & -4 & -8 & 7 & 6 \\ 0 & 0 & 5 & -3 & 8 \\ 0 & 0 & 0 & -1 & -5 \\ 0 & 0 & 0 & 0 & 1 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.8 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 0.1 \\ 1.9 \\ -3 \\ 2.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 3 \\ 1 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r} 0.8 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 4.6 \\ -1 \\ 0 \\ 1.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 2 \\ 1 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r} 0.8 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr}1 & 5 & -3 & 2 & 9 \\ - 0.2 & -1 & 3 & 7.2 & - 0.6 \\ 0.4 & 2 & - 7.2 & 1.8 & - 4.4 \\ 0.9 & 4.5 & 1.5 & 7.5 & 7.1 \end{array}\right) \sim \left(\begin{array}{rrrrr}1 & 5 & -3 & 2 & 9 \\ 0 & 0 & -6 & 1 & -8 \\ 0 & 0 & 0 & 8 & -2 \\ 0 & 0 & 0 & 0 & -5 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-9 \\ 13 \\ - 6.6 \\ 7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 2 \\ 1 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}-5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-10 \\ 13.2 \\ -7 \\ 6.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 2 \\ 1 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}-5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr} 0.1 & - 1.3 & 3.2 & 2.2 & - 10.8 \\ 0.3 & - 3.9 & 4 & 9.4 & - 9.6 \\ 0.5 & - 6.5 & 0 & 6.5 & - 8.5 \\ - 0.1 & 1.3 & - 1.6 & - 1.4 & - 0.7 \end{array}\right) \sim \left(\begin{array}{rrrrr} 0.5 & - 6.5 & 0 & 6.5 & - 8.5 \\ 0 & 0 & 4 & 5.5 & - 4.5 \\ 0 & 0 & 0 & - 3.5 & - 5.5 \\ 0 & 0 & 0 & 0 & - 7.5 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}13 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}10 \\ -4 \\ - 2.5 \\ 1.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 1 \\ -2 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}13 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 9.9 \\ - 4.3 \\ -3 \\ 1.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 1 \\ -2 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}13 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr} 0.4 & 1.6 & -8 & 8.4 & - 2.6 \\ 0.2 & 0.8 & 2.2 & 6.2 & - 7.3 \\ 0.2 & 0.8 & - 0.9 & 2.8 & - 11.7 \\ - 0.2 & - 0.8 & 7.1 & -2 & - 1.6 \end{array}\right) \sim \left(\begin{array}{rrrrr} 0.4 & 1.6 & -8 & 8.4 & - 2.6 \\ 0 & 0 & 6.2 & 2 & -6 \\ 0 & 0 & 0 & - 2.4 & - 7.4 \\ 0 & 0 & 0 & 0 & - 3.6 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-5 \\ - 14.9 \\ 5.9 \\ - 7.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ -2 \\ -3 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}-4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 5.4 \\ - 15.1 \\ 5.7 \\ - 7.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ -2 \\ -3 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}-4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right)

4 \times 5 with two free variables

      v v
* * * * *
0 * * * *
0 0 * * *
0 0 0 0 0

Example matrix

A = \left(\begin{array}{rrrrr} 0.9 & - 4.4 & 1.4 & 8.3 & - 2.7 \\ 0.5 & - 2.8 & 1.2 & 7.9 & - 9.5 \\ 1 & -6 & 6 & 7 & -3 \\ - 0.1 & 1 & - 2.8 & 2.5 & - 4.5 \end{array}\right) \sim \left(\begin{array}{rrrrr}1 & -6 & 6 & 7 & -3 \\ 0 & 1 & -4 & 2 & 0 \\ 0 & 0 & -1 & 4 & -8 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}53 \\ 14 \\ 4 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}-141 \\ -32 \\ -8 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 3.8 \\ - 2.4 \\ -2 \\ - 1.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 2 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}53 \\ 14 \\ 4 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}-141 \\ -32 \\ -8 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 4.7 \\ - 2.9 \\ -3 \\ - 1.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 2 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}53 \\ 14 \\ 4 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}-141 \\ -32 \\ -8 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr} 0.9 & 6.2 & 8.9 & 6.9 & - 11.6 \\ 1 & 6 & 5 & 7 & -6 \\ - 0.1 & 0.4 & 7.5 & 1.3 & - 8.4 \\ - 0.3 & - 2.6 & - 8.5 & -4 & 9.3 \end{array}\right) \sim \left(\begin{array}{rrrrr}1 & 6 & 5 & 7 & -6 \\ 0 & 1 & 8 & 2 & -9 \\ 0 & 0 & -2 & -1 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 16.5 \\ 2 \\ - 0.5 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}- 26.5 \\ 5 \\ 0.5 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 0.1 \\ -3 \\ 6.3 \\ - 4.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -2 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 16.5 \\ 2 \\ - 0.5 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 26.5 \\ 5 \\ 0.5 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.8 \\ -4 \\ 6.4 \\ - 4.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ -2 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 16.5 \\ 2 \\ - 0.5 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 26.5 \\ 5 \\ 0.5 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr} 0.5 & -1 & 4 & 9.3 & - 6.8 \\ - 0.5 & - 0.8 & 5.4 & 11.8 & - 8.1 \\ - 2.5 & 0 & 7.5 & 1 & 6.5 \\ - 1.5 & 0.6 & 1.6 & - 8.3 & 11.2 \end{array}\right) \sim \left(\begin{array}{rrrrr}- 2.5 & 0 & 7.5 & 1 & 6.5 \\ 0 & -1 & 5.5 & 9.5 & - 5.5 \\ 0 & 0 & - 0.5 & 4 & -5 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 24.4 \\ 53.5 \\ 8 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}- 27.4 \\ - 60.5 \\ -10 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 1.5 \\ 0.7 \\ 0 \\ - 0.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 4 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 24.4 \\ 53.5 \\ 8 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 27.4 \\ - 60.5 \\ -10 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}1 \\ 1.2 \\ 2.5 \\ 1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 4 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 24.4 \\ 53.5 \\ 8 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 27.4 \\ - 60.5 \\ -10 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr}- 0.1 & - 3.6 & - 6.1 & - 9.8 & - 0.7 \\ 0.2 & 6.8 & - 3.4 & 8 & - 6.2 \\ 0.1 & 3.3 & - 6.6 & - 5.3 & - 9.4 \\ - 0.1 & - 3.3 & 6.1 & 2.1 & 7.2 \end{array}\right) \sim \left(\begin{array}{rrrrr} 0.2 & 6.8 & - 3.4 & 8 & - 6.2 \\ 0 & - 0.2 & - 7.8 & - 5.8 & - 3.8 \\ 0 & 0 & -1 & - 6.4 & - 4.4 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 7649.2 \\ 220.6 \\ - 6.4 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}- 5232.2 \\ 152.6 \\ - 4.4 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 12.9 \\ - 9.4 \\ 0.4 \\ 0.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -2 \\ -1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 7649.2 \\ 220.6 \\ - 6.4 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 5232.2 \\ 152.6 \\ - 4.4 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}13 \\ - 9.6 \\ 0.3 \\ 0.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ -2 \\ -1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 7649.2 \\ 220.6 \\ - 6.4 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 5232.2 \\ 152.6 \\ - 4.4 \\ 0 \\ 1 \end{array}\right)

    v   v
* * * * *
0 * * * *
0 0 0 * *
0 0 0 0 0

Example matrix

A = \left(\begin{array}{rrrrr}-5 & 4 & 6 & -9 & -7 \\ - 3.5 & - 0.4 & - 2.2 & - 4.3 & 8.5 \\ 4 & - 7.2 & - 12.8 & 7.2 & 13.6 \\ - 4.5 & 3.2 & 4.6 & - 9.5 & - 10.4 \end{array}\right) \sim \left(\begin{array}{rrrrr}-5 & 4 & 6 & -9 & -7 \\ 0 & -4 & -8 & 0 & 8 \\ 0 & 0 & 0 & 2 & 7 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 0.4 \\ -2 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 6.5 \\ 2 \\ 0 \\ - 3.5 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-1 \\ - 1.5 \\ 4.8 \\ 2.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ -1 \\ 0 \\ -2 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 0.4 \\ -2 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 6.5 \\ 2 \\ 0 \\ - 3.5 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-2 \\ - 6.6 \\ - 2.4 \\ - 0.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 1 \\ 0 \\ -1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 0.4 \\ -2 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 6.5 \\ 2 \\ 0 \\ - 3.5 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr} 0.4 & - 5.1 & - 0.8 & - 10.1 & - 9.6 \\ -2 & 3 & 4 & 9 & 5 \\ - 0.4 & - 4.4 & 0.8 & - 5.2 & -3 \\ - 1.4 & 5.6 & 2.8 & 12 & 8.3 \end{array}\right) \sim \left(\begin{array}{rrrrr}-2 & 3 & 4 & 9 & 5 \\ 0 & -5 & 0 & -7 & -4 \\ 0 & 0 & 0 & -2 & -5 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 4.7 \\ 2.7 \\ 0 \\ - 2.5 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 0.9 \\ -1 \\ 2.8 \\ -2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ -2 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 4.7 \\ 2.7 \\ 0 \\ - 2.5 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 3.3 \\ 1 \\ 1.2 \\ 1.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -3 \\ 0 \\ 2 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 4.7 \\ 2.7 \\ 0 \\ - 2.5 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr}-3 & - 3.5 & 8.6 & - 12.1 & - 3.9 \\ -2 & - 2.6 & 3.6 & - 2.2 & 1 \\ 5 & 7.5 & -1 & 8.5 & - 3.5 \\ 4 & 5.8 & - 2.4 & 5 & - 3.2 \end{array}\right) \sim \left(\begin{array}{rrrrr}5 & 7.5 & -1 & 8.5 & - 3.5 \\ 0 & 1 & 8 & -7 & -6 \\ 0 & 0 & 0 & 4 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 12.2 \\ -8 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 2.2 \\ 2.5 \\ 0 \\ - 0.5 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 7.1 \\ - 0.6 \\ - 3.5 \\ - 0.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -2 \\ 0 \\ -1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 12.2 \\ -8 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 2.2 \\ 2.5 \\ 0 \\ - 0.5 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 7.6 \\ 1.6 \\ -4 \\ - 4.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ -3 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 12.2 \\ -8 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 2.2 \\ 2.5 \\ 0 \\ - 0.5 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr}1 & 1.8 & 2.2 & - 8.6 & 3.8 \\ 0.5 & - 1.1 & - 3.9 & - 5.5 & - 4.9 \\ 0.5 & 1.9 & 3.6 & - 4.1 & 14.7 \\ 0.5 & 1.9 & 3.6 & - 3.9 & 10 \end{array}\right) \sim \left(\begin{array}{rrrrr}1 & 1.8 & 2.2 & - 8.6 & 3.8 \\ 0 & -2 & -5 & - 1.2 & - 6.8 \\ 0 & 0 & 0 & - 0.4 & 9.4 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 2.3 \\ - 2.5 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 229.8 \\ - 17.5 \\ 0 \\ 23.5 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-1 \\ - 5.7 \\ 1.7 \\ 1.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 2 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 2.3 \\ - 2.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 229.8 \\ - 17.5 \\ 0 \\ 23.5 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-2 \\ - 6.2 \\ 1.2 \\ 1.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 2 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 2.3 \\ - 2.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 229.8 \\ - 17.5 \\ 0 \\ 23.5 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr}-5 & 4 & 6 & -9 & -7 \\ - 3.5 & - 0.4 & - 2.2 & - 4.3 & 8.5 \\ 4 & - 7.2 & - 12.8 & 7.2 & 13.6 \\ - 4.5 & 3.2 & 4.6 & - 9.5 & - 10.4 \end{array}\right) \sim \left(\begin{array}{rrrrr}-5 & 4 & 6 & -9 & -7 \\ 0 & -4 & -8 & 0 & 8 \\ 0 & 0 & 0 & 2 & 7 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 0.4 \\ -2 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 6.5 \\ 2 \\ 0 \\ - 3.5 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-1 \\ - 1.5 \\ 4.8 \\ 2.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ -1 \\ 0 \\ -2 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 0.4 \\ -2 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 6.5 \\ 2 \\ 0 \\ - 3.5 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-2 \\ - 6.6 \\ - 2.4 \\ - 0.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 1 \\ 0 \\ -1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 0.4 \\ -2 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 6.5 \\ 2 \\ 0 \\ - 3.5 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr} 0.4 & - 5.1 & - 0.8 & - 10.1 & - 9.6 \\ -2 & 3 & 4 & 9 & 5 \\ - 0.4 & - 4.4 & 0.8 & - 5.2 & -3 \\ - 1.4 & 5.6 & 2.8 & 12 & 8.3 \end{array}\right) \sim \left(\begin{array}{rrrrr}-2 & 3 & 4 & 9 & 5 \\ 0 & -5 & 0 & -7 & -4 \\ 0 & 0 & 0 & -2 & -5 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 4.7 \\ 2.7 \\ 0 \\ - 2.5 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 0.9 \\ -1 \\ 2.8 \\ -2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ -2 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 4.7 \\ 2.7 \\ 0 \\ - 2.5 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 3.3 \\ 1 \\ 1.2 \\ 1.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -3 \\ 0 \\ 2 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 4.7 \\ 2.7 \\ 0 \\ - 2.5 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr}-3 & - 3.5 & 8.6 & - 12.1 & - 3.9 \\ -2 & - 2.6 & 3.6 & - 2.2 & 1 \\ 5 & 7.5 & -1 & 8.5 & - 3.5 \\ 4 & 5.8 & - 2.4 & 5 & - 3.2 \end{array}\right) \sim \left(\begin{array}{rrrrr}5 & 7.5 & -1 & 8.5 & - 3.5 \\ 0 & 1 & 8 & -7 & -6 \\ 0 & 0 & 0 & 4 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 12.2 \\ -8 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 2.2 \\ 2.5 \\ 0 \\ - 0.5 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 7.1 \\ - 0.6 \\ - 3.5 \\ - 0.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -2 \\ 0 \\ -1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 12.2 \\ -8 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 2.2 \\ 2.5 \\ 0 \\ - 0.5 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 7.6 \\ 1.6 \\ -4 \\ - 4.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ -3 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 12.2 \\ -8 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 2.2 \\ 2.5 \\ 0 \\ - 0.5 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr}1 & 1.8 & 2.2 & - 8.6 & 3.8 \\ 0.5 & - 1.1 & - 3.9 & - 5.5 & - 4.9 \\ 0.5 & 1.9 & 3.6 & - 4.1 & 14.7 \\ 0.5 & 1.9 & 3.6 & - 3.9 & 10 \end{array}\right) \sim \left(\begin{array}{rrrrr}1 & 1.8 & 2.2 & - 8.6 & 3.8 \\ 0 & -2 & -5 & - 1.2 & - 6.8 \\ 0 & 0 & 0 & - 0.4 & 9.4 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 2.3 \\ - 2.5 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 229.8 \\ - 17.5 \\ 0 \\ 23.5 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-1 \\ - 5.7 \\ 1.7 \\ 1.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 2 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 2.3 \\ - 2.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 229.8 \\ - 17.5 \\ 0 \\ 23.5 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-2 \\ - 6.2 \\ 1.2 \\ 1.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 2 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 2.3 \\ - 2.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 229.8 \\ - 17.5 \\ 0 \\ 23.5 \\ 1 \end{array}\right)

    v v  
* * * * *
0 * * * *
0 0 0 0 *
0 0 0 0 0

Example matrix

A = \left(\begin{array}{rrrrr} 0.4 & - 3.5 & 2.7 & 5.2 & 12.4 \\ - 0.4 & 5 & - 4.2 & - 7.6 & - 7.6 \\ 1 & 0 & -2 & -1 & 9 \\ 0.1 & 1 & - 1.2 & - 1.7 & - 0.5 \end{array}\right) \sim \left(\begin{array}{rrrrr}1 & 0 & -2 & -1 & 9 \\ 0 & 5 & -5 & -8 & -4 \\ 0 & 0 & 0 & 0 & 6 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}2 \\ 1 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}1 \\ 1.6 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 3.8 \\ -4 \\ -5 \\ - 1.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -2 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_3 \left(\begin{array}{r}2 \\ 1 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}1 \\ 1.6 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 4.2 \\ - 3.6 \\ -6 \\ - 1.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ -2 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_3 \left(\begin{array}{r}2 \\ 1 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}1 \\ 1.6 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr}5 & -7 & 9 & -9 & 0 \\ 3 & - 5.2 & 3.4 & - 3.4 & 8 \\ -4 & 5.5 & - 7.4 & 7.4 & 3.8 \\ - 2.5 & 4.4 & - 2.7 & 2.7 & - 7.5 \end{array}\right) \sim \left(\begin{array}{rrrrr}5 & -7 & 9 & -9 & 0 \\ 0 & -1 & -2 & 2 & 8 \\ 0 & 0 & 0 & 0 & 3 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 4.6 \\ -2 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 4.6 \\ 2 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-1 \\ 4.4 \\ 4.3 \\ - 4.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 3 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_3 \left(\begin{array}{r}- 4.6 \\ -2 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 4.6 \\ 2 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}1 \\ 6.6 \\ 2.8 \\ - 6.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 2 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_3 \left(\begin{array}{r}- 4.6 \\ -2 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 4.6 \\ 2 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr} 0.5 & 6.5 & -7 & -8 & -2 \\ - 0.4 & - 4.2 & 1.6 & 12.4 & 8.6 \\ - 0.1 & - 0.9 & - 0.2 & 4 & - 0.3 \\ 0.1 & 1.9 & - 3.8 & 2 & 1 \end{array}\right) \sim \left(\begin{array}{rrrrr} 0.5 & 6.5 & -7 & -8 & -2 \\ 0 & 1 & -4 & 6 & 7 \\ 0 & 0 & 0 & 0 & - 3.5 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-38 \\ 4 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}94 \\ -6 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-9 \\ - 1.8 \\ 1.7 \\ - 4.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -2 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_3 \left(\begin{array}{r}-38 \\ 4 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}94 \\ -6 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 9.5 \\ - 1.4 \\ 1.8 \\ - 4.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ -2 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_3 \left(\begin{array}{r}-38 \\ 4 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}94 \\ -6 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr} 2.5 & 0.6 & 5.8 & 1.1 & 1.7 \\ -5 & - 0.8 & - 5.4 & - 6.8 & - 4.8 \\ - 2.5 & 0 & 3.5 & -8 & - 2.6 \\ - 2.5 & - 0.6 & - 5.8 & - 1.1 & -2 \end{array}\right) \sim \left(\begin{array}{rrrrr}-5 & - 0.8 & - 5.4 & - 6.8 & - 4.8 \\ 0 & 0.4 & 6.2 & - 4.6 & - 0.2 \\ 0 & 0 & 0 & 0 & - 0.6 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 1.4 \\ - 15.5 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 3.2 \\ 11.5 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}1 \\ 0.4 \\ 0.2 \\ - 0.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ -1 \\ 0 \\ 0 \\ -2 \end{array}\right) + u_3 \left(\begin{array}{r} 1.4 \\ - 15.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 3.2 \\ 11.5 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.4 \\ 1.4 \\ 0.1 \\ 0.7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ -2 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_3 \left(\begin{array}{r} 1.4 \\ - 15.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 3.2 \\ 11.5 \\ 0 \\ 1 \\ 0 \end{array}\right)

  v     v
* * * * *
0 0 * * *
0 0 0 * *
0 0 0 0 0

Example matrix

A = \left(\begin{array}{rrrrr} 1.2 & - 1.8 & - 5.6 & 10.4 & 12.4 \\ -1 & 1.5 & 5.5 & - 9.5 & -7 \\ 2 & -3 & -9 & 7 & 6 \\ - 0.2 & 0.3 & 0.2 & 0.5 & - 2.6 \end{array}\right) \sim \left(\begin{array}{rrrrr}2 & -3 & -9 & 7 & 6 \\ 0 & 0 & 1 & -6 & -4 \\ 0 & 0 & 0 & 5 & 8 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 1.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 22.6 \\ 0 \\ - 5.6 \\ - 1.6 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}4 \\ - 2.5 \\ -3 \\ 0.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 2 \\ 1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 1.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 22.6 \\ 0 \\ - 5.6 \\ - 1.6 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 2.8 \\ - 1.5 \\ -5 \\ 0.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 2 \\ 1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 1.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 22.6 \\ 0 \\ - 5.6 \\ - 1.6 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr}2 & -1 & -4 & -8 & 9 \\ 1.4 & - 0.7 & - 1.8 & - 11.6 & 14.3 \\ - 0.8 & 0.4 & 2.4 & - 6.6 & 0.8 \\ 0.4 & - 0.2 & - 0.4 & - 3.5 & 5.2 \end{array}\right) \sim \left(\begin{array}{rrrrr}2 & -1 & -4 & -8 & 9 \\ 0 & 0 & 1 & -6 & 8 \\ 0 & 0 & 0 & -5 & -2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 26.9 \\ 0 \\ - 10.4 \\ - 0.4 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}0 \\ -7 \\ - 10.6 \\ - 2.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ -1 \\ 1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 0.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 26.9 \\ 0 \\ - 10.4 \\ - 0.4 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}2 \\ - 5.6 \\ - 11.4 \\ - 1.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ -1 \\ 1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 0.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 26.9 \\ 0 \\ - 10.4 \\ - 0.4 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr}-2 & -5 & 9.5 & -8 & 1.5 \\ - 0.8 & -2 & 3.3 & - 9.2 & 3.1 \\ 0.8 & 2 & - 4.2 & - 4.1 & - 8.1 \\ 0.4 & 1 & - 1.5 & 7.9 & 3.4 \end{array}\right) \sim \left(\begin{array}{rrrrr}-2 & -5 & 9.5 & -8 & 1.5 \\ 0 & 0 & - 0.5 & -6 & 2.5 \\ 0 & 0 & 0 & - 2.5 & - 9.5 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 2.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 256.3 \\ 0 \\ 50.6 \\ - 3.8 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}3 \\ - 5.8 \\ - 9.3 \\ 6.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 2 \\ 1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 2.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 256.3 \\ 0 \\ 50.6 \\ - 3.8 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}5 \\ -5 \\ - 10.1 \\ 6.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 2 \\ 1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 2.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 256.3 \\ 0 \\ 50.6 \\ - 3.8 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr} 0.2 & 5.8 & -1 & -5 & - 1.8 \\ - 0.1 & - 2.9 & 0.1 & - 6.5 & 9.1 \\ 0.1 & 2.9 & - 0.3 & 3 & 3.6 \\ - 0.1 & - 2.9 & 0.7 & 7.5 & 1.1 \end{array}\right) \sim \left(\begin{array}{rrrrr} 0.2 & 5.8 & -1 & -5 & - 1.8 \\ 0 & 0 & - 0.4 & -9 & 8.2 \\ 0 & 0 & 0 & 1 & 8.6 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-29 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}864 \\ 0 \\ 214 \\ - 8.6 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 1.6 \\ 6.6 \\ - 3.9 \\ -5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 4 \\ -1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-29 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}864 \\ 0 \\ 214 \\ - 8.6 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 1.4 \\ 6.7 \\ -4 \\ - 4.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 4 \\ -1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-29 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}864 \\ 0 \\ 214 \\ - 8.6 \\ 1 \end{array}\right)

  v   v  
* * * * *
0 0 * * *
0 0 0 0 *
0 0 0 0 0

Example matrix

A = \left(\begin{array}{rrrrr}1 & -3 & 5 & -7 & 0 \\ 0.7 & - 2.1 & 4.7 & - 7.3 & 0.8 \\ - 0.8 & 2.4 & -8 & 13.6 & -6 \\ 0.1 & - 0.3 & 1.3 & - 2.3 & 1.3 \end{array}\right) \sim \left(\begin{array}{rrrrr}1 & -3 & 5 & -7 & 0 \\ 0 & 0 & -4 & 8 & -6 \\ 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}3 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-3 \\ 0 \\ 2 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-1 \\ - 1.1 \\ - 1.2 \\ 0.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ -1 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}3 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-3 \\ 0 \\ 2 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-2 \\ - 1.8 \\ - 0.4 \\ 0.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ -1 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}3 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-3 \\ 0 \\ 2 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr}- 0.1 & - 0.6 & - 0.2 & - 7.7 & - 2.4 \\ 1 & 6 & -7 & -4 & 8 \\ 0.4 & 2.4 & - 1.8 & 7.4 & 7.2 \\ - 0.6 & - 3.6 & 4.3 & 3.3 & -6 \end{array}\right) \sim \left(\begin{array}{rrrrr}1 & 6 & -7 & -4 & 8 \\ 0 & 0 & 1 & 9 & 4 \\ 0 & 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-6 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-59 \\ 0 \\ -9 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 3.2 \\ -2 \\ 5.2 \\ 0.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 2 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}-6 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-59 \\ 0 \\ -9 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 3.1 \\ -3 \\ 4.8 \\ 0.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 2 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}-6 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-59 \\ 0 \\ -9 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr}- 0.8 & 4.8 & - 1.5 & 13.1 & - 2.2 \\ 1 & -6 & 2.5 & - 4.5 & 9 \\ 0.4 & - 2.4 & 0.7 & - 7.5 & - 4.4 \\ 0.2 & - 1.2 & 0.4 & - 2.8 & - 2.2 \end{array}\right) \sim \left(\begin{array}{rrrrr}1 & -6 & 2.5 & - 4.5 & 9 \\ 0 & 0 & 0.5 & 9.5 & 5 \\ 0 & 0 & 0 & 0 & -5 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}6 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}52 \\ 0 \\ -19 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 0.6 \\ 3 \\ - 5.6 \\ -3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ -4 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}6 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}52 \\ 0 \\ -19 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 1.4 \\ 2 \\ -6 \\ - 3.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ -4 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}6 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}52 \\ 0 \\ -19 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr}-1 & - 5.8 & 6.2 & - 8.8 & 8.4 \\ - 0.5 & - 2.9 & 2.7 & - 5.6 & 14.3 \\ - 0.5 & - 2.9 & 3.9 & -2 & - 2.8 \\ 0.5 & 2.9 & - 3.5 & 3.2 & 2.6 \end{array}\right) \sim \left(\begin{array}{rrrrr}-1 & - 5.8 & 6.2 & - 8.8 & 8.4 \\ 0 & 0 & 0.8 & 2.4 & -7 \\ 0 & 0 & 0 & 0 & 6.6 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 5.8 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 27.4 \\ 0 \\ -3 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 0.2 \\ 10.6 \\ - 7.7 \\ 7.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ -1 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}- 5.8 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 27.4 \\ 0 \\ -3 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.8 \\ 10.1 \\ - 8.2 \\ 7.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ -1 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}- 5.8 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 27.4 \\ 0 \\ -3 \\ 1 \\ 0 \end{array}\right)

  v v    
* * * * *
0 0 0 * *
0 0 0 0 *
0 0 0 0 0

Example matrix

A = \left(\begin{array}{rrrrr} 0.5 & - 0.5 & 0.4 & - 0.8 & - 8.4 \\ -5 & 5 & -4 & -8 & 2 \\ 1.5 & - 1.5 & 1.2 & 6.4 & 2.4 \\ - 2.5 & 2.5 & -2 & - 7.6 & - 3.8 \end{array}\right) \sim \left(\begin{array}{rrrrr}-5 & 5 & -4 & -8 & 2 \\ 0 & 0 & 0 & 4 & 3 \\ 0 & 0 & 0 & 0 & -7 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}1 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 0.8 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 6.6 \\ 0 \\ -1 \\ - 1.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ -1 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}1 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 0.8 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 4.8 \\ -2 \\ - 4.4 \\ 1.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 0 \\ -2 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}1 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 0.8 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr} 1.8 & 2.7 & - 3.6 & - 4.4 & 3.6 \\ - 0.4 & - 0.6 & 0.8 & 0.4 & 6.2 \\ -2 & -3 & 4 & 6 & -4 \\ 1 & 1.5 & -2 & - 3.4 & - 2.9 \end{array}\right) \sim \left(\begin{array}{rrrrr}-2 & -3 & 4 & 6 & -4 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 7 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 1.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}2 \\ 5.4 \\ 0 \\ - 5.7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 0 \\ 2 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}- 1.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.8 \\ - 7.4 \\ 2 \\ 3.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 0 \\ 1 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}- 1.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr} 0.5 & 6.5 & -4 & -7 & - 8.5 \\ 0.3 & 3.9 & - 2.4 & - 2.2 & - 8.6 \\ 0.1 & 1.3 & - 0.8 & - 1.8 & -6 \\ 0.2 & 2.6 & - 1.6 & - 1.2 & - 2.2 \end{array}\right) \sim \left(\begin{array}{rrrrr} 0.5 & 6.5 & -4 & -7 & - 8.5 \\ 0 & 0 & 0 & 2 & - 3.5 \\ 0 & 0 & 0 & 0 & -5 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-13 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}8 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 0.5 \\ - 5.2 \\ - 3.8 \\ - 0.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 0 \\ -1 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}-13 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}8 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}0 \\ - 5.5 \\ - 3.9 \\ - 0.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ -1 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}-13 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}8 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr} 0.2 & - 1.3 & - 2.4 & 1 & - 15.8 \\ 0.2 & - 1.3 & - 2.4 & - 0.5 & - 10.7 \\ 0.4 & - 2.6 & - 4.8 & 5 & - 7.4 \\ 0.2 & - 1.3 & - 2.4 & 1 & - 2.9 \end{array}\right) \sim \left(\begin{array}{rrrrr} 0.4 & - 2.6 & - 4.8 & 5 & - 7.4 \\ 0 & 0 & 0 & -3 & -7 \\ 0 & 0 & 0 & 0 & - 8.6 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 6.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}12 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-13 \\ - 10.9 \\ 4.2 \\ - 0.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 0 \\ 2 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r} 6.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}12 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 13.2 \\ - 11.1 \\ 3.8 \\ - 0.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 2 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r} 6.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}12 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)

4 \times 5 with three free variables

    v v v
* * * * *
0 * * * *
0 0 0 0 0
0 0 0 0 0

Example matrix

A = \left(\begin{array}{rrrrr} 0.8 & - 2.8 & -4 & 6.2 & - 2.6 \\ 1 & - 0.4 & 1.2 & - 6.2 & 4.5 \\ -2 & 2 & 0 & 7 & -6 \\ - 0.2 & - 1.4 & - 3.2 & 7.9 & - 4.6 \end{array}\right) \sim \left(\begin{array}{rrrrr}-2 & 2 & 0 & 7 & -6 \\ 0 & -2 & -4 & 9 & -5 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-2 \\ -2 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}8 \\ 4.5 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}- 5.5 \\ - 2.5 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.2 \\ 1.6 \\ -2 \\ - 1.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-2 \\ -2 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}8 \\ 4.5 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 5.5 \\ - 2.5 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.4 \\ 2.6 \\ -4 \\ -2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-2 \\ -2 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}8 \\ 4.5 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 5.5 \\ - 2.5 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr}-1 & 7 & 1 & 9 & -7 \\ - 0.4 & 2.6 & 1 & 3.6 & - 2.5 \\ - 0.9 & 4.3 & 6.9 & 8.1 & - 3.3 \\ 0.2 & - 0.2 & - 3.8 & - 1.8 & - 0.4 \end{array}\right) \sim \left(\begin{array}{rrrrr}-1 & 7 & 1 & 9 & -7 \\ 0 & -2 & 6 & 0 & 3 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}22 \\ 3 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}9 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 3.5 \\ 1.5 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}3 \\ 1 \\ 0.7 \\ 0.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}22 \\ 3 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}9 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 3.5 \\ 1.5 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}4 \\ 1.4 \\ 1.6 \\ 0.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}22 \\ 3 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}9 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 3.5 \\ 1.5 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr}-1 & - 0.3 & -12 & 0.5 & - 0.1 \\ - 2.5 & -2 & - 7.5 & -5 & -4 \\ -2 & -2 & 1.2 & -6 & - 4.4 \\ - 1.5 & - 1.3 & - 2.7 & - 3.5 & - 2.7 \end{array}\right) \sim \left(\begin{array}{rrrrr}- 2.5 & -2 & - 7.5 & -5 & -4 \\ 0 & 0.5 & -9 & 2.5 & 1.5 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 17.4 \\ 18 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}2 \\ -5 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 0.8 \\ -3 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.7 \\ - 0.5 \\ 0 \\ - 0.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ -1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 17.4 \\ 18 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}2 \\ -5 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 0.8 \\ -3 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.4 \\ -1 \\ 0 \\ - 0.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ -2 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 17.4 \\ 18 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}2 \\ -5 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 0.8 \\ -3 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr}- 0.1 & - 1.2 & - 8.9 & -8 & - 8.1 \\ - 0.1 & - 0.7 & - 4.8 & - 5.8 & -6 \\ - 0.2 & - 0.4 & - 1.4 & - 7.2 & - 7.8 \\ - 0.1 & 0.3 & 3.4 & - 1.4 & - 1.8 \end{array}\right) \sim \left(\begin{array}{rrrrr}- 0.2 & - 0.4 & - 1.4 & - 7.2 & - 7.8 \\ 0 & -1 & - 8.2 & - 4.4 & - 4.2 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 9.4 \\ - 8.2 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 27.2 \\ - 4.4 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}- 30.6 \\ - 4.2 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 0.9 \\ 0.4 \\ - 0.2 \\ - 0.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ -1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 9.4 \\ - 8.2 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 27.2 \\ - 4.4 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 30.6 \\ - 4.2 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 0.8 \\ 0.3 \\ - 0.4 \\ - 0.7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 9.4 \\ - 8.2 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 27.2 \\ - 4.4 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 30.6 \\ - 4.2 \\ 0 \\ 0 \\ 1 \end{array}\right)

  v   v v
* * * * *
0 0 * * *
0 0 0 0 0
0 0 0 0 0

Example matrix

A = \left(\begin{array}{rrrrr}- 0.2 & 0.4 & 1.2 & 3.3 & - 3.9 \\ 2 & -4 & -2 & -3 & 4 \\ 0.8 & - 1.6 & 1.2 & 4.8 & - 5.4 \\ 0.4 & - 0.8 & - 1.4 & - 3.6 & 4.3 \end{array}\right) \sim \left(\begin{array}{rrrrr}2 & -4 & -2 & -3 & 4 \\ 0 & 0 & 2 & 6 & -7 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 1.5 \\ 0 \\ -3 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 1.5 \\ 0 \\ 3.5 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 0.8 \\ 2 \\ 2.8 \\ - 0.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 1.5 \\ 0 \\ -3 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 1.5 \\ 0 \\ 3.5 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.4 \\ 4 \\ - 0.4 \\ 1.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ -1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 1.5 \\ 0 \\ -3 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 1.5 \\ 0 \\ 3.5 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr}-5 & -6 & -9 & -7 & 5 \\ 1 & 1.2 & 1 & - 1.4 & - 3.4 \\ 2.5 & 3 & 6.5 & 10.5 & 3.5 \\ 3 & 3.6 & 4.8 & 2.1 & - 4.8 \end{array}\right) \sim \left(\begin{array}{rrrrr}-5 & -6 & -9 & -7 & 5 \\ 0 & 0 & 2 & 7 & 6 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 1.2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 4.9 \\ 0 \\ - 3.5 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 6.4 \\ 0 \\ -3 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-1 \\ 1 \\ - 1.5 \\ 1.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ -1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 1.2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 4.9 \\ 0 \\ - 3.5 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 6.4 \\ 0 \\ -3 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-2 \\ 2 \\ -3 \\ 2.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ -2 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 1.2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 4.9 \\ 0 \\ - 3.5 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 6.4 \\ 0 \\ -3 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr} 0.4 & 2.6 & - 3.3 & - 3.3 & - 7.7 \\ 0.2 & 1.3 & - 1.3 & 0.1 & 2.8 \\ 1 & 6.5 & -7 & -2 & 4.5 \\ 0.8 & 5.2 & - 5.2 & 0.4 & 11.2 \end{array}\right) \sim \left(\begin{array}{rrrrr}1 & 6.5 & -7 & -2 & 4.5 \\ 0 & 0 & - 0.5 & - 2.5 & - 9.5 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 6.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-33 \\ 0 \\ -5 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}- 137.5 \\ 0 \\ -19 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.7 \\ - 0.5 \\ -3 \\ -2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 6.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-33 \\ 0 \\ -5 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 137.5 \\ 0 \\ -19 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 2.1 \\ - 0.7 \\ -4 \\ - 2.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 6.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-33 \\ 0 \\ -5 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 137.5 \\ 0 \\ -19 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr} 0.5 & - 0.5 & 0.2 & 2.2 & 5.3 \\ 1 & -1 & 0 & - 2.6 & 2 \\ 0.5 & - 0.5 & - 0.4 & - 8.3 & - 7.6 \\ - 0.5 & 0.5 & - 0.2 & - 2.2 & - 5.3 \end{array}\right) \sim \left(\begin{array}{rrrrr}1 & -1 & 0 & - 2.6 & 2 \\ 0 & 0 & - 0.4 & -7 & - 8.6 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}1 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 2.6 \\ 0 \\ - 17.5 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}-2 \\ 0 \\ - 21.5 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 0.3 \\ 1 \\ 0.9 \\ - 0.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ -1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}1 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 2.6 \\ 0 \\ - 17.5 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}-2 \\ 0 \\ - 21.5 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 0.9 \\ 1 \\ - 0.3 \\ - 0.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 2 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}1 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 2.6 \\ 0 \\ - 17.5 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}-2 \\ 0 \\ - 21.5 \\ 0 \\ 1 \end{array}\right)

  v v   v
* * * * *
0 0 0 * *
0 0 0 0 0
0 0 0 0 0

Example matrix

A = \left(\begin{array}{rrrrr}-1 & 6 & 2 & -8 & 4 \\ 0.5 & -3 & -1 & - 0.5 & 3.4 \\ 0.3 & - 1.8 & - 0.6 & 7.4 & - 7.2 \\ 0.4 & - 2.4 & - 0.8 & 2.7 & -1 \end{array}\right) \sim \left(\begin{array}{rrrrr}-1 & 6 & 2 & -8 & 4 \\ 0 & 0 & 0 & 5 & -6 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}6 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 5.6 \\ 0 \\ 0 \\ 1.2 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}4 \\ 2.5 \\ - 6.2 \\ - 1.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 0 \\ -1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}6 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 5.6 \\ 0 \\ 0 \\ 1.2 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}5 \\ 2 \\ - 6.5 \\ - 1.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ -1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}6 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 5.6 \\ 0 \\ 0 \\ 1.2 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr} 0.2 & 0.6 & - 1.2 & 4 & 0 \\ 1 & 3 & -6 & 6 & 7 \\ 0.9 & 2.7 & - 5.4 & 1.4 & 8.3 \\ 0.7 & 2.1 & - 4.2 & 5 & 4.5 \end{array}\right) \sim \left(\begin{array}{rrrrr}1 & 3 & -6 & 6 & 7 \\ 0 & 0 & 0 & -4 & 2 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-3 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}6 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-10 \\ 0 \\ 0 \\ 0.5 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 3.2 \\ -2 \\ 2.2 \\ - 2.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 0 \\ -1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-3 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}6 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}-10 \\ 0 \\ 0 \\ 0.5 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 3.4 \\ -3 \\ 1.3 \\ - 2.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ -1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-3 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}6 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}-10 \\ 0 \\ 0 \\ 0.5 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr}- 0.4 & 1 & - 1.6 & - 1.6 & 0.8 \\ 0.8 & -2 & 3.2 & - 3.8 & 1.2 \\ -1 & 2.5 & -4 & 8.5 & -3 \\ 0.2 & - 0.5 & 0.8 & 0.3 & - 0.2 \end{array}\right) \sim \left(\begin{array}{rrrrr}-1 & 2.5 & -4 & 8.5 & -3 \\ 0 & 0 & 0 & -5 & 2 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 2.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-4 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 0.4 \\ 0 \\ 0 \\ 0.4 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 3.2 \\ - 0.6 \\ 4.5 \\ 1.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 2.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-4 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 0.4 \\ 0 \\ 0 \\ 0.4 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 2.8 \\ - 1.4 \\ 5.5 \\ 0.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 2.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-4 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 0.4 \\ 0 \\ 0 \\ 0.4 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr}- 0.1 & 4.2 & 0.3 & - 0.5 & 8.1 \\ 0.2 & - 8.4 & - 0.6 & -1 & -8 \\ 0.1 & - 4.2 & - 0.3 & 1.5 & - 12.2 \\ 0.1 & - 4.2 & - 0.3 & 0.5 & - 8.1 \end{array}\right) \sim \left(\begin{array}{rrrrr} 0.2 & - 8.4 & - 0.6 & -1 & -8 \\ 0 & 0 & 0 & 2 & - 8.2 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}42 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}3 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 60.5 \\ 0 \\ 0 \\ 4.1 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 0.4 \\ 1.2 \\ - 1.4 \\ - 0.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 0 \\ -1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}42 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}3 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 60.5 \\ 0 \\ 0 \\ 4.1 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 0.3 \\ 1.4 \\ - 1.3 \\ - 0.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ -1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}42 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}3 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 60.5 \\ 0 \\ 0 \\ 4.1 \\ 1 \end{array}\right)

  v v v  
* * * * *
0 0 0 0 *
0 0 0 0 0
0 0 0 0 0

Example matrix

A = \left(\begin{array}{rrrrr} 0.6 & - 0.3 & 1.5 & - 0.9 & - 1.3 \\ 0.2 & - 0.1 & 0.5 & - 0.3 & - 4.7 \\ 2 & -1 & 5 & -3 & -7 \\ 0.4 & - 0.2 & 1 & - 0.6 & - 3.4 \end{array}\right) \sim \left(\begin{array}{rrrrr}2 & -1 & 5 & -3 & -7 \\ 0 & 0 & 0 & 0 & -4 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 2.5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 1.5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 1.1 \\ - 3.9 \\ 1 \\ - 1.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r} 0.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 2.5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 1.5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 0.5 \\ - 4.1 \\ -1 \\ - 2.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r} 0.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 2.5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 1.5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr}2 & -4 & 9 & -5 & -1 \\ 0.6 & - 1.2 & 2.7 & - 1.5 & 5.7 \\ - 0.6 & 1.2 & - 2.7 & 1.5 & - 2.1 \\ - 0.4 & 0.8 & - 1.8 & 1 & 3.2 \end{array}\right) \sim \left(\begin{array}{rrrrr}2 & -4 & 9 & -5 & -1 \\ 0 & 0 & 0 & 0 & 6 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 4.5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 2.5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}3 \\ - 5.1 \\ 1.5 \\ - 3.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 4.5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 2.5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}5 \\ - 4.5 \\ 0.9 \\ -4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 4.5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 2.5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr} 0.5 & - 0.4 & - 1.9 & - 1.4 & - 8.9 \\ 2.5 & -2 & - 9.5 & -7 & - 4.5 \\ 1.5 & - 1.2 & - 5.7 & - 4.2 & 3.7 \\ 2 & - 1.6 & - 7.6 & - 5.6 & - 6.8 \end{array}\right) \sim \left(\begin{array}{rrrrr} 2.5 & -2 & - 9.5 & -7 & - 4.5 \\ 0 & 0 & 0 & 0 & -8 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.8 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 3.8 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 2.8 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 7.9 \\ 0.5 \\ 6.7 \\ - 2.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r} 0.8 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 3.8 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 2.8 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 7.4 \\ 3 \\ 8.2 \\ - 0.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r} 0.8 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 3.8 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 2.8 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr}-2 & -8 & 3.2 & - 6.2 & - 6.6 \\ 1 & 4 & - 1.6 & 3.1 & 1.7 \\ 1 & 4 & - 1.6 & 3.1 & 4.1 \\ -1 & -4 & 1.6 & - 3.1 & - 4.1 \end{array}\right) \sim \left(\begin{array}{rrrrr}-2 & -8 & 3.2 & - 6.2 & - 6.6 \\ 0 & 0 & 0 & 0 & - 1.6 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 1.6 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 3.1 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 0.6 \\ 1.3 \\ - 1.1 \\ 1.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}-4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 1.6 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 3.1 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.4 \\ 2.3 \\ - 0.1 \\ 0.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}-4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 1.6 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 3.1 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr} 0.6 & - 0.3 & 1.5 & - 0.9 & - 1.3 \\ 0.2 & - 0.1 & 0.5 & - 0.3 & - 4.7 \\ 2 & -1 & 5 & -3 & -7 \\ 0.4 & - 0.2 & 1 & - 0.6 & - 3.4 \end{array}\right) \sim \left(\begin{array}{rrrrr}2 & -1 & 5 & -3 & -7 \\ 0 & 0 & 0 & 0 & -4 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 2.5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 1.5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 1.1 \\ - 3.9 \\ 1 \\ - 1.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r} 0.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 2.5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 1.5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 0.5 \\ - 4.1 \\ -1 \\ - 2.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r} 0.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 2.5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 1.5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr}2 & -4 & 9 & -5 & -1 \\ 0.6 & - 1.2 & 2.7 & - 1.5 & 5.7 \\ - 0.6 & 1.2 & - 2.7 & 1.5 & - 2.1 \\ - 0.4 & 0.8 & - 1.8 & 1 & 3.2 \end{array}\right) \sim \left(\begin{array}{rrrrr}2 & -4 & 9 & -5 & -1 \\ 0 & 0 & 0 & 0 & 6 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 4.5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 2.5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}3 \\ - 5.1 \\ 1.5 \\ - 3.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 4.5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 2.5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}5 \\ - 4.5 \\ 0.9 \\ -4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 4.5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 2.5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr} 0.5 & - 0.4 & - 1.9 & - 1.4 & - 8.9 \\ 2.5 & -2 & - 9.5 & -7 & - 4.5 \\ 1.5 & - 1.2 & - 5.7 & - 4.2 & 3.7 \\ 2 & - 1.6 & - 7.6 & - 5.6 & - 6.8 \end{array}\right) \sim \left(\begin{array}{rrrrr} 2.5 & -2 & - 9.5 & -7 & - 4.5 \\ 0 & 0 & 0 & 0 & -8 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.8 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 3.8 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 2.8 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 7.9 \\ 0.5 \\ 6.7 \\ - 2.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r} 0.8 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 3.8 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 2.8 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 7.4 \\ 3 \\ 8.2 \\ - 0.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r} 0.8 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 3.8 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 2.8 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr}-2 & -8 & 3.2 & - 6.2 & - 6.6 \\ 1 & 4 & - 1.6 & 3.1 & 1.7 \\ 1 & 4 & - 1.6 & 3.1 & 4.1 \\ -1 & -4 & 1.6 & - 3.1 & - 4.1 \end{array}\right) \sim \left(\begin{array}{rrrrr}-2 & -8 & 3.2 & - 6.2 & - 6.6 \\ 0 & 0 & 0 & 0 & - 1.6 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 1.6 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 3.1 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 0.6 \\ 1.3 \\ - 1.1 \\ 1.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}-4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 1.6 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 3.1 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.4 \\ 2.3 \\ - 0.1 \\ 0.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}-4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 1.6 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 3.1 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

4 \times 5 with four free variables

  v v v v
* * * * *
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0

Example matrix

A = \left(\begin{array}{rrrrr}- 0.4 & 1.4 & 1 & - 1.8 & 0.2 \\ -1 & 3.5 & 2.5 & - 4.5 & 0.5 \\ -2 & 7 & 5 & -9 & 1 \\ 0.2 & - 0.7 & - 0.5 & 0.9 & - 0.1 \end{array}\right) \sim \left(\begin{array}{rrrrr}-2 & 7 & 5 & -9 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 3.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 2.5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 4.5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 0.5 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.4 \\ -1 \\ -2 \\ 0.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 3.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 2.5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 4.5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 0.5 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.8 \\ -2 \\ -4 \\ 0.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 3.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 2.5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 4.5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 0.5 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr}-5 & 2 & 4 & 7 & 3 \\ 4.5 & - 1.8 & - 3.6 & - 6.3 & - 2.7 \\ 2 & - 0.8 & - 1.6 & - 2.8 & - 1.2 \\ 2.5 & -1 & -2 & - 3.5 & - 1.5 \end{array}\right) \sim \left(\begin{array}{rrrrr}-5 & 2 & 4 & 7 & 3 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 0.8 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 1.4 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 0.6 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-5 \\ 4.5 \\ 2 \\ 2.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 0.4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 0.8 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 1.4 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 0.6 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-10 \\ 9 \\ 4 \\ 5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 0.4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 0.8 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 1.4 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 0.6 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr}2 & 1.4 & - 0.4 & - 0.6 & - 3.4 \\ -3 & - 2.1 & 0.6 & 0.9 & 5.1 \\ 5 & 3.5 & -1 & - 1.5 & - 8.5 \\ 4 & 2.8 & - 0.8 & - 1.2 & - 6.8 \end{array}\right) \sim \left(\begin{array}{rrrrr}5 & 3.5 & -1 & - 1.5 & - 8.5 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 0.7 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 0.2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 0.3 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 1.7 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}2 \\ -3 \\ 5 \\ 4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 0.7 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 0.2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 0.3 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 1.7 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}4 \\ -6 \\ 10 \\ 8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 0.7 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 0.2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 0.3 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 1.7 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrr}- 0.2 & 2.4 & 3.8 & - 7.6 & -5 \\ - 0.1 & 1.2 & 1.9 & - 3.8 & - 2.5 \\ 0.1 & - 1.2 & - 1.9 & 3.8 & 2.5 \\ - 0.1 & 1.2 & 1.9 & - 3.8 & - 2.5 \end{array}\right) \sim \left(\begin{array}{rrrrr}- 0.2 & 2.4 & 3.8 & - 7.6 & -5 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}12 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}19 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-38 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}-25 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.2 \\ - 0.1 \\ 0.1 \\ - 0.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}12 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}19 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-38 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}-25 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.4 \\ - 0.2 \\ 0.2 \\ - 0.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}12 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}19 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-38 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}-25 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

4 \times 6

  1. 4 \times 6 with 2 free variables
  2. 4 \times 6 with 3 free variables
  3. 4 \times 6 with 4 free variables
  4. 4 \times 6 with 5 free variables

4 \times 6 with two free variables

        v v
* * * * * *
0 * * * * *
0 0 * * * *
0 0 0 * * *

Example matrix

A = \left(\begin{array}{rrrrrr}-5 & 7 & 8 & -4 & -1 & 0 \\ -3 & 3.2 & 3.8 & - 6.9 & 8.4 & -6 \\ 3.5 & - 2.9 & - 1.6 & 5.8 & - 5.3 & -2 \\ - 3.5 & 6.5 & 8 & -3 & - 5.3 & -4 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-5 & 7 & 8 & -4 & -1 & 0 \\ 0 & 2 & 4 & 3 & -6 & -2 \\ 0 & 0 & 1 & -3 & 6 & -7 \\ 0 & 0 & 0 & -5 & 5 & -8 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 4.7 \\ 7.5 \\ -3 \\ 1 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 3.4 \\ -1 \\ 2.2 \\ - 1.6 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-1 \\ 2.9 \\ - 4.3 \\ - 0.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 3 \\ -2 \\ -1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 4.7 \\ 7.5 \\ -3 \\ 1 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 3.4 \\ -1 \\ 2.2 \\ - 1.6 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}1 \\ 3.1 \\ - 3.7 \\ 2.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 4 \\ -2 \\ -1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 4.7 \\ 7.5 \\ -3 \\ 1 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 3.4 \\ -1 \\ 2.2 \\ - 1.6 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr} 0.2 & -2 & - 8.3 & 3.4 & - 5.2 & 2.2 \\ 1.8 & - 1.4 & 1 & 4.4 & - 13.3 & - 7.1 \\ -2 & 0 & -7 & 6 & 2 & 8 \\ 1 & 0.8 & 7.9 & 6 & - 11.4 & - 9.8 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-2 & 0 & -7 & 6 & 2 & 8 \\ 0 & -2 & -9 & 4 & -5 & 3 \\ 0 & 0 & 1 & 7 & -8 & -2 \\ 0 & 0 & 0 & 5 & -6 & -3 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}6 \\ 1.7 \\ - 0.4 \\ 1.2 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 13.5 \\ 12.6 \\ - 2.2 \\ 0.6 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 4.3 \\ 3.2 \\ 7 \\ 4.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 4 \\ -1 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}6 \\ 1.7 \\ - 0.4 \\ 1.2 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 13.5 \\ 12.6 \\ - 2.2 \\ 0.6 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 4.1 \\ 1.4 \\ 9 \\ 3.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 4 \\ -1 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}6 \\ 1.7 \\ - 0.4 \\ 1.2 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 13.5 \\ 12.6 \\ - 2.2 \\ 0.6 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}-3 & 0.2 & - 4.6 & - 0.3 & 13 & 11 \\ 3 & -2 & 3 & 1 & 5.8 & - 2.9 \\ -5 & 2 & - 8.5 & -3 & 7.5 & 2.5 \\ -1 & 1.2 & - 1.6 & - 5.4 & -2 & 2.7 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-5 & 2 & - 8.5 & -3 & 7.5 & 2.5 \\ 0 & -1 & 0.5 & 1.5 & 8.5 & 9.5 \\ 0 & 0 & - 2.5 & -2 & 3.5 & -9 \\ 0 & 0 & 0 & -4 & 4 & 8 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}4 \\ 10.3 \\ 0.6 \\ 1 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 12.1 \\ 9.9 \\ - 5.2 \\ 2 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.1 \\ -2 \\ 1.5 \\ - 2.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 3 \\ -1 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}4 \\ 10.3 \\ 0.6 \\ 1 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 12.1 \\ 9.9 \\ - 5.2 \\ 2 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 2.5 \\ 1 \\ 0 \\ - 2.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 3 \\ -2 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}4 \\ 10.3 \\ 0.6 \\ 1 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 12.1 \\ 9.9 \\ - 5.2 \\ 2 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}- 0.5 & 3.5 & 10.3 & 0.6 & 3.4 & 6.3 \\ 0.5 & - 2.5 & - 6.9 & - 6.1 & 3.1 & 2.4 \\ 1 & -3 & - 8.6 & - 9.6 & 5.6 & 3.8 \\ 0.5 & - 2.5 & - 7.1 & - 4.6 & 2 & 0.1 \end{array}\right) \sim \left(\begin{array}{rrrrrr}1 & -3 & - 8.6 & - 9.6 & 5.6 & 3.8 \\ 0 & 2 & 6 & - 4.2 & 6.2 & 8.2 \\ 0 & 0 & 0.4 & - 3.4 & 3.4 & 4.6 \\ 0 & 0 & 0 & - 0.2 & 0.6 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}26 \\ - 47.8 \\ 17 \\ 3 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}- 11.5 \\ 30.4 \\ - 11.5 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 4.7 \\ - 2.2 \\ -3 \\ - 0.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 2 \\ -1 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}26 \\ - 47.8 \\ 17 \\ 3 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 11.5 \\ 30.4 \\ - 11.5 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 4.2 \\ - 2.7 \\ -4 \\ -1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 2 \\ -1 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}26 \\ - 47.8 \\ 17 \\ 3 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 11.5 \\ 30.4 \\ - 11.5 \\ 0 \\ 0 \\ 1 \end{array}\right)

      v   v
* * * * * *
0 * * * * *
0 0 * * * *
0 0 0 0 * *

Example matrix

A = \left(\begin{array}{rrrrrr}- 4.5 & 7.3 & - 1.4 & 1.5 & 8.1 & 1.2 \\ 5 & -7 & 6 & -5 & -9 & -8 \\ 0.5 & -1 & - 1.6 & 2.4 & - 2.9 & 8 \\ 4.5 & -7 & 2.8 & - 2.8 & - 11.7 & 3.6 \end{array}\right) \sim \left(\begin{array}{rrrrrr}5 & -7 & 6 & -5 & -9 & -8 \\ 0 & 1 & 4 & -3 & 0 & -6 \\ 0 & 0 & -1 & 2 & -2 & 7 \\ 0 & 0 & 0 & 0 & -4 & 8 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 8.4 \\ -5 \\ 2 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 6.8 \\ -6 \\ 3 \\ 0 \\ 2 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 3.4 \\ 4 \\ - 0.7 \\ 2.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 3 \\ 1 \\ 0 \\ -1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 8.4 \\ -5 \\ 2 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 6.8 \\ -6 \\ 3 \\ 0 \\ 2 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 6.2 \\ 2 \\ - 1.2 \\ 0 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 4 \\ 1 \\ 0 \\ -1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 8.4 \\ -5 \\ 2 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 6.8 \\ -6 \\ 3 \\ 0 \\ 2 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}-4 & 10.6 & - 3.8 & 4.2 & 10.2 & 4 \\ 5 & -7 & 1 & 6 & -9 & 0 \\ - 1.5 & 6.1 & - 0.7 & - 0.6 & 13.1 & 2.2 \\ -2 & 5.3 & - 1.7 & 1.5 & 10.9 & - 3.1 \end{array}\right) \sim \left(\begin{array}{rrrrrr}5 & -7 & 1 & 6 & -9 & 0 \\ 0 & 5 & -3 & 9 & 3 & 4 \\ 0 & 0 & 2 & -6 & 8 & -1 \\ 0 & 0 & 0 & 0 & 5 & -5 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 1.8 \\ 0 \\ 3 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 2.4 \\ - 3.5 \\ - 3.5 \\ 0 \\ 1 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 2.4 \\ -3 \\ 0.9 \\ - 3.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 3 \\ 4 \\ 0 \\ -1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 1.8 \\ 0 \\ 3 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 2.4 \\ - 3.5 \\ - 3.5 \\ 0 \\ 1 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.6 \\ 2 \\ - 0.6 \\ - 5.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 3 \\ 4 \\ 0 \\ -1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 1.8 \\ 0 \\ 3 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 2.4 \\ - 3.5 \\ - 3.5 \\ 0 \\ 1 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr} 1.6 & 0.2 & 5.6 & - 3.8 & 4 & 0.4 \\ - 2.4 & - 1.7 & 4.7 & 5.4 & 9.8 & 0.4 \\ -4 & 0.5 & - 5.5 & -1 & 5.5 & 9 \\ 3.2 & - 0.8 & 8 & 0.8 & 2.4 & -5 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-4 & 0.5 & - 5.5 & -1 & 5.5 & 9 \\ 0 & -2 & 8 & 6 & 6.5 & -5 \\ 0 & 0 & 5 & -3 & 7.5 & 3 \\ 0 & 0 & 0 & 0 & 2.5 & 2 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 0.4 \\ 5.4 \\ 0.6 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}0 \\ - 2.7 \\ 0.6 \\ 0 \\ - 0.8 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.6 \\ - 2.4 \\ - 1.5 \\ 2.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -4 \\ -2 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 0.4 \\ 5.4 \\ 0.6 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}0 \\ - 2.7 \\ 0.6 \\ 0 \\ - 0.8 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 3.2 \\ 0 \\ 2.5 \\ - 0.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ -4 \\ -2 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 0.4 \\ 5.4 \\ 0.6 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}0 \\ - 2.7 \\ 0.6 \\ 0 \\ - 0.8 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}1 & 0.8 & 7.2 & 7.7 & 1.3 & - 8.6 \\ -2 & - 0.8 & - 5.2 & - 2.6 & 7.4 & 0.8 \\ -1 & - 0.6 & - 3.9 & - 0.5 & 10.2 & - 1.5 \\ -1 & - 0.2 & 0.2 & 3.9 & 9.7 & - 6.7 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-2 & - 0.8 & - 5.2 & - 2.6 & 7.4 & 0.8 \\ 0 & 0.4 & 4.6 & 6.4 & 5 & - 8.2 \\ 0 & 0 & 1 & 4 & 9 & -6 \\ 0 & 0 & 0 & 0 & -1 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 2.9 \\ 30 \\ -4 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 4.2 \\ - 48.5 \\ 6 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 0.5 \\ 2.2 \\ 8.3 \\ 4.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 3 \\ -1 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 2.9 \\ 30 \\ -4 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 4.2 \\ - 48.5 \\ 6 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.3 \\ 3 \\ 8.9 \\ 5.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 2 \\ -1 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 2.9 \\ 30 \\ -4 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 4.2 \\ - 48.5 \\ 6 \\ 0 \\ 0 \\ 1 \end{array}\right)

      v v  
* * * * * *
0 * * * * *
0 0 * * * *
0 0 0 0 0 *

Example matrix

A = \left(\begin{array}{rrrrrr}-2 & -7 & 2 & 3 & 9 & -4 \\ 1 & 2.5 & - 2.2 & - 1.3 & 1.9 & 6.4 \\ - 1.4 & - 9.9 & 5.4 & 3.1 & 3.3 & - 10.8 \\ 0.6 & 3.1 & 0.4 & - 1.1 & - 8.4 & 2.4 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-2 & -7 & 2 & 3 & 9 & -4 \\ 0 & -5 & 4 & 1 & -3 & -8 \\ 0 & 0 & -2 & 0 & 7 & 6 \\ 0 & 0 & 0 & 0 & 0 & 5 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.8 \\ 0.2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 0.3 \\ 2.2 \\ 3.5 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-1 \\ 3.5 \\ 4.3 \\ 2.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -1 \\ 2 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_4 \left(\begin{array}{r} 0.8 \\ 0.2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 0.3 \\ 2.2 \\ 3.5 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}1 \\ 2.5 \\ 5.7 \\ 1.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ -1 \\ 2 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_4 \left(\begin{array}{r} 0.8 \\ 0.2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 0.3 \\ 2.2 \\ 3.5 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}-5 & -2 & 6 & -9 & -3 & 9 \\ -4 & - 2.8 & 4.4 & -19 & - 10.4 & 6.8 \\ 1.5 & 2.6 & 2.2 & 10.7 & 0.9 & - 8.7 \\ -3 & -1 & 4.6 & - 6.7 & - 4.2 & 5.6 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-5 & -2 & 6 & -9 & -3 & 9 \\ 0 & 2 & 4 & 8 & 0 & -6 \\ 0 & 0 & 2 & -7 & -8 & -4 \\ 0 & 0 & 0 & 0 & 0 & 2 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 6.8 \\ -11 \\ 3.5 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 7.4 \\ -8 \\ 4 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}3 \\ 2.4 \\ - 0.9 \\ 3.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -1 \\ 2 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_4 \left(\begin{array}{r} 6.8 \\ -11 \\ 3.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 7.4 \\ -8 \\ 4 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}1 \\ - 5.2 \\ - 0.3 \\ 2.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 4 \\ 1 \\ 0 \\ 0 \\ 2 \end{array}\right) + u_4 \left(\begin{array}{r} 6.8 \\ -11 \\ 3.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 7.4 \\ -8 \\ 4 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}- 0.4 & 5.6 & - 4.8 & 1.6 & 15.6 & 5.9 \\ 1.2 & - 9.8 & 10.4 & 1.8 & - 12.2 & - 2.8 \\ 2 & -8 & 9 & 5.5 & - 9.5 & -3 \\ - 1.2 & 6.8 & - 6.6 & - 1.5 & 15.1 & - 2.7 \end{array}\right) \sim \left(\begin{array}{rrrrrr}2 & -8 & 9 & 5.5 & - 9.5 & -3 \\ 0 & -5 & 5 & - 1.5 & - 6.5 & -1 \\ 0 & 0 & 1 & 1.5 & 8.5 & 4.5 \\ 0 & 0 & 0 & 0 & 0 & - 8.5 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 3.2 \\ - 1.8 \\ - 1.5 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 3.8 \\ - 9.8 \\ - 8.5 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.1 \\ - 1.6 \\ 4 \\ 4.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 2 \\ 1 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_4 \left(\begin{array}{r}- 3.2 \\ - 1.8 \\ - 1.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 3.8 \\ - 9.8 \\ - 8.5 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.7 \\ - 2.8 \\ 2 \\ 6.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 2 \\ 1 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_4 \left(\begin{array}{r}- 3.2 \\ - 1.8 \\ - 1.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 3.8 \\ - 9.8 \\ - 8.5 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}1 & - 3.6 & - 2.8 & - 2.4 & - 5.9 & - 1.4 \\ 1 & 2.4 & 3.4 & -13 & 7.4 & 13.9 \\ -2 & - 0.8 & - 3.2 & 7.2 & 6.6 & - 6.4 \\ 1 & 2.4 & 3.6 & - 8.6 & 2.7 & 5.9 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-2 & - 0.8 & - 3.2 & 7.2 & 6.6 & - 6.4 \\ 0 & -4 & - 4.4 & 1.2 & - 2.6 & - 4.6 \\ 0 & 0 & - 0.4 & - 8.8 & 9.4 & 8.4 \\ 0 & 0 & 0 & 0 & 0 & - 3.8 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}29 \\ 24.5 \\ -22 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 23.7 \\ - 26.5 \\ 23.5 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 1.2 \\ - 7.5 \\ - 3.6 \\ 1.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ -2 \\ 3 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_4 \left(\begin{array}{r}29 \\ 24.5 \\ -22 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 23.7 \\ - 26.5 \\ 23.5 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 2.2 \\ - 6.5 \\ - 5.6 \\ 2.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ -2 \\ 3 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_4 \left(\begin{array}{r}29 \\ 24.5 \\ -22 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 23.7 \\ - 26.5 \\ 23.5 \\ 0 \\ 1 \\ 0 \end{array}\right)

    v     v
* * * * * *
0 * * * * *
0 0 0 * * *
0 0 0 0 * *

Example matrix

A = \left(\begin{array}{rrrrrr}-1 & - 0.6 & - 1.4 & - 5.6 & - 5.6 & 3.7 \\ 4 & - 6.4 & 5.6 & 1.2 & - 5.2 & 7.4 \\ 5 & -3 & 7 & 9 & -9 & 8 \\ 0.5 & 1.7 & 0.7 & 5.1 & 0.9 & 1.3 \end{array}\right) \sim \left(\begin{array}{rrrrrr}5 & -3 & 7 & 9 & -9 & 8 \\ 0 & -4 & 0 & -6 & 2 & 1 \\ 0 & 0 & 0 & -2 & -8 & 5 \\ 0 & 0 & 0 & 0 & -2 & 4 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 1.4 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 17.6 \\ 9.5 \\ 0 \\ - 5.5 \\ 2 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 5.2 \\ - 3.2 \\ -4 \\ 1.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 2 \\ 0 \\ -1 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 1.4 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 17.6 \\ 9.5 \\ 0 \\ - 5.5 \\ 2 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}11 \\ - 0.4 \\ 2 \\ -4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 3 \\ 0 \\ -2 \\ -1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 1.4 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 17.6 \\ 9.5 \\ 0 \\ - 5.5 \\ 2 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}- 0.9 & - 1.8 & - 4.5 & - 15.8 & 0 & 9.8 \\ - 0.6 & 2 & - 7.8 & 2.4 & - 5.2 & 9.2 \\ -1 & 0 & -8 & -6 & 3 & 7 \\ - 0.2 & - 1.2 & 0.2 & - 2.8 & 12.4 & - 8.8 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-1 & 0 & -8 & -6 & 3 & 7 \\ 0 & 2 & -3 & 6 & -7 & 5 \\ 0 & 0 & 0 & -5 & -9 & 8 \\ 0 & 0 & 0 & 0 & 4 & -4 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-8 \\ 1.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 11.2 \\ 1.6 \\ 0 \\ - 0.2 \\ 1 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 5.9 \\ - 1.4 \\ 6 \\ 9.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 4 \\ 0 \\ -1 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-8 \\ 1.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 11.2 \\ 1.6 \\ 0 \\ - 0.2 \\ 1 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 6.8 \\ - 0.8 \\ 7 \\ 10 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 4 \\ 0 \\ -1 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-8 \\ 1.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 11.2 \\ 1.6 \\ 0 \\ - 0.2 \\ 1 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}-2 & -1 & 8 & -6 & - 6.5 & 8.5 \\ 0.8 & - 4.6 & - 1.2 & - 7.1 & - 1.9 & - 10.4 \\ 0.4 & 3.2 & - 2.8 & 7.9 & -5 & 2.5 \\ - 0.8 & - 4.4 & 4.8 & - 10.8 & - 1.5 & - 1.7 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-2 & -1 & 8 & -6 & - 6.5 & 8.5 \\ 0 & -5 & 2 & - 9.5 & - 4.5 & -7 \\ 0 & 0 & 0 & 1 & -9 & 0 \\ 0 & 0 & 0 & 0 & - 2.5 & 0.5 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 3.8 \\ 0.4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 0.7 \\ -5 \\ 0 \\ 1.8 \\ 0.2 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 2.5 \\ - 4.5 \\ - 7.2 \\ 0.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 4 \\ 0 \\ -2 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 3.8 \\ 0.4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 0.7 \\ -5 \\ 0 \\ 1.8 \\ 0.2 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 4.5 \\ - 3.7 \\ - 6.8 \\ 0.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 4 \\ 0 \\ -2 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 3.8 \\ 0.4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 0.7 \\ -5 \\ 0 \\ 1.8 \\ 0.2 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}2 & 2.6 & -2 & 0.7 & 8.9 & - 0.9 \\ 4 & 5.6 & - 6.4 & 7 & 8.2 & - 7.4 \\ 2 & 2.7 & - 2.6 & 0.1 & 1.7 & 3.1 \\ 2 & 2.7 & - 2.6 & 3.1 & 4.9 & -3 \end{array}\right) \sim \left(\begin{array}{rrrrrr}4 & 5.6 & - 6.4 & 7 & 8.2 & - 7.4 \\ 0 & - 0.2 & 1.2 & - 2.8 & 4.8 & 2.8 \\ 0 & 0 & 0 & -2 & - 4.8 & 5.4 \\ 0 & 0 & 0 & 0 & -4 & 2 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 6.8 \\ 6 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 8.8 \\ 5 \\ 0 \\ 1.5 \\ 0.5 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 5.4 \\ 3.6 \\ 1 \\ 0.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -2 \\ 0 \\ 1 \\ -1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 6.8 \\ 6 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 8.8 \\ 5 \\ 0 \\ 1.5 \\ 0.5 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}7 \\ -2 \\ 0.2 \\ 0.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ -2 \\ 0 \\ -1 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 6.8 \\ 6 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 8.8 \\ 5 \\ 0 \\ 1.5 \\ 0.5 \\ 1 \end{array}\right)

    v   v  
* * * * * *
0 * * * * *
0 0 0 * * *
0 0 0 0 0 *

Example matrix

A = \left(\begin{array}{rrrrrr}- 3.2 & - 2.4 & - 9.6 & 1.2 & 3.4 & - 4.6 \\ - 3.6 & - 6.5 & - 3.2 & - 0.9 & 12.2 & 14.1 \\ -4 & -5 & -8 & -1 & 8 & 9 \\ 1.2 & 0.1 & 5.2 & - 0.9 & 0.5 & - 3.3 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-4 & -5 & -8 & -1 & 8 & 9 \\ 0 & -2 & 4 & 0 & 5 & 6 \\ 0 & 0 & 0 & 2 & 1 & -7 \\ 0 & 0 & 0 & 0 & 0 & -9 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 4.5 \\ 2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-1 \\ 2.5 \\ 0 \\ - 0.5 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 2.6 \\ -2 \\ 0 \\ 7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ -2 \\ 0 \\ -3 \\ 0 \\ -1 \end{array}\right) + u_3 \left(\begin{array}{r}- 4.5 \\ 2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}-1 \\ 2.5 \\ 0 \\ - 0.5 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 7.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ -3 \\ 0 \\ -2 \\ 0 \\ -1 \end{array}\right) + u_3 \left(\begin{array}{r}- 4.5 \\ 2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}-1 \\ 2.5 \\ 0 \\ - 0.5 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr} 1.5 & - 0.2 & 2.7 & - 1.4 & - 3.1 & 4.2 \\ -5 & 2 & 3 & -6 & 1 & 8 \\ - 3.5 & 0.4 & - 6.9 & - 6.2 & 7.7 & 11.6 \\ - 4.5 & 1.4 & - 0.9 & -7 & 3.7 & 15.4 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-5 & 2 & 3 & -6 & 1 & 8 \\ 0 & -1 & -9 & -2 & 7 & 6 \\ 0 & 0 & 0 & -4 & 0 & 9 \\ 0 & 0 & 0 & 0 & 0 & 4 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-3 \\ -9 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}3 \\ 7 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 1.9 \\ 1 \\ 2.7 \\ - 1.7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ -2 \\ 0 \\ -3 \\ 0 \\ -1 \end{array}\right) + u_3 \left(\begin{array}{r}-3 \\ -9 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}3 \\ 7 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 2.8 \\ 2 \\ 0.4 \\ -2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 1 \\ 0 \\ -3 \\ 0 \\ -1 \end{array}\right) + u_3 \left(\begin{array}{r}-3 \\ -9 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}3 \\ 7 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr} 0.5 & 0.5 & 4.4 & - 2.2 & 1 & - 7.4 \\ 2.5 & 6.5 & -6 & 9.5 & -1 & 1.5 \\ - 1.5 & - 2.9 & - 3.4 & - 1.2 & - 0.9 & 3.1 \\ 1.5 & 3.7 & - 2.2 & 4.6 & - 0.3 & - 4.7 \end{array}\right) \sim \left(\begin{array}{rrrrrr} 2.5 & 6.5 & -6 & 9.5 & -1 & 1.5 \\ 0 & 1 & -7 & 4.5 & - 1.5 & 4 \\ 0 & 0 & 0 & - 0.5 & 0 & - 4.5 \\ 0 & 0 & 0 & 0 & 0 & -3 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 15.8 \\ 7 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 3.5 \\ 1.5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 2.5 \\ -1 \\ 1.6 \\ 3.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ -4 \\ 0 \\ 2 \\ 0 \\ -1 \end{array}\right) + u_3 \left(\begin{array}{r}- 15.8 \\ 7 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 3.5 \\ 1.5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}2 \\ 0.5 \\ 1.7 \\ 4.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ -3 \\ 0 \\ 2 \\ 0 \\ -1 \end{array}\right) + u_3 \left(\begin{array}{r}- 15.8 \\ 7 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 3.5 \\ 1.5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}-2 & 0.8 & 11.4 & - 2.8 & - 4.4 & - 7.2 \\ 4 & - 1.2 & - 7.2 & 6.4 & - 6.8 & 7.2 \\ 2 & - 0.7 & - 7.5 & 3.8 & - 7.5 & 10.4 \\ -2 & 0.5 & - 0.3 & -3 & 3.3 & 6.1 \end{array}\right) \sim \left(\begin{array}{rrrrrr}4 & - 1.2 & - 7.2 & 6.4 & - 6.8 & 7.2 \\ 0 & 0.2 & 7.8 & 0.4 & - 7.8 & - 3.6 \\ 0 & 0 & 0 & 0.8 & -8 & 5 \\ 0 & 0 & 0 & 0 & 0 & 5.4 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 9.9 \\ -39 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 8.6 \\ 19 \\ 0 \\ 10 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 4.4 \\ 0.4 \\ 4.9 \\ 8.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 3 \\ 0 \\ -3 \\ 0 \\ 1 \end{array}\right) + u_3 \left(\begin{array}{r}- 9.9 \\ -39 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 8.6 \\ 19 \\ 0 \\ 10 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 4.4 \\ 1.6 \\ 6 \\ 8.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 4 \\ 0 \\ -2 \\ 0 \\ 1 \end{array}\right) + u_3 \left(\begin{array}{r}- 9.9 \\ -39 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 8.6 \\ 19 \\ 0 \\ 10 \\ 1 \\ 0 \end{array}\right)

    v v    
* * * * * *
0 * * * * *
0 0 0 0 * *
0 0 0 0 0 *

Example matrix

A = \left(\begin{array}{rrrrrr}4 & 1.8 & 7 & - 13.4 & - 12.2 & 9.4 \\ - 4.5 & -1 & - 0.7 & 7.9 & 14.6 & - 9.5 \\ -5 & -1 & 0 & 8 & 9 & -8 \\ - 2.5 & - 1.1 & - 4.2 & 8.2 & 9.9 & - 4.6 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-5 & -1 & 0 & 8 & 9 & -8 \\ 0 & 1 & 7 & -7 & -5 & 3 \\ 0 & 0 & 0 & 0 & 6 & -2 \\ 0 & 0 & 0 & 0 & 0 & 2 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 1.4 \\ -7 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 0.2 \\ 7 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-4 \\ - 1.6 \\ 0 \\ - 2.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ -3 \\ 0 \\ 0 \\ -1 \\ -2 \end{array}\right) + u_3 \left(\begin{array}{r} 1.4 \\ -7 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 0.2 \\ 7 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 2.2 \\ - 0.6 \\ -1 \\ - 4.7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 1 \\ 0 \\ 0 \\ -1 \\ -3 \end{array}\right) + u_3 \left(\begin{array}{r} 1.4 \\ -7 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 0.2 \\ 7 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr} 0.9 & - 8.4 & - 1.2 & 6.3 & - 14.3 & - 7.3 \\ 0.2 & - 6.2 & -2 & - 1.2 & - 9.2 & 3.6 \\ 1 & -6 & 0 & 9 & -1 & -7 \\ - 0.7 & 5.2 & 0.4 & - 5.7 & 7.3 & 13.3 \end{array}\right) \sim \left(\begin{array}{rrrrrr}1 & -6 & 0 & 9 & -1 & -7 \\ 0 & -5 & -2 & -3 & -9 & 5 \\ 0 & 0 & 0 & 0 & -8 & -4 \\ 0 & 0 & 0 & 0 & 0 & 7 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 2.4 \\ - 0.4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 12.6 \\ - 0.6 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 7.1 \\ 4.6 \\ 13 \\ 9.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -3 \\ 0 \\ 0 \\ 2 \\ 1 \end{array}\right) + u_3 \left(\begin{array}{r}- 2.4 \\ - 0.4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 12.6 \\ - 0.6 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-8 \\ 4.4 \\ 12 \\ 10.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ -3 \\ 0 \\ 0 \\ 2 \\ 1 \end{array}\right) + u_3 \left(\begin{array}{r}- 2.4 \\ - 0.4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 12.6 \\ - 0.6 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr} 0.8 & 0.9 & 3.9 & 1.6 & - 7.5 & 1 \\ 1.6 & - 1.7 & 2.9 & 3.2 & - 8.9 & 3.3 \\ 2 & -4 & 1 & 4 & - 2.5 & -5 \\ - 0.4 & - 0.2 & - 1.6 & - 0.8 & 2.5 & - 0.6 \end{array}\right) \sim \left(\begin{array}{rrrrrr}2 & -4 & 1 & 4 & - 2.5 & -5 \\ 0 & 2.5 & 3.5 & 0 & - 6.5 & 3 \\ 0 & 0 & 0 & 0 & -3 & 5.5 \\ 0 & 0 & 0 & 0 & 0 & - 1.5 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 3.3 \\ - 1.4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-2 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-4 \\ 4.2 \\ - 2.5 \\ 0.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ -1 \\ 0 \\ 0 \\ 1 \\ 2 \end{array}\right) + u_3 \left(\begin{array}{r}- 3.3 \\ - 1.4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-2 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 3.2 \\ 5.8 \\ - 0.5 \\ - 0.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -1 \\ 0 \\ 0 \\ 1 \\ 2 \end{array}\right) + u_3 \left(\begin{array}{r}- 3.3 \\ - 1.4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-2 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr} 0.2 & 2.4 & - 1.9 & - 6.6 & -8 & - 4.8 \\ 0.4 & - 3.2 & - 9.4 & - 3.6 & - 2.8 & - 4.4 \\ - 0.2 & - 0.4 & 3.3 & 4.2 & 9.9 & - 1.9 \\ 0.2 & 0.4 & - 3.3 & - 4.2 & - 2.1 & - 1.4 \end{array}\right) \sim \left(\begin{array}{rrrrrr} 0.4 & - 3.2 & - 9.4 & - 3.6 & - 2.8 & - 4.4 \\ 0 & 4 & 2.8 & - 4.8 & - 6.6 & - 2.6 \\ 0 & 0 & 0 & 0 & 5.2 & - 5.4 \\ 0 & 0 & 0 & 0 & 0 & 4.8 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 17.9 \\ - 0.7 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 18.6 \\ 1.2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 2.4 \\ - 3.2 \\ 10.2 \\ 0.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 2 \\ 0 \\ 0 \\ 1 \\ -1 \end{array}\right) + u_3 \left(\begin{array}{r} 17.9 \\ - 0.7 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 18.6 \\ 1.2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 2.2 \\ - 3.6 \\ 10.4 \\ 0.7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 2 \\ 0 \\ 0 \\ 1 \\ -1 \end{array}\right) + u_3 \left(\begin{array}{r} 17.9 \\ - 0.7 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 18.6 \\ 1.2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)

  v       v
* * * * * *
0 0 * * * *
0 0 0 * * *
0 0 0 0 * *

Example matrix

A = \left(\begin{array}{rrrrrr} 1.8 & 2.7 & - 6.7 & 11.9 & - 9.2 & - 2.6 \\ -2 & -3 & 7 & -7 & 6 & 0 \\ 1.4 & 2.1 & - 2.9 & - 3.1 & - 5.2 & 3 \\ - 1.4 & - 2.1 & 4.7 & - 2.1 & 7.3 & 0.7 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-2 & -3 & 7 & -7 & 6 & 0 \\ 0 & 0 & 2 & -8 & -1 & 3 \\ 0 & 0 & 0 & 4 & -4 & -2 \\ 0 & 0 & 0 & 0 & 5 & 2 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 1.5 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 6.1 \\ 0 \\ - 1.3 \\ 0.1 \\ - 0.4 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 4.6 \\ 4 \\ - 3.8 \\ 1.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 3 \\ 1 \\ -1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 1.5 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 6.1 \\ 0 \\ - 1.3 \\ 0.1 \\ - 0.4 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 8.2 \\ 0 \\ -1 \\ - 0.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 3 \\ 1 \\ -1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 1.5 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 6.1 \\ 0 \\ - 1.3 \\ 0.1 \\ - 0.4 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}-5 & -9 & -2 & 3 & -8 & -7 \\ - 0.5 & - 0.9 & 3 & - 0.9 & 3.2 & 9.1 \\ 0.5 & 0.9 & - 3.8 & 3.7 & 5.8 & - 0.3 \\ 4.5 & 8.1 & 1 & - 3.3 & 4.6 & 5.8 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-5 & -9 & -2 & 3 & -8 & -7 \\ 0 & 0 & -4 & 4 & 5 & -1 \\ 0 & 0 & 0 & 2 & 8 & 9 \\ 0 & 0 & 0 & 0 & 2 & 6 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 1.8 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 6.5 \\ 0 \\ 3.5 \\ 7.5 \\ -3 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}1 \\ - 2.3 \\ 2.9 \\ - 2.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 2 \\ 4 \\ -1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 1.8 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 6.5 \\ 0 \\ 3.5 \\ 7.5 \\ -3 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-3 \\ - 3.5 \\ - 0.7 \\ 3.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 1 \\ 2 \\ -1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 1.8 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 6.5 \\ 0 \\ 3.5 \\ 7.5 \\ -3 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}- 0.3 & 5.7 & - 6.4 & - 11.7 & 8.8 & - 5.3 \\ - 0.4 & 7.6 & - 5.7 & - 12.1 & 4.2 & 0.4 \\ 0.5 & - 9.5 & 6.5 & 7 & -8 & 3 \\ - 0.1 & 1.9 & - 0.3 & - 2.4 & - 6.4 & 11.6 \end{array}\right) \sim \left(\begin{array}{rrrrrr} 0.5 & - 9.5 & 6.5 & 7 & -8 & 3 \\ 0 & 0 & - 2.5 & - 7.5 & 4 & - 3.5 \\ 0 & 0 & 0 & -5 & -3 & 3.5 \\ 0 & 0 & 0 & 0 & -4 & 8 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}19 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 9.9 \\ 0 \\ 3.3 \\ - 0.5 \\ 2 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 2.5 \\ - 0.8 \\ - 2.5 \\ 4.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ -3 \\ 1 \\ -1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}19 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 9.9 \\ 0 \\ 3.3 \\ - 0.5 \\ 2 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 2.2 \\ - 0.4 \\ -3 \\ 4.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ -3 \\ 1 \\ -1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}19 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 9.9 \\ 0 \\ 3.3 \\ - 0.5 \\ 2 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr} 0.4 & 8.4 & - 3.4 & - 4.2 & 3.8 & - 2.6 \\ 0.2 & 4.2 & - 0.7 & 6.9 & - 3.5 & - 1.5 \\ - 0.2 & - 4.2 & 2.2 & 2.6 & - 12.4 & 3.2 \\ - 0.2 & - 4.2 & 2.2 & 8.6 & - 1.7 & 0.2 \end{array}\right) \sim \left(\begin{array}{rrrrrr} 0.4 & 8.4 & - 3.4 & - 4.2 & 3.8 & - 2.6 \\ 0 & 0 & 1 & 9 & - 5.4 & - 0.2 \\ 0 & 0 & 0 & -4 & - 7.8 & 2 \\ 0 & 0 & 0 & 0 & -1 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-21 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 24.8 \\ 0 \\ - 4.3 \\ 0.5 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 4.4 \\ - 12.6 \\ - 6.8 \\ - 2.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 4 \\ -1 \\ 1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-21 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 24.8 \\ 0 \\ - 4.3 \\ 0.5 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 4.8 \\ - 12.8 \\ - 6.6 \\ - 1.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 4 \\ -1 \\ 1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-21 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 24.8 \\ 0 \\ - 4.3 \\ 0.5 \\ 0 \\ 1 \end{array}\right)

  v     v  
* * * * * *
0 0 * * * *
0 0 0 * * *
0 0 0 0 0 *

Example matrix

A = \left(\begin{array}{rrrrrr}5 & -4 & 9 & -9 & -2 & 3 \\ - 0.5 & 0.4 & - 1.1 & 3.7 & 6.8 & - 6.9 \\ -4 & 3.2 & - 8.2 & 1.2 & - 5.4 & - 10.4 \\ 3 & - 2.4 & 5 & - 10.6 & - 9.6 & 1.1 \end{array}\right) \sim \left(\begin{array}{rrrrrr}5 & -4 & 9 & -9 & -2 & 3 \\ 0 & 0 & -1 & -6 & -7 & -8 \\ 0 & 0 & 0 & 4 & 8 & -5 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.8 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 12.2 \\ 0 \\ 5 \\ -2 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}0 \\ - 8.8 \\ 1 \\ 5.7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ -3 \\ -1 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r} 0.8 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 12.2 \\ 0 \\ 5 \\ -2 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-4 \\ - 8.2 \\ 5.2 \\ 3.7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ -4 \\ -1 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r} 0.8 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 12.2 \\ 0 \\ 5 \\ -2 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr} 1.5 & - 0.3 & 4.6 & - 8.5 & 6.6 & - 6.6 \\ 4 & - 0.8 & 2 & - 5.7 & 1.5 & 7.5 \\ -5 & 1 & -2 & 5 & 8 & -8 \\ -3 & 0.6 & - 2.8 & 5.6 & 2.6 & - 4.8 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-5 & 1 & -2 & 5 & 8 & -8 \\ 0 & 0 & 4 & -7 & 9 & -9 \\ 0 & 0 & 0 & -1 & 7 & 2 \\ 0 & 0 & 0 & 0 & 0 & -4 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.2 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 4.6 \\ 0 \\ 10 \\ 7 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 0.2 \\ - 0.9 \\ -4 \\ 1.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 1 \\ 2 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r} 0.2 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 4.6 \\ 0 \\ 10 \\ 7 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.3 \\ - 4.9 \\ 1 \\ 4.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 1 \\ 2 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r} 0.2 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 4.6 \\ 0 \\ 10 \\ 7 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}- 2.5 & 4 & - 7.5 & 9 & 3.5 & -2 \\ 0.5 & - 0.8 & 2 & - 5.9 & 2.5 & - 0.2 \\ 1 & - 1.6 & 5.5 & - 4.1 & 4.6 & - 7.2 \\ -1 & 1.6 & -1 & 5.6 & 5 & - 12.3 \end{array}\right) \sim \left(\begin{array}{rrrrrr}- 2.5 & 4 & - 7.5 & 9 & 3.5 & -2 \\ 0 & 0 & 2.5 & - 0.5 & 6 & -8 \\ 0 & 0 & 0 & -4 & 2 & 1 \\ 0 & 0 & 0 & 0 & 0 & - 4.5 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 1.6 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 10.1 \\ 0 \\ - 2.3 \\ 0.5 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 1.5 \\ 7 \\ 0.9 \\ 2.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ -3 \\ -2 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r} 1.6 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 10.1 \\ 0 \\ - 2.3 \\ 0.5 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}4 \\ 6 \\ - 2.6 \\ 1.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ -4 \\ -2 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r} 1.6 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 10.1 \\ 0 \\ - 2.3 \\ 0.5 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr} 0.2 & - 1.1 & 3.3 & - 1.6 & 0.9 & - 2.5 \\ - 0.4 & 2.2 & - 6.4 & - 3.6 & 4.8 & 6.6 \\ 0.2 & - 1.1 & 3 & 4.6 & - 5.4 & - 8.5 \\ - 0.2 & 1.1 & - 3.1 & - 4.2 & 4.8 & 9 \end{array}\right) \sim \left(\begin{array}{rrrrrr}- 0.4 & 2.2 & - 6.4 & - 3.6 & 4.8 & 6.6 \\ 0 & 0 & - 0.2 & 2.8 & -3 & - 5.2 \\ 0 & 0 & 0 & -2 & 1.8 & - 1.8 \\ 0 & 0 & 0 & 0 & 0 & 4 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 5.5 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 42.3 \\ 0 \\ - 2.4 \\ 0.9 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 6.7 \\ 3.6 \\ 0.4 \\ - 2.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ -1 \\ -3 \\ 0 \\ -2 \end{array}\right) + u_2 \left(\begin{array}{r} 5.5 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 42.3 \\ 0 \\ - 2.4 \\ 0.9 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 6.9 \\ 3.2 \\ 0.6 \\ - 2.7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ -1 \\ -3 \\ 0 \\ -2 \end{array}\right) + u_2 \left(\begin{array}{r} 5.5 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 42.3 \\ 0 \\ - 2.4 \\ 0.9 \\ 1 \\ 0 \end{array}\right)

  v   v    
* * * * * *
0 0 * * * *
0 0 0 0 * *
0 0 0 0 0 *

Example matrix

A = \left(\begin{array}{rrrrrr}- 0.8 & - 3.6 & 0.8 & - 3.2 & - 5.4 & - 5.6 \\ - 0.2 & - 0.9 & 2.3 & 7.6 & - 4.6 & - 1.7 \\ -2 & -9 & 3 & -4 & 4 & -7 \\ 0.2 & 0.9 & - 1.3 & - 3.6 & 7.7 & - 2.7 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-2 & -9 & 3 & -4 & 4 & -7 \\ 0 & 0 & 2 & 8 & -5 & -1 \\ 0 & 0 & 0 & 0 & -8 & -3 \\ 0 & 0 & 0 & 0 & 0 & -6 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 4.5 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-8 \\ 0 \\ -4 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.4 \\ 0.9 \\ 9 \\ 8.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 2 \\ 0 \\ 1 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}- 4.5 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-8 \\ 0 \\ -4 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 11.8 \\ 0.5 \\ -5 \\ 2.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ -3 \\ 0 \\ -1 \\ -2 \end{array}\right) + u_2 \left(\begin{array}{r}- 4.5 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-8 \\ 0 \\ -4 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}- 1.6 & - 2.4 & 0.4 & 6.4 & - 0.2 & 10.7 \\ 0.2 & 0.3 & 3.6 & - 0.8 & - 1.3 & - 4.9 \\ 2 & 3 & -4 & -8 & 7 & 1 \\ - 1.2 & - 1.8 & 5.6 & 4.8 & - 2.2 & - 6.4 \end{array}\right) \sim \left(\begin{array}{rrrrrr}2 & 3 & -4 & -8 & 7 & 1 \\ 0 & 0 & 4 & 0 & -2 & -5 \\ 0 & 0 & 0 & 0 & 4 & 8 \\ 0 & 0 & 0 & 0 & 0 & -9 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 1.5 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}4 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 5.3 \\ 4.4 \\ -6 \\ 2.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 2 \\ 0 \\ -1 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}- 1.5 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}4 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 6.9 \\ 4.2 \\ -8 \\ 3.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 2 \\ 0 \\ -1 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}- 1.5 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}4 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr} 2.5 & - 1.5 & -8 & 7 & 3.5 & - 8.5 \\ - 1.5 & 0.9 & 5.4 & - 7.8 & - 11.8 & 7.8 \\ -1 & 0.6 & 4.2 & - 8.8 & - 5.9 & 7.9 \\ 1 & - 0.6 & -3 & 1.6 & 1.9 & 3 \end{array}\right) \sim \left(\begin{array}{rrrrrr} 2.5 & - 1.5 & -8 & 7 & 3.5 & - 8.5 \\ 0 & 0 & 1 & -6 & - 4.5 & 4.5 \\ 0 & 0 & 0 & 0 & -7 & 0 \\ 0 & 0 & 0 & 0 & 0 & 5.5 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.6 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 16.4 \\ 0 \\ 6 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-2 \\ - 9.4 \\ - 5.2 \\ - 6.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 3 \\ 0 \\ 1 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r} 0.6 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 16.4 \\ 0 \\ 6 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-7 \\ - 6.4 \\ - 3.2 \\ - 8.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 3 \\ 0 \\ 1 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r} 0.6 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 16.4 \\ 0 \\ 6 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr} 0.5 & 3.1 & - 5.5 & 6.4 & 0 & 6.9 \\ -1 & - 6.2 & 7 & - 6.8 & 4.8 & - 8.6 \\ 0.5 & 3.1 & - 7.5 & 9.4 & 6.8 & 6.7 \\ 0.5 & 3.1 & - 1.5 & 0.4 & - 5.9 & - 1.8 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-1 & - 6.2 & 7 & - 6.8 & 4.8 & - 8.6 \\ 0 & 0 & -4 & 6 & 9.2 & 2.4 \\ 0 & 0 & 0 & 0 & - 2.2 & 1.4 \\ 0 & 0 & 0 & 0 & 0 & - 4.2 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 6.2 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 3.7 \\ 0 \\ 1.5 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.7 \\ 4.6 \\ - 0.3 \\ -12 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 3 \\ 0 \\ 1 \\ 2 \end{array}\right) + u_2 \left(\begin{array}{r}- 6.2 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 3.7 \\ 0 \\ 1.5 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 6.3 \\ 11 \\ 2.9 \\ 1.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ -1 \\ 0 \\ 1 \\ -2 \end{array}\right) + u_2 \left(\begin{array}{r}- 6.2 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 3.7 \\ 0 \\ 1.5 \\ 1 \\ 0 \\ 0 \end{array}\right)

  v v      
* * * * * *
0 0 0 * * *
0 0 0 0 * *
0 0 0 0 0 *

Example matrix

A = \left(\begin{array}{rrrrrr} 3.5 & 0.7 & - 4.2 & - 13.9 & - 2.1 & - 12.3 \\ -1 & - 0.2 & 1.2 & 3.2 & 2.6 & -5 \\ -5 & -1 & 6 & 7 & 3 & 9 \\ 3 & 0.6 & - 3.6 & - 6.9 & - 0.4 & - 15.8 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-5 & -1 & 6 & 7 & 3 & 9 \\ 0 & 0 & 0 & -9 & 0 & -6 \\ 0 & 0 & 0 & 0 & 2 & -8 \\ 0 & 0 & 0 & 0 & 0 & -3 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 0.2 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 1.2 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 3.7 \\ - 0.8 \\ -1 \\ - 1.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ -1 \\ 4 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}- 0.2 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 1.2 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 2.3 \\ - 2.4 \\ 1 \\ - 4.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ -1 \\ 3 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}- 0.2 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 1.2 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}-1 & -4 & - 0.5 & - 1.3 & - 0.1 & 2 \\ - 1.2 & - 4.8 & - 0.6 & -7 & - 6.6 & 5.6 \\ -2 & -8 & -1 & -5 & -6 & 1 \\ 1.4 & 5.6 & 0.7 & 6.7 & 8.4 & -4 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-2 & -8 & -1 & -5 & -6 & 1 \\ 0 & 0 & 0 & -4 & -3 & 5 \\ 0 & 0 & 0 & 0 & 2 & 3 \\ 0 & 0 & 0 & 0 & 0 & -2 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-4 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 0.5 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 0.7 \\ 1 \\ 0 \\ 2.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 0 \\ -3 \\ 2 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}-4 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 0.5 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.5 \\ - 0.6 \\ -1 \\ 1.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ -2 \\ 1 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}-4 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 0.5 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr} 0.4 & 1.4 & - 1.8 & 2.1 & - 11.2 & 10.9 \\ - 0.2 & - 0.7 & 0.9 & 1.7 & 6.3 & 4.3 \\ -1 & - 3.5 & 4.5 & 1 & 8 & - 8.5 \\ - 0.8 & - 2.8 & 3.6 & 0.3 & 0.4 & - 11.5 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-1 & - 3.5 & 4.5 & 1 & 8 & - 8.5 \\ 0 & 0 & 0 & 2.5 & -8 & 7.5 \\ 0 & 0 & 0 & 0 & 9.5 & 1.5 \\ 0 & 0 & 0 & 0 & 0 & -2 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 3.5 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 4.5 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 12.4 \\ 14.6 \\ 3.5 \\ - 13.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ -1 \\ 2 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}- 3.5 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 4.5 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-12 \\ 14.4 \\ 2.5 \\ - 14.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 0 \\ -1 \\ 2 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}- 3.5 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 4.5 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}- 0.1 & 1.2 & -2 & - 6.1 & 7.2 & -1 \\ 0.2 & - 2.4 & 4 & - 5.4 & -6 & - 9.2 \\ 0.1 & - 1.2 & 2 & - 7.1 & - 8.5 & - 15.8 \\ - 0.1 & 1.2 & -2 & - 1.7 & 8.9 & 15.2 \end{array}\right) \sim \left(\begin{array}{rrrrrr} 0.2 & - 2.4 & 4 & - 5.4 & -6 & - 9.2 \\ 0 & 0 & 0 & - 8.8 & 4.2 & - 5.6 \\ 0 & 0 & 0 & 0 & - 7.6 & - 8.4 \\ 0 & 0 & 0 & 0 & 0 & 9.2 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}12 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-20 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 8.9 \\ - 7.4 \\ - 7.9 \\ 0.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 0 \\ 1 \\ 2 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}12 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-20 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}9 \\ - 7.6 \\ -8 \\ 0.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 1 \\ 2 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}12 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-20 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right)

4 \times 6 with three free variables

      v v v
* * * * * *
0 * * * * *
0 0 * * * *
0 0 0 0 0 0

Example matrix

A = \left(\begin{array}{rrrrrr} 0.2 & - 1.5 & 1.1 & 4.3 & 2.5 & 2.7 \\ 0.6 & - 3.4 & 8.4 & 8.2 & - 3.1 & 1 \\ 2 & -8 & -2 & 4 & 3 & 0 \\ 1.2 & - 5.7 & 10.9 & 9.5 & - 5.8 & - 0.7 \end{array}\right) \sim \left(\begin{array}{rrrrrr}2 & -8 & -2 & 4 & 3 & 0 \\ 0 & -1 & 9 & 7 & -4 & 1 \\ 0 & 0 & -5 & -1 & 5 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 18.6 \\ 5.2 \\ - 0.2 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 19.5 \\ 5 \\ 1 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 18.8 \\ 4.6 \\ 0.4 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.1 \\ 4 \\ -10 \\ 4.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 18.6 \\ 5.2 \\ - 0.2 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 19.5 \\ 5 \\ 1 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 18.8 \\ 4.6 \\ 0.4 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.3 \\ 3.4 \\ -12 \\ 3.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 18.6 \\ 5.2 \\ - 0.2 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 19.5 \\ 5 \\ 1 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 18.8 \\ 4.6 \\ 0.4 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}- 1.5 & 2.3 & - 9.2 & - 5.3 & 7.9 & - 5.8 \\ -5 & 1 & -4 & -1 & 3 & 4 \\ -1 & 0 & 2 & 9.3 & - 3.1 & 7.5 \\ - 0.5 & 1.5 & - 6.4 & - 5.4 & 5.8 & - 5.7 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-5 & 1 & -4 & -1 & 3 & 4 \\ 0 & 2 & -8 & -5 & 7 & -7 \\ 0 & 0 & 2 & 9 & -3 & 6 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.3 \\ - 15.5 \\ - 4.5 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 0.1 \\ 2.5 \\ 1.5 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 1.5 \\ - 8.5 \\ -3 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 0.8 \\ -4 \\ -3 \\ 1.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ -3 \\ -1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 0.3 \\ - 15.5 \\ - 4.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 0.1 \\ 2.5 \\ 1.5 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 1.5 \\ - 8.5 \\ -3 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 3.1 \\ -3 \\ -3 \\ 2.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ -2 \\ -1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 0.3 \\ - 15.5 \\ - 4.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 0.1 \\ 2.5 \\ 1.5 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 1.5 \\ - 8.5 \\ -3 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}-1 & -1 & 2 & -12 & 8 & - 2.7 \\ 2.5 & 1.5 & 8.5 & - 6.5 & - 8.5 & -1 \\ - 0.5 & - 0.8 & - 1.2 & - 5.7 & 3.7 & 8.2 \\ 0.5 & 0.1 & - 2.1 & 3.1 & - 3.3 & 10.6 \end{array}\right) \sim \left(\begin{array}{rrrrrr} 2.5 & 1.5 & 8.5 & - 6.5 & - 8.5 & -1 \\ 0 & - 0.5 & 0.5 & -7 & 2 & 8 \\ 0 & 0 & 5 & -9 & 3 & - 9.5 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 3.8 \\ - 12.2 \\ 1.8 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 3.4 \\ 3.4 \\ - 0.6 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}- 16.8 \\ 17.9 \\ 1.9 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-4 \\ - 1.5 \\ 0.8 \\ 3.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -2 \\ -1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 3.8 \\ - 12.2 \\ 1.8 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 3.4 \\ 3.4 \\ - 0.6 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 16.8 \\ 17.9 \\ 1.9 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-3 \\ -3 \\ 1.6 \\ 3.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -3 \\ -1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 3.8 \\ - 12.2 \\ 1.8 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 3.4 \\ 3.4 \\ - 0.6 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 16.8 \\ 17.9 \\ 1.9 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}-1 & 1.4 & -1 & - 9.4 & -10 & - 2.6 \\ -1 & 2.6 & -2 & 3.1 & 7 & - 12.7 \\ -2 & 3.6 & - 3.2 & -7 & - 1.2 & -9 \\ 1 & - 2.2 & 2 & - 1.1 & -6 & 7.5 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-2 & 3.6 & - 3.2 & -7 & - 1.2 & -9 \\ 0 & 0.8 & - 0.4 & 6.6 & 7.6 & - 8.2 \\ 0 & 0 & 0.4 & - 2.6 & - 5.6 & - 2.2 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 22.9 \\ -5 \\ 6.5 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 27.5 \\ - 2.5 \\ 14 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 10.1 \\ 13 \\ 5.5 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.8 \\ - 0.2 \\ 0.4 \\ - 0.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ -2 \\ -3 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 22.9 \\ -5 \\ 6.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 27.5 \\ - 2.5 \\ 14 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 10.1 \\ 13 \\ 5.5 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.4 \\ 0.4 \\ 0.8 \\ - 0.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ -1 \\ -2 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 22.9 \\ -5 \\ 6.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 27.5 \\ - 2.5 \\ 14 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 10.1 \\ 13 \\ 5.5 \\ 0 \\ 0 \\ 1 \end{array}\right)

    v   v v
* * * * * *
0 * * * * *
0 0 0 * * *
0 0 0 0 0 0

Example matrix

A = \left(\begin{array}{rrrrrr} 1.6 & - 1.8 & - 0.6 & 11.8 & 5.6 & 8.4 \\ -2 & 1 & -8 & -6 & 3 & -3 \\ - 0.2 & - 0.8 & - 7.1 & 0.7 & 11.5 & 5.1 \\ 0.6 & 0.2 & 5.9 & - 3.7 & - 3.3 & - 2.1 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-2 & 1 & -8 & -6 & 3 & -3 \\ 0 & -1 & -7 & 7 & 8 & 6 \\ 0 & 0 & 0 & -5 & 4 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 7.5 \\ -7 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 5.9 \\ 13.6 \\ 0 \\ 0.8 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 1.5 \\ 6 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-3 \\ 0 \\ 2.3 \\ 3.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ -4 \\ 0 \\ -1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 7.5 \\ -7 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 5.9 \\ 13.6 \\ 0 \\ 0.8 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 1.5 \\ 6 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.4 \\ -2 \\ 2.1 \\ 4.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ -4 \\ 0 \\ -1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 7.5 \\ -7 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 5.9 \\ 13.6 \\ 0 \\ 0.8 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 1.5 \\ 6 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}- 2.5 & - 1.7 & - 0.9 & 1.3 & - 7.7 & 8.3 \\ -5 & -7 & 0 & -2 & -9 & 6 \\ 2 & 4.8 & -1 & 7.8 & - 4.4 & - 5.4 \\ 1.5 & 3.7 & - 0.8 & 9.8 & - 7.3 & - 11.4 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-5 & -7 & 0 & -2 & -9 & 6 \\ 0 & 2 & -1 & 7 & -8 & -3 \\ 0 & 0 & 0 & -4 & 4 & 8 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 0.7 \\ 0.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 2.9 \\ 0.5 \\ 0 \\ 1 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 8.1 \\ - 5.5 \\ 0 \\ 2 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.1 \\ 4 \\ - 0.6 \\ 3.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ -3 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 0.7 \\ 0.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 2.9 \\ 0.5 \\ 0 \\ 1 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 8.1 \\ - 5.5 \\ 0 \\ 2 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 3.6 \\ -1 \\ 1.4 \\ 4.7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -3 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 0.7 \\ 0.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 2.9 \\ 0.5 \\ 0 \\ 1 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 8.1 \\ - 5.5 \\ 0 \\ 2 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}- 0.6 & - 1.3 & - 6.7 & 2.1 & - 2.4 & 1.8 \\ - 0.8 & 1.1 & -1 & - 1.3 & 8.4 & - 1.7 \\ -1 & 2 & 0.5 & -4 & 3.5 & 8 \\ 0.4 & 1.2 & 5.4 & - 2.2 & 0.9 & 0.7 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-1 & 2 & 0.5 & -4 & 3.5 & 8 \\ 0 & - 2.5 & -7 & 4.5 & - 4.5 & -3 \\ 0 & 0 & 0 & 1 & 6.5 & - 7.5 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 5.1 \\ - 2.8 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 2.5 \\ - 13.5 \\ 0 \\ - 6.5 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 2.6 \\ 12.3 \\ 0 \\ 7.5 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.1 \\ - 1.7 \\ -1 \\ 0.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ -2 \\ 0 \\ -1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 5.1 \\ - 2.8 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 2.5 \\ - 13.5 \\ 0 \\ - 6.5 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 2.6 \\ 12.3 \\ 0 \\ 7.5 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.6 \\ 1 \\ -1 \\ 0.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 4 \\ 0 \\ 2 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 5.1 \\ - 2.8 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 2.5 \\ - 13.5 \\ 0 \\ - 6.5 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 2.6 \\ 12.3 \\ 0 \\ 7.5 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}- 0.5 & - 0.9 & 0.2 & 2.8 & - 1.9 & 14 \\ 1 & 1.4 & - 5.8 & 0.8 & - 8.2 & - 9.8 \\ 0.5 & 1.1 & 2.5 & 2 & - 8.1 & - 7.1 \\ - 0.5 & - 0.9 & 0.2 & 0.8 & 2.1 & 10 \end{array}\right) \sim \left(\begin{array}{rrrrrr}1 & 1.4 & - 5.8 & 0.8 & - 8.2 & - 9.8 \\ 0 & 0.4 & 5.4 & 1.6 & -4 & - 2.2 \\ 0 & 0 & 0 & 4 & -8 & 8 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 24.7 \\ - 13.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 3.8 \\ 2 \\ 0 \\ 2 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}- 7.5 \\ 13.5 \\ 0 \\ -2 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 2.7 \\ 1.4 \\ 1.9 \\ 0.7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ -1 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 24.7 \\ - 13.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 3.8 \\ 2 \\ 0 \\ 2 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 7.5 \\ 13.5 \\ 0 \\ -2 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 3.1 \\ 1 \\ 1.3 \\ 1.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ -2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 24.7 \\ - 13.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 3.8 \\ 2 \\ 0 \\ 2 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 7.5 \\ 13.5 \\ 0 \\ -2 \\ 0 \\ 1 \end{array}\right)

    v v   v
* * * * * *
0 * * * * *
0 0 0 0 * *
0 0 0 0 0 0

Example matrix

A = \left(\begin{array}{rrrrrr}-3 & 2.4 & - 2.4 & 4.2 & 0.8 & 9.7 \\ 5 & -2 & 0 & -3 & 6 & -6 \\ 3.5 & - 5.4 & 8 & - 10.1 & 6.2 & - 1.2 \\ 0.5 & 1.4 & - 3.2 & 2.9 & 4.3 & 4.5 \end{array}\right) \sim \left(\begin{array}{rrrrrr}5 & -2 & 0 & -3 & 6 & -6 \\ 0 & -4 & 8 & -8 & 2 & 3 \\ 0 & 0 & 0 & 0 & 5 & 7 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.8 \\ 2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 0.2 \\ -2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 2.9 \\ 0 \\ 0 \\ 0 \\ - 1.4 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 4.4 \\ 2 \\ - 4.6 \\ - 1.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 1 \\ 0 \\ 0 \\ -1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 0.8 \\ 2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 0.2 \\ -2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 2.9 \\ 0 \\ 0 \\ 0 \\ - 1.4 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-5 \\ 5 \\ - 6.5 \\ 0 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 2 \\ 0 \\ 0 \\ -1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 0.8 \\ 2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 0.2 \\ -2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 2.9 \\ 0 \\ 0 \\ 0 \\ - 1.4 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}- 3.5 & 6 & 7.3 & 3 & 1.7 & 14.6 \\ 5 & -6 & -4 & 6 & 0 & -9 \\ - 0.5 & 2.6 & 5.4 & 7.4 & 3 & 7.9 \\ -1 & 1.4 & 1.3 & - 0.4 & 0.6 & 1.9 \end{array}\right) \sim \left(\begin{array}{rrrrrr}5 & -6 & -4 & 6 & 0 & -9 \\ 0 & 2 & 5 & 8 & 3 & 7 \\ 0 & 0 & 0 & 0 & -1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 2.2 \\ - 2.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-6 \\ -4 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-6 \\ - 6.5 \\ 0 \\ 0 \\ 2 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 0.6 \\ 2 \\ - 0.2 \\ -1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 3 \\ 0 \\ 0 \\ -2 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 2.2 \\ - 2.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-6 \\ -4 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}-6 \\ - 6.5 \\ 0 \\ 0 \\ 2 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.2 \\ 3 \\ 0.7 \\ - 0.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 2 \\ 0 \\ 0 \\ -1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 2.2 \\ - 2.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-6 \\ -4 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}-6 \\ - 6.5 \\ 0 \\ 0 \\ 2 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}- 0.6 & - 0.1 & - 4.3 & 3.1 & 4.9 & - 5.7 \\ 0.2 & 5.7 & 3.7 & - 6.7 & - 0.8 & - 6.2 \\ 1 & 3.5 & 8.5 & - 8.5 & -4 & 4 \\ - 0.4 & 2.6 & - 1.8 & - 0.6 & - 0.4 & - 6.8 \end{array}\right) \sim \left(\begin{array}{rrrrrr}1 & 3.5 & 8.5 & - 8.5 & -4 & 4 \\ 0 & 5 & 2 & -5 & 0 & -7 \\ 0 & 0 & 0 & 0 & 2.5 & - 0.5 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 7.1 \\ - 0.4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}5 \\ 1 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 8.1 \\ 1.4 \\ 0 \\ 0 \\ 0.2 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 2.6 \\ - 5.7 \\ - 3.5 \\ - 4.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -1 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 7.1 \\ - 0.4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}5 \\ 1 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 8.1 \\ 1.4 \\ 0 \\ 0 \\ 0.2 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 7.3 \\ 4.9 \\ - 0.5 \\ 0.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 1 \\ 0 \\ 0 \\ 2 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 7.1 \\ - 0.4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}5 \\ 1 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 8.1 \\ 1.4 \\ 0 \\ 0 \\ 0.2 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}1 & 1 & - 3.7 & 7.5 & 3.8 & 6.5 \\ -1 & 5 & - 1.7 & 6.3 & 9 & 0.7 \\ -2 & 2 & 3.8 & - 5.8 & 0.4 & - 7.4 \\ 1 & 1 & - 3.7 & 7.5 & 4.4 & 5.6 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-2 & 2 & 3.8 & - 5.8 & 0.4 & - 7.4 \\ 0 & 4 & - 3.6 & 9.2 & 8.8 & 4.4 \\ 0 & 0 & 0 & 0 & - 0.4 & 0.6 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 2.8 \\ 0.9 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 5.2 \\ - 2.3 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 7.8 \\ - 4.4 \\ 0 \\ 0 \\ 1.5 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.8 \\ 0 \\ 1.6 \\ - 1.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 2 \\ 0 \\ 0 \\ -1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 2.8 \\ 0.9 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 5.2 \\ - 2.3 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 7.8 \\ - 4.4 \\ 0 \\ 0 \\ 1.5 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.6 \\ -1 \\ 1.2 \\ - 1.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 4 \\ 0 \\ 0 \\ -2 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 2.8 \\ 0.9 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 5.2 \\ - 2.3 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 7.8 \\ - 4.4 \\ 0 \\ 0 \\ 1.5 \\ 1 \end{array}\right)

    v v v  
* * * * * *
0 * * * * *
0 0 0 0 0 *
0 0 0 0 0 0

Example matrix

A = \left(\begin{array}{rrrrrr}3 & -5 & 1.4 & 11.6 & - 3.2 & 1 \\ -5 & 5 & -4 & -6 & -3 & 0 \\ -3 & 2 & - 2.9 & 0.4 & - 4.3 & 2.5 \\ 2.5 & - 2.9 & 1.8 & 4.6 & 0.5 & 1 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-5 & 5 & -4 & -6 & -3 & 0 \\ 0 & -2 & -1 & 8 & -5 & 1 \\ 0 & 0 & 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 1.3 \\ - 0.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 2.8 \\ 4 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 3.1 \\ - 2.5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-1 \\ -5 \\ -1 \\ 3.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 3 \\ 0 \\ 0 \\ 0 \\ 2 \end{array}\right) + u_3 \left(\begin{array}{r}- 1.3 \\ - 0.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 2.8 \\ 4 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 3.1 \\ - 2.5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}0 \\ -5 \\ - 2.5 \\ 2.7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 2 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_3 \left(\begin{array}{r}- 1.3 \\ - 0.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 2.8 \\ 4 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 3.1 \\ - 2.5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr} 1.5 & 5 & - 6.7 & - 1.1 & - 0.3 & - 8.7 \\ -5 & 0 & -1 & 7 & -9 & 9 \\ -4 & 4.5 & - 7.1 & 6.5 & - 9.9 & 5.8 \\ 2 & - 4.5 & 6.7 & - 3.7 & 6.3 & 1 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-5 & 0 & -1 & 7 & -9 & 9 \\ 0 & 5 & -7 & 1 & -3 & -6 \\ 0 & 0 & 0 & 0 & 0 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 0.2 \\ 1.4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 1.4 \\ - 0.2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 1.8 \\ 0.6 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 6.4 \\ -2 \\ 0.1 \\ 5.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 1 \\ 0 \\ 0 \\ 0 \\ 2 \end{array}\right) + u_3 \left(\begin{array}{r}- 0.2 \\ 1.4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 1.4 \\ - 0.2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 1.8 \\ 0.6 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 5.8 \\ -6 \\ 2.8 \\ -2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 2 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_3 \left(\begin{array}{r}- 0.2 \\ 1.4 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 1.4 \\ - 0.2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 1.8 \\ 0.6 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}4 & 1.3 & - 4.6 & 4.5 & - 7.3 & 17.4 \\ -3 & - 0.4 & 4.6 & - 9.7 & 2.6 & 2.9 \\ 5 & 1.5 & -6 & 7 & - 8.5 & 8.5 \\ -4 & - 0.9 & 5.4 & - 8.9 & 5.3 & 1.6 \end{array}\right) \sim \left(\begin{array}{rrrrrr}5 & 1.5 & -6 & 7 & - 8.5 & 8.5 \\ 0 & 0.5 & 1 & - 5.5 & - 2.5 & 8 \\ 0 & 0 & 0 & 0 & 0 & 9 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 1.8 \\ -2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 4.7 \\ 11 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 0.2 \\ 5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 9.5 \\ - 7.1 \\ 1 \\ - 8.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 3 \\ 0 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_3 \left(\begin{array}{r} 1.8 \\ -2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 4.7 \\ 11 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 0.2 \\ 5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 8.2 \\ - 7.5 \\ 2.5 \\ - 9.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 4 \\ 0 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_3 \left(\begin{array}{r} 1.8 \\ -2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 4.7 \\ 11 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 0.2 \\ 5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr} 0.2 & - 2.8 & -7 & - 0.2 & 2.1 & - 4.7 \\ - 0.2 & - 3.2 & - 6.2 & - 4.6 & 3.9 & - 3.6 \\ 0.4 & 2.4 & 3.6 & 6 & - 3.8 & - 5.8 \\ 0.2 & 3.2 & 6.2 & 4.6 & - 3.9 & 0.8 \end{array}\right) \sim \left(\begin{array}{rrrrrr} 0.4 & 2.4 & 3.6 & 6 & - 3.8 & - 5.8 \\ 0 & -4 & - 8.8 & - 3.2 & 4 & - 1.8 \\ 0 & 0 & 0 & 0 & 0 & - 5.6 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 4.2 \\ - 2.2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 10.2 \\ - 0.8 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 3.5 \\ 1 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.1 \\ - 1.2 \\ - 6.6 \\ - 1.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -1 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_3 \left(\begin{array}{r} 4.2 \\ - 2.2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 10.2 \\ - 0.8 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 3.5 \\ 1 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.3 \\ -1 \\ -7 \\ - 1.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ -1 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_3 \left(\begin{array}{r} 4.2 \\ - 2.2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 10.2 \\ - 0.8 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 3.5 \\ 1 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

  v     v v
* * * * * *
0 0 * * * *
0 0 0 * * *
0 0 0 0 0 0

Example matrix

A = \left(\begin{array}{rrrrrr}2 & 0 & -8 & 1 & 3 & 4 \\ - 1.2 & 0 & 9.8 & - 2.6 & - 7.8 & 5.6 \\ 1.4 & 0 & - 4.1 & - 0.9 & - 6.7 & - 3.8 \\ -1 & 0 & 8 & - 1.9 & - 4.9 & 6.2 \end{array}\right) \sim \left(\begin{array}{rrrrrr}2 & 0 & -8 & 1 & 3 & 4 \\ 0 & 0 & 5 & -2 & -6 & 8 \\ 0 & 0 & 0 & -1 & -7 & -9 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 4.4 \\ 0 \\ - 1.6 \\ -7 \\ 1 \end{array}\right), \left(\begin{array}{r}- 18.3 u_{6} \\ 0 \\ - 5.2 u_{6} \\ -9 u_{6} \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}2 \\ - 0.2 \\ - 0.3 \\ 0.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 1 \\ 2 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 4.4 \\ 0 \\ - 1.6 \\ -7 \\ 1 \end{array}\right) + u_6 \left(\begin{array}{r}- 18.3 u_{6} \\ 0 \\ - 5.2 u_{6} \\ -9 u_{6} \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}0 \\ 1 \\ - 1.7 \\ 1.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 1 \\ 2 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 4.4 \\ 0 \\ - 1.6 \\ -7 \\ 1 \end{array}\right) + u_6 \left(\begin{array}{r}- 18.3 u_{6} \\ 0 \\ - 5.2 u_{6} \\ -9 u_{6} \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr} 0.2 & 1.8 & 2.2 & - 1.4 & 6.8 & -12 \\ -1 & -9 & 1 & 6 & 5 & 2 \\ 0.9 & 8.1 & 3.1 & 2.6 & - 6.5 & - 7.8 \\ 0.8 & 7.2 & - 3.2 & - 9.1 & - 3.7 & 2.8 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-1 & -9 & 1 & 6 & 5 & 2 \\ 0 & 0 & 4 & 8 & -2 & -6 \\ 0 & 0 & 0 & -5 & 9 & -8 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-9 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 12.7 \\ 0 \\ - 3.1 \\ 1.8 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}- 2.9 \\ 0 \\ 4.7 \\ - 1.6 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-5 \\ 0 \\ 0 \\ 0.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ -2 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-9 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 12.7 \\ 0 \\ - 3.1 \\ 1.8 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 2.9 \\ 0 \\ 4.7 \\ - 1.6 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 2.8 \\ 1 \\ 3.1 \\ - 2.7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ -1 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-9 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 12.7 \\ 0 \\ - 3.1 \\ 1.8 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 2.9 \\ 0 \\ 4.7 \\ - 1.6 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr} 0.3 & - 3.6 & 2 & 4.2 & 1.3 & 2.9 \\ - 0.4 & 4.8 & - 1.4 & - 2.6 & - 2.6 & 0.7 \\ - 0.5 & 6 & -3 & - 9.5 & -7 & -1 \\ - 0.2 & 2.4 & - 1.6 & - 5.3 & - 3.3 & - 1.4 \end{array}\right) \sim \left(\begin{array}{rrrrrr}- 0.5 & 6 & -3 & - 9.5 & -7 & -1 \\ 0 & 0 & 1 & 5 & 3 & 1.5 \\ 0 & 0 & 0 & - 2.5 & - 3.5 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}12 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 11.4 \\ 0 \\ 4 \\ - 1.4 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 15.8 \\ 0 \\ - 5.5 \\ 0.8 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.2 \\ 0.8 \\ - 1.5 \\ - 0.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ -3 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}12 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 11.4 \\ 0 \\ 4 \\ - 1.4 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 15.8 \\ 0 \\ - 5.5 \\ 0.8 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.9 \\ 0.4 \\ -2 \\ - 1.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ -3 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}12 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 11.4 \\ 0 \\ 4 \\ - 1.4 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 15.8 \\ 0 \\ - 5.5 \\ 0.8 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}- 0.1 & - 3.2 & 0.6 & 2.1 & 5 & - 2.6 \\ 0.1 & 3.2 & - 0.1 & - 0.5 & 0.2 & - 0.9 \\ - 0.2 & - 6.4 & - 0.8 & - 1.4 & - 0.4 & - 3.2 \\ - 0.1 & - 3.2 & 0.1 & 0.8 & 3.7 & - 3.6 \end{array}\right) \sim \left(\begin{array}{rrrrrr}- 0.2 & - 6.4 & - 0.8 & - 1.4 & - 0.4 & - 3.2 \\ 0 & 0 & 1 & 2.8 & 5.2 & -1 \\ 0 & 0 & 0 & 0.2 & 2.6 & -3 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-32 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 35.8 \\ 0 \\ 31.2 \\ -13 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}43 \\ 0 \\ -41 \\ 15 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.1 \\ 0.2 \\ 0.2 \\ 0.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ -3 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-32 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 35.8 \\ 0 \\ 31.2 \\ -13 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}43 \\ 0 \\ -41 \\ 15 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}0 \\ 0.1 \\ 0.4 \\ 0.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ -3 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-32 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 35.8 \\ 0 \\ 31.2 \\ -13 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}43 \\ 0 \\ -41 \\ 15 \\ 0 \\ 1 \end{array}\right)

  v   v   v
* * * * * *
0 0 * * * *
0 0 0 0 * *
0 0 0 0 0 0

Example matrix

A = \left(\begin{array}{rrrrrr} 0.8 & - 3.2 & 5.8 & - 5.8 & -1 & - 7.2 \\ 0.7 & - 2.8 & 9.2 & - 9.2 & - 3.4 & - 11.1 \\ -1 & 4 & -6 & 6 & 2 & 3 \\ 0.2 & - 0.8 & 4.7 & - 4.7 & - 2.7 & - 4.2 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-1 & 4 & -6 & 6 & 2 & 3 \\ 0 & 0 & 5 & -5 & -2 & -9 \\ 0 & 0 & 0 & 0 & 1 & -3 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}4 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}0 \\ 0 \\ 1 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-9 \\ 0 \\ 3 \\ 0 \\ 3 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.4 \\ - 0.3 \\ -1 \\ 1.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ -1 \\ 0 \\ -2 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}4 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}0 \\ 0 \\ 1 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}-9 \\ 0 \\ 3 \\ 0 \\ 3 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.6 \\ 0.4 \\ -2 \\ 1.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ -1 \\ 0 \\ -2 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}4 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}0 \\ 0 \\ 1 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}-9 \\ 0 \\ 3 \\ 0 \\ 3 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}-1 & -7 & 0 & 8 & 5 & -6 \\ - 0.4 & - 2.8 & 0.8 & 2.8 & 7.6 & -14 \\ 0.6 & 4.2 & -2 & - 3.8 & -7 & 12.6 \\ 0.5 & 3.5 & - 0.4 & - 3.8 & - 2.1 & 2.4 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-1 & -7 & 0 & 8 & 5 & -6 \\ 0 & 0 & -2 & 1 & -4 & 9 \\ 0 & 0 & 0 & 0 & 4 & -8 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-7 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}8 \\ 0 \\ 0.5 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}4 \\ 0 \\ 0.5 \\ 0 \\ 2 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}1 \\ 3.6 \\ 1.4 \\ 1.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ -3 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-7 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}8 \\ 0 \\ 0.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}4 \\ 0 \\ 0.5 \\ 0 \\ 2 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}2 \\ 4 \\ 0.8 \\ 0.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ -3 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-7 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}8 \\ 0 \\ 0.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}4 \\ 0 \\ 0.5 \\ 0 \\ 2 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}- 0.1 & 0.1 & - 1.7 & 3.1 & 7.7 & - 2.4 \\ - 0.5 & 0.5 & 1.5 & 6.5 & 8.5 & 3 \\ - 0.3 & 0.3 & 5.9 & - 0.6 & 0.1 & 4.3 \\ 0.3 & - 0.3 & 1.1 & - 5.7 & - 3.9 & - 2.4 \end{array}\right) \sim \left(\begin{array}{rrrrrr}- 0.5 & 0.5 & 1.5 & 6.5 & 8.5 & 3 \\ 0 & 0 & 5 & - 4.5 & -5 & 2.5 \\ 0 & 0 & 0 & 0 & 4 & -2 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}1 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 15.7 \\ 0 \\ 0.9 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 14.5 \\ 0 \\ 0 \\ 0 \\ 0.5 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}9 \\ 5 \\ -7 \\ - 3.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ -1 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}1 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 15.7 \\ 0 \\ 0.9 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 14.5 \\ 0 \\ 0 \\ 0 \\ 0.5 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 9.1 \\ 5.5 \\ - 6.7 \\ - 4.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ -1 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}1 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 15.7 \\ 0 \\ 0.9 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 14.5 \\ 0 \\ 0 \\ 0 \\ 0.5 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}- 0.1 & - 3.1 & 1.4 & 5.4 & - 6.7 & 12.9 \\ - 0.1 & - 3.1 & 0.9 & 2.8 & - 4.7 & 6.4 \\ 0.2 & 6.2 & - 0.8 & - 0.4 & - 2.6 & - 8.6 \\ - 0.1 & - 3.1 & - 0.1 & - 2.4 & 6.3 & 1.1 \end{array}\right) \sim \left(\begin{array}{rrrrrr} 0.2 & 6.2 & - 0.8 & - 0.4 & - 2.6 & - 8.6 \\ 0 & 0 & 1 & 5.2 & -8 & 8.6 \\ 0 & 0 & 0 & 0 & -2 & - 2.2 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-31 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 18.8 \\ 0 \\ - 5.2 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 40.9 \\ 0 \\ - 17.4 \\ 0 \\ - 1.1 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.4 \\ - 1.4 \\ - 5.2 \\ 5.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 4 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-31 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 18.8 \\ 0 \\ - 5.2 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 40.9 \\ 0 \\ - 17.4 \\ 0 \\ - 1.1 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 2.9 \\ - 2.4 \\ - 4.2 \\ 5.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 3 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-31 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 18.8 \\ 0 \\ - 5.2 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 40.9 \\ 0 \\ - 17.4 \\ 0 \\ - 1.1 \\ 1 \end{array}\right)

  v   v v  
* * * * * *
0 0 * * * *
0 0 0 0 0 *
0 0 0 0 0 0

Example matrix

A = \left(\begin{array}{rrrrrr} 0.6 & 2.4 & 2 & - 7.8 & 10 & -2 \\ 0.7 & 2.8 & - 0.8 & - 4.4 & 0.7 & - 10.9 \\ 1 & 4 & 0 & -8 & 5 & -5 \\ - 0.8 & - 3.2 & - 0.2 & 6.7 & - 4.7 & 7.4 \end{array}\right) \sim \left(\begin{array}{rrrrrr}1 & 4 & 0 & -8 & 5 & -5 \\ 0 & 0 & 2 & -3 & 7 & 1 \\ 0 & 0 & 0 & 0 & 0 & -7 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-4 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}8 \\ 0 \\ 1.5 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-5 \\ 0 \\ - 3.5 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.6 \\ - 7.3 \\ -1 \\ 4.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ -1 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}-4 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}8 \\ 0 \\ 1.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}-5 \\ 0 \\ - 3.5 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 3.6 \\ - 6.5 \\ -1 \\ 4.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ -2 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}-4 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}8 \\ 0 \\ 1.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}-5 \\ 0 \\ - 3.5 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}5 & 6 & 9 & -6 & -1 & 8 \\ 3.5 & 4.2 & 7.3 & - 11.2 & 3.3 & 0.6 \\ 2 & 2.4 & 4.1 & - 5.9 & 1.6 & 2.7 \\ - 3.5 & - 4.2 & - 6.7 & 7 & - 0.9 & -2 \end{array}\right) \sim \left(\begin{array}{rrrrrr}5 & 6 & 9 & -6 & -1 & 8 \\ 0 & 0 & 1 & -7 & 4 & -5 \\ 0 & 0 & 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 1.2 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 11.4 \\ 0 \\ 7 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 7.4 \\ 0 \\ -4 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}5 \\ - 3.5 \\ 0.5 \\ 0.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ -2 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}- 1.2 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 11.4 \\ 0 \\ 7 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 7.4 \\ 0 \\ -4 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-6 \\ - 1.2 \\ - 2.9 \\ 1.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ -2 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}- 1.2 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 11.4 \\ 0 \\ 7 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 7.4 \\ 0 \\ -4 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}- 0.5 & 3.5 & - 2.5 & - 5.5 & 2 & 0 \\ - 0.3 & 2.1 & 3.5 & 4.7 & - 0.8 & 1 \\ 0.4 & - 2.8 & -1 & - 0.4 & - 0.4 & 1.9 \\ 0.1 & - 0.7 & - 1.5 & - 2.1 & 0.4 & 0.1 \end{array}\right) \sim \left(\begin{array}{rrrrrr}- 0.5 & 3.5 & - 2.5 & - 5.5 & 2 & 0 \\ 0 & 0 & 5 & 8 & -2 & 1 \\ 0 & 0 & 0 & 0 & 0 & 2.5 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}7 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-3 \\ 0 \\ - 1.6 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}2 \\ 0 \\ 0.4 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 3.5 \\ 1.9 \\ - 2.1 \\ - 1.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 1 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}7 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-3 \\ 0 \\ - 1.6 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}2 \\ 0 \\ 0.4 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-4 \\ 1.6 \\ - 1.7 \\ - 1.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 1 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}7 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-3 \\ 0 \\ - 1.6 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}2 \\ 0 \\ 0.4 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}- 0.5 & - 3.9 & 0 & 0.5 & 2.7 & 0.5 \\ 1 & 7.8 & 2 & 7.6 & 1.4 & -5 \\ 0.5 & 3.9 & -1 & - 4.8 & - 6.1 & - 7.3 \\ - 0.5 & - 3.9 & -2 & - 8.1 & - 4.1 & - 2.1 \end{array}\right) \sim \left(\begin{array}{rrrrrr}1 & 7.8 & 2 & 7.6 & 1.4 & -5 \\ 0 & 0 & -2 & - 8.6 & - 6.8 & - 4.8 \\ 0 & 0 & 0 & 0 & 0 & - 4.4 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 7.8 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}1 \\ 0 \\ - 4.3 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 5.4 \\ 0 \\ - 3.4 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.5 \\ -3 \\ - 4.3 \\ - 2.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ -1 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}- 7.8 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}1 \\ 0 \\ - 4.3 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 5.4 \\ 0 \\ - 3.4 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.5 \\ -5 \\ - 3.3 \\ - 0.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ -2 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}- 7.8 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}1 \\ 0 \\ - 4.3 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 5.4 \\ 0 \\ - 3.4 \\ 0 \\ 1 \\ 0 \end{array}\right)

  v v     v
* * * * * *
0 0 0 * * *
0 0 0 0 * *
0 0 0 0 0 0

Example matrix

A = \left(\begin{array}{rrrrrr}- 4.5 & 3.6 & - 2.7 & - 7.7 & 10.6 & 6.5 \\ -5 & 4 & -3 & -7 & 8 & 7 \\ 2.5 & -2 & 1.5 & 5.5 & -6 & - 9.5 \\ 1.5 & - 1.2 & 0.9 & 3.1 & - 3.2 & - 5.5 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-5 & 4 & -3 & -7 & 8 & 7 \\ 0 & 0 & 0 & 2 & -2 & -6 \\ 0 & 0 & 0 & 0 & 2 & -4 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.8 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 0.6 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 2.4 \\ 0 \\ 0 \\ 5 \\ 2 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 0.3 \\ 1 \\ - 2.5 \\ - 1.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 0 \\ -2 \\ -1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 0.8 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 0.6 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 2.4 \\ 0 \\ 0 \\ 5 \\ 2 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 2.6 \\ 0 \\ -2 \\ - 1.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 0 \\ -3 \\ -2 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 0.8 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 0.6 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 2.4 \\ 0 \\ 0 \\ 5 \\ 2 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}- 4.5 & 2.7 & - 5.4 & - 4.7 & - 8.1 & 9.8 \\ -5 & 3 & -6 & -3 & -9 & 2 \\ 4.5 & - 2.7 & 5.4 & 3.7 & 9.1 & 0.2 \\ -4 & 2.4 & - 4.8 & - 1.8 & - 7.8 & - 4.4 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-5 & 3 & -6 & -3 & -9 & 2 \\ 0 & 0 & 0 & -2 & 0 & 8 \\ 0 & 0 & 0 & 0 & 1 & 6 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.6 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 1.2 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 8.8 \\ 0 \\ 0 \\ 4 \\ -6 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-2 \\ 0 \\ -1 \\ 1.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 1 \\ -2 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 0.6 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 1.2 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 8.8 \\ 0 \\ 0 \\ 4 \\ -6 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}4 \\ 0 \\ -3 \\ - 0.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ -2 \\ -1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 0.6 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 1.2 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 8.8 \\ 0 \\ 0 \\ 4 \\ -6 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr} 0.5 & - 5.5 & - 7.5 & 3 & - 2.5 & -8 \\ - 0.4 & 4.4 & 6 & - 1.9 & 1.3 & 6.1 \\ 0.2 & - 2.2 & -3 & 3.7 & - 9.5 & 0.3 \\ - 0.1 & 1.1 & 1.5 & 0.9 & -5 & 4.1 \end{array}\right) \sim \left(\begin{array}{rrrrrr} 0.5 & - 5.5 & - 7.5 & 3 & - 2.5 & -8 \\ 0 & 0 & 0 & 2.5 & - 8.5 & 3.5 \\ 0 & 0 & 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}11 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}15 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}9 \\ 0 \\ 0 \\ 2 \\ 1 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.5 \\ 0.9 \\ 2.9 \\ 2.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 0 \\ -2 \\ -1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}11 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}15 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}9 \\ 0 \\ 0 \\ 2 \\ 1 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-2 \\ 1.3 \\ 2.7 \\ 2.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ -2 \\ -1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}11 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}15 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}9 \\ 0 \\ 0 \\ 2 \\ 1 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr} 0.4 & 5.4 & - 1.2 & - 7.6 & 2.4 & 4.4 \\ 0.2 & 2.7 & - 0.6 & - 2.8 & 0.9 & 2.3 \\ - 0.2 & - 2.7 & 0.6 & 5.8 & - 2.2 & 5.2 \\ 0.2 & 2.7 & - 0.6 & - 2.8 & 0.6 & 7.7 \end{array}\right) \sim \left(\begin{array}{rrrrrr} 0.4 & 5.4 & - 1.2 & - 7.6 & 2.4 & 4.4 \\ 0 & 0 & 0 & 2 & -1 & 7.4 \\ 0 & 0 & 0 & 0 & 0.2 & - 3.6 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 13.5 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}3 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 18.3 \\ 0 \\ 0 \\ 5.3 \\ 18 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 0.4 \\ 0.3 \\ - 1.2 \\ - 0.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 1 \\ 3 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 13.5 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}3 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 18.3 \\ 0 \\ 0 \\ 5.3 \\ 18 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.2 \\ - 0.2 \\ 0.6 \\ - 0.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 0 \\ 1 \\ 2 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 13.5 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}3 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 18.3 \\ 0 \\ 0 \\ 5.3 \\ 18 \\ 1 \end{array}\right)

  v v   v  
* * * * * *
0 0 0 * * *
0 0 0 0 0 *
0 0 0 0 0 0

Example matrix

A = \left(\begin{array}{rrrrrr}- 0.4 & - 1.6 & 2.4 & - 5.8 & 10.6 & 5.4 \\ - 0.9 & - 3.6 & 5.4 & 2.2 & 2.5 & - 4.1 \\ -1 & -4 & 6 & -2 & 9 & 1 \\ - 0.5 & -2 & 3 & -5 & 10.1 & 3.8 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-1 & -4 & 6 & -2 & 9 & 1 \\ 0 & 0 & 0 & -5 & 7 & 5 \\ 0 & 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-4 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}6 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 6.2 \\ 0 \\ 0 \\ 1.4 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 4.6 \\ - 3.9 \\ -1 \\ 4.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 0 \\ -2 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}-4 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}6 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 6.2 \\ 0 \\ 0 \\ 1.4 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}5 \\ -3 \\ 0 \\ 4.7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ -2 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}-4 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}6 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 6.2 \\ 0 \\ 0 \\ 1.4 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr} 3.5 & - 5.6 & - 2.1 & -2 & - 9.7 & 0.1 \\ -5 & 8 & 3 & 0 & 1 & 7 \\ -3 & 4.8 & 1.8 & -1 & - 3.9 & 15.7 \\ 1 & - 1.6 & - 0.6 & - 0.4 & -2 & 3.2 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-5 & 8 & 3 & 0 & 1 & 7 \\ 0 & 0 & 0 & -2 & -9 & 5 \\ 0 & 0 & 0 & 0 & 0 & 9 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 1.6 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 0.6 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 0.2 \\ 0 \\ 0 \\ - 4.5 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.9 \\ -3 \\ 5.7 \\ 3.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 4 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r} 1.6 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 0.6 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 0.2 \\ 0 \\ 0 \\ - 4.5 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 1.1 \\ -3 \\ 6.7 \\ 4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 3 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r} 1.6 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 0.6 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 0.2 \\ 0 \\ 0 \\ - 4.5 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr} 0.3 & 2.7 & - 2.1 & 3.6 & 4.5 & - 6.5 \\ 0.5 & 4.5 & - 3.5 & 1 & - 0.5 & 6.5 \\ 0.2 & 1.8 & - 1.4 & - 4.6 & - 8.2 & 4.1 \\ 0.4 & 3.6 & - 2.8 & - 0.2 & -2 & 7.4 \end{array}\right) \sim \left(\begin{array}{rrrrrr} 0.5 & 4.5 & - 3.5 & 1 & - 0.5 & 6.5 \\ 0 & 0 & 0 & -5 & -8 & 1.5 \\ 0 & 0 & 0 & 0 & 0 & - 9.5 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-9 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}7 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 4.2 \\ 0 \\ 0 \\ - 1.6 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 0.5 \\ - 6.5 \\ 5.9 \\ - 5.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 0 \\ -2 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}-9 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}7 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 4.2 \\ 0 \\ 0 \\ - 1.6 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 0.2 \\ -7 \\ 5.7 \\ - 5.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ -2 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}-9 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}7 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 4.2 \\ 0 \\ 0 \\ - 1.6 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}1 & -4 & - 4.3 & 0.3 & - 3.1 & - 1.4 \\ 2 & -8 & - 8.6 & 0.8 & - 6.4 & 8.2 \\ 1 & -4 & - 4.3 & 0.6 & - 3.4 & 7.9 \\ 1 & -4 & - 4.3 & 0.3 & - 3.1 & 0.4 \end{array}\right) \sim \left(\begin{array}{rrrrrr}2 & -8 & - 8.6 & 0.8 & - 6.4 & 8.2 \\ 0 & 0 & 0 & 0.2 & - 0.2 & 3.8 \\ 0 & 0 & 0 & 0 & 0 & - 3.6 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}4 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 4.3 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 2.8 \\ 0 \\ 0 \\ 1 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 4.6 \\ -1 \\ - 3.5 \\ 2.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 4 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}4 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 4.3 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 2.8 \\ 0 \\ 0 \\ 1 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 4.3 \\ - 1.8 \\ - 4.1 \\ 2.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 3 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}4 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 4.3 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 2.8 \\ 0 \\ 0 \\ 1 \\ 1 \\ 0 \end{array}\right)

  v v v    
* * * * * *
0 0 0 0 * *
0 0 0 0 0 *
0 0 0 0 0 0

Example matrix

A = \left(\begin{array}{rrrrrr}2 & -7 & -5 & 1 & 7 & -8 \\ -1 & 3.5 & 2.5 & - 0.5 & - 9.5 & 2 \\ 1.6 & - 5.6 & -4 & 0.8 & 3.8 & -4 \\ 0.2 & - 0.7 & - 0.5 & 0.1 & 6.1 & - 0.8 \end{array}\right) \sim \left(\begin{array}{rrrrrr}2 & -7 & -5 & 1 & 7 & -8 \\ 0 & 0 & 0 & 0 & -6 & -2 \\ 0 & 0 & 0 & 0 & 0 & 3 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 3.5 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 2.5 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 0.5 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-3 \\ - 8.5 \\ 0.6 \\ 5.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 0 \\ 1 \\ 2 \end{array}\right) + u_2 \left(\begin{array}{r} 3.5 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 2.5 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 0.5 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-1 \\ - 9.5 \\ 2.2 \\ 5.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 0 \\ 0 \\ 1 \\ 2 \end{array}\right) + u_2 \left(\begin{array}{r} 3.5 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 2.5 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 0.5 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}-5 & 3 & 2 & 8 & 7 & 5 \\ 1 & - 0.6 & - 0.4 & - 1.6 & 6.7 & - 2.9 \\ - 3.5 & 2.1 & 1.4 & 5.6 & - 4.1 & 4.5 \\ -2 & 1.2 & 0.8 & 3.2 & 7.3 & 0.9 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-5 & 3 & 2 & 8 & 7 & 5 \\ 0 & 0 & 0 & 0 & -9 & 1 \\ 0 & 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.6 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 0.4 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 1.6 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}2 \\ 2 \\ - 4.6 \\ 2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 0 \\ 0 \\ 1 \\ 3 \end{array}\right) + u_2 \left(\begin{array}{r} 0.6 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 0.4 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 1.6 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}2 \\ 3.9 \\ - 5.6 \\ 3.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 0 \\ 1 \\ 2 \end{array}\right) + u_2 \left(\begin{array}{r} 0.6 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 0.4 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 1.6 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}- 0.2 & 0 & - 2.8 & 1.2 & - 6.2 & - 1.1 \\ - 0.4 & 0 & - 5.6 & 2.4 & - 5.4 & - 8.2 \\ - 0.5 & 0 & -7 & 3 & -3 & -9 \\ 0.2 & 0 & 2.8 & - 1.2 & 2.2 & 4.6 \end{array}\right) \sim \left(\begin{array}{rrrrrr}- 0.5 & 0 & -7 & 3 & -3 & -9 \\ 0 & 0 & 0 & 0 & -5 & 2.5 \\ 0 & 0 & 0 & 0 & 0 & - 2.5 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-14 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}6 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 5.9 \\ 1.2 \\ 4 \\ - 1.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-14 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}6 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 5.7 \\ 1.6 \\ 4.5 \\ - 1.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-14 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}6 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}2 & 7.2 & - 7.4 & - 4.8 & - 6.2 & -5 \\ -1 & - 3.6 & 3.7 & 2.4 & 4 & 8.6 \\ 1 & 3.6 & - 3.7 & - 2.4 & - 4.9 & - 9.1 \\ -1 & - 3.6 & 3.7 & 2.4 & 2.2 & 0.6 \end{array}\right) \sim \left(\begin{array}{rrrrrr}2 & 7.2 & - 7.4 & - 4.8 & - 6.2 & -5 \\ 0 & 0 & 0 & 0 & - 1.8 & - 6.6 \\ 0 & 0 & 0 & 0 & 0 & 2.8 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 3.6 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 3.7 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 2.4 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.4 \\ - 3.6 \\ 2.3 \\ 0.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 0 \\ 2 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}- 3.6 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 3.7 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 2.4 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 0.6 \\ - 4.6 \\ 3.3 \\ - 0.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 0 \\ 0 \\ 2 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}- 3.6 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 3.7 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 2.4 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)

4 \times 6 with four free variables

    v v v v
* * * * * *
0 * * * * *
0 0 0 0 0 0
0 0 0 0 0 0

Example matrix

A = \left(\begin{array}{rrrrrr}- 1.8 & 5.6 & - 2.2 & - 4.3 & 4.8 & - 12.6 \\ 0.8 & - 2.4 & 1.2 & 1.6 & -2 & 5.2 \\ 2 & -4 & 8 & -3 & -2 & 4 \\ - 1.6 & 2.2 & - 8.9 & 5.9 & 0.1 & 1.3 \end{array}\right) \sim \left(\begin{array}{rrrrrr}2 & -4 & 8 & -3 & -2 & 4 \\ 0 & 2 & 5 & -7 & 3 & -9 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-9 \\ - 2.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 8.5 \\ 3.5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-2 \\ - 1.5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}7 \\ 4.5 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}2 \\ - 0.8 \\ 0 \\ -1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-9 \\ - 2.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 8.5 \\ 3.5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}-2 \\ - 1.5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}7 \\ 4.5 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 0.2 \\ 0 \\ 2 \\ - 2.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-9 \\ - 2.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 8.5 \\ 3.5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}-2 \\ - 1.5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}7 \\ 4.5 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}- 4.5 & - 8.3 & 0.3 & 9.1 & 3.1 & 12.4 \\ -5 & -7 & -3 & 9 & -1 & 6 \\ - 2.5 & - 2.3 & - 3.3 & 3.9 & - 2.9 & - 1.2 \\ 3.5 & 5.1 & 1.8 & - 6.4 & 0.3 & - 4.9 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-5 & -7 & -3 & 9 & -1 & 6 \\ 0 & -2 & 3 & 1 & 4 & 7 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 2.7 \\ 1.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 1.1 \\ 0.5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-3 \\ 2 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}- 3.7 \\ 3.5 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 3.1 \\ -1 \\ - 2.9 \\ 0.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ -2 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 2.7 \\ 1.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 1.1 \\ 0.5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}-3 \\ 2 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 3.7 \\ 3.5 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.7 \\ -3 \\ - 2.7 \\ 1.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ -1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 2.7 \\ 1.5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 1.1 \\ 0.5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}-3 \\ 2 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 3.7 \\ 3.5 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}-1 & - 0.5 & - 1.5 & 0 & 8 & 3 \\ 0.8 & 0.9 & 5.7 & - 3.5 & - 8.4 & 4.1 \\ - 0.2 & - 0.2 & - 1.2 & 0.7 & 2 & - 0.7 \\ - 0.8 & - 0.6 & -3 & 1.4 & 7.2 & - 0.2 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-1 & - 0.5 & - 1.5 & 0 & 8 & 3 \\ 0 & 0.5 & 4.5 & - 3.5 & -2 & 6.5 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}3 \\ -9 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 3.5 \\ 7 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}6 \\ 4 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 9.5 \\ -13 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.5 \\ - 0.1 \\ 0 \\ - 0.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ -1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}3 \\ -9 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 3.5 \\ 7 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}6 \\ 4 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 9.5 \\ -13 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}0 \\ -1 \\ 0.2 \\ 0.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ -2 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}3 \\ -9 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 3.5 \\ 7 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}6 \\ 4 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 9.5 \\ -13 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr} 0.2 & 1.9 & - 1.8 & - 5.4 & - 3.3 & 3.1 \\ 0.4 & 1.8 & 7.6 & 0.8 & 6.6 & 1.4 \\ 0.2 & 1.4 & 1 & - 2.5 & 0 & 1.9 \\ 0.2 & 0.4 & 6.6 & 3.3 & 6.6 & - 0.5 \end{array}\right) \sim \left(\begin{array}{rrrrrr} 0.4 & 1.8 & 7.6 & 0.8 & 6.6 & 1.4 \\ 0 & 1 & - 5.6 & - 5.8 & - 6.6 & 2.4 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 44.2 \\ 5.6 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 28.1 \\ 5.8 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 46.2 \\ 6.6 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 7.3 \\ - 2.4 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.1 \\ - 0.2 \\ - 0.6 \\ 0.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 44.2 \\ 5.6 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 28.1 \\ 5.8 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 46.2 \\ 6.6 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 7.3 \\ - 2.4 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.3 \\ - 0.6 \\ - 0.8 \\ 0.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ -1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 44.2 \\ 5.6 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 28.1 \\ 5.8 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 46.2 \\ 6.6 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 7.3 \\ - 2.4 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

  v   v v v
* * * * * *
0 0 * * * *
0 0 0 0 0 0
0 0 0 0 0 0

Example matrix

A = \left(\begin{array}{rrrrrr}1 & 0 & 3 & -1 & 5 & 8 \\ 0.2 & 0 & - 0.8 & 4.7 & 3.1 & 3 \\ - 0.8 & 0 & - 0.4 & - 6.2 & -7 & - 8.4 \\ - 0.2 & 0 & - 1.6 & 3.7 & 0.5 & - 0.6 \end{array}\right) \sim \left(\begin{array}{rrrrrr}1 & 0 & 3 & -1 & 5 & 8 \\ 0 & 0 & 2 & -7 & -3 & -2 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 9.5 \\ 0 \\ 3.5 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}- 9.5 \\ 0 \\ 1.5 \\ 0 \\ 1 \end{array}\right), \left(\begin{array}{r}-11 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-1 \\ 1.2 \\ - 1.2 \\ 1.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ -1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 9.5 \\ 0 \\ 3.5 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 9.5 \\ 0 \\ 1.5 \\ 0 \\ 1 \end{array}\right) + u_6 \left(\begin{array}{r}-11 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}0 \\ 1.4 \\ -2 \\ 1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ -1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 9.5 \\ 0 \\ 3.5 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 9.5 \\ 0 \\ 1.5 \\ 0 \\ 1 \end{array}\right) + u_6 \left(\begin{array}{r}-11 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}1 & 9 & 8 & -9 & 2 & -8 \\ 0.9 & 8.1 & 10.7 & - 7.4 & - 2.4 & - 5.1 \\ - 0.2 & - 1.8 & - 6.6 & 0.8 & 5.6 & - 1.4 \\ 0.5 & 4.5 & 6.5 & -4 & -2 & - 2.5 \end{array}\right) \sim \left(\begin{array}{rrrrrr}1 & 9 & 8 & -9 & 2 & -8 \\ 0 & 0 & -5 & -1 & 6 & -3 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-9 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 10.6 \\ 0 \\ - 0.2 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 11.6 \\ 0 \\ 1.2 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 12.8 \\ 0 \\ - 0.6 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-4 \\ - 7.1 \\ 5.8 \\ - 4.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ -1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-9 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 10.6 \\ 0 \\ - 0.2 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 11.6 \\ 0 \\ 1.2 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 12.8 \\ 0 \\ - 0.6 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-5 \\ -8 \\ 6 \\ -5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ -1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-9 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 10.6 \\ 0 \\ - 0.2 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 11.6 \\ 0 \\ 1.2 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 12.8 \\ 0 \\ - 0.6 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}- 2.5 & 1.5 & 2 & 5 & 0 & 6.5 \\ - 0.5 & 0.3 & 1.4 & 8.5 & 0.5 & 6.8 \\ 1 & - 0.6 & - 1.6 & -8 & - 0.4 & -7 \\ 0.5 & - 0.3 & -1 & - 5.5 & - 0.3 & - 4.6 \end{array}\right) \sim \left(\begin{array}{rrrrrr}- 2.5 & 1.5 & 2 & 5 & 0 & 6.5 \\ 0 & 0 & 1 & 7.5 & 0.5 & 5.5 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.6 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-4 \\ 0 \\ - 7.5 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 0.4 \\ 0 \\ - 0.5 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}- 1.8 \\ 0 \\ - 5.5 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-3 \\ 0.4 \\ 0.4 \\ 0 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 0.6 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-4 \\ 0 \\ - 7.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 0.4 \\ 0 \\ - 0.5 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 1.8 \\ 0 \\ - 5.5 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 3.5 \\ 1.3 \\ - 0.2 \\ - 0.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 2 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 0.6 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-4 \\ 0 \\ - 7.5 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 0.4 \\ 0 \\ - 0.5 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 1.8 \\ 0 \\ - 5.5 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}-2 & 5.4 & 1.6 & 3.4 & - 9.2 & 1.2 \\ -1 & 2.7 & 1 & - 1.9 & - 2.5 & - 4.3 \\ -1 & 2.7 & 0.4 & 8.9 & - 8.8 & 10.4 \\ 1 & - 2.7 & -1 & 1.9 & 2.5 & 4.3 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-2 & 5.4 & 1.6 & 3.4 & - 9.2 & 1.2 \\ 0 & 0 & - 0.4 & 7.2 & - 4.2 & 9.8 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 2.7 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 16.1 \\ 0 \\ 18 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-13 \\ 0 \\ - 10.5 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 20.2 \\ 0 \\ 24.5 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 0.8 \\ 1 \\ - 0.8 \\ -1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 3 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 2.7 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 16.1 \\ 0 \\ 18 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}-13 \\ 0 \\ - 10.5 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 20.2 \\ 0 \\ 24.5 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 1.2 \\ 1 \\ - 0.2 \\ -1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 2 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 2.7 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 16.1 \\ 0 \\ 18 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}-13 \\ 0 \\ - 10.5 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 20.2 \\ 0 \\ 24.5 \\ 0 \\ 0 \\ 1 \end{array}\right)

  v v   v v
* * * * * *
0 0 0 * * *
0 0 0 0 0 0
0 0 0 0 0 0

Example matrix

A = \left(\begin{array}{rrrrrr}-1 & 8 & -8 & -3 & 4 & -4 \\ 0.7 & - 5.6 & 5.6 & 1.5 & 0.2 & - 0.8 \\ - 0.1 & 0.8 & - 0.8 & - 1.3 & 5.4 & - 6.4 \\ 0.6 & - 4.8 & 4.8 & 1.9 & - 2.9 & 3 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-1 & 8 & -8 & -3 & 4 & -4 \\ 0 & 0 & 0 & -1 & 5 & -6 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}8 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-8 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-11 \\ 0 \\ 0 \\ 5 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}14 \\ 0 \\ 0 \\ -6 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}0 \\ 0.6 \\ 1 \\ - 0.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ -1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}8 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-8 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}-11 \\ 0 \\ 0 \\ 5 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}14 \\ 0 \\ 0 \\ -6 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}1 \\ - 0.1 \\ 1.1 \\ - 0.7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ -1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}8 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-8 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}-11 \\ 0 \\ 0 \\ 5 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}14 \\ 0 \\ 0 \\ -6 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}-1 & 2 & 4 & 9 & 7 & -2 \\ 0.3 & - 0.6 & - 1.2 & - 4.3 & - 5.3 & 0.6 \\ 0.7 & - 1.4 & - 2.8 & - 10.3 & - 12.9 & 1.4 \\ 0.2 & - 0.4 & - 0.8 & 1.4 & 5 & 0.4 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-1 & 2 & 4 & 9 & 7 & -2 \\ 0 & 0 & 0 & -4 & -8 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}2 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}4 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-11 \\ 0 \\ 0 \\ -2 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}-2 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}5 \\ - 3.1 \\ - 7.5 \\ 2.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}2 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}4 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}-11 \\ 0 \\ 0 \\ -2 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}-2 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}6 \\ - 3.4 \\ - 8.2 \\ 2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}2 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}4 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}-11 \\ 0 \\ 0 \\ -2 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}-2 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr} 2.5 & 1 & - 0.5 & -3 & 8.5 & 8 \\ 1.5 & 0.6 & - 0.3 & - 2.8 & 1.6 & 4.8 \\ 0.5 & 0.2 & - 0.1 & - 0.4 & 2.4 & 1.6 \\ -1 & - 0.4 & 0.2 & 2 & - 0.6 & - 3.2 \end{array}\right) \sim \left(\begin{array}{rrrrrr} 2.5 & 1 & - 0.5 & -3 & 8.5 & 8 \\ 0 & 0 & 0 & -1 & - 3.5 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 0.4 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 0.2 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 7.6 \\ 0 \\ 0 \\ - 3.5 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}- 3.2 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}2 \\ 0.2 \\ 0.6 \\ 0 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 0.4 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 0.2 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 7.6 \\ 0 \\ 0 \\ - 3.5 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 3.2 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 1.5 \\ - 1.1 \\ 0.7 \\ 1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 2 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 0.4 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 0.2 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 7.6 \\ 0 \\ 0 \\ - 3.5 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 3.2 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}- 0.5 & 3.8 & - 4.9 & - 0.5 & - 0.5 & - 4.1 \\ - 0.5 & 3.8 & - 4.9 & 0.5 & 0.3 & - 4.8 \\ 1 & - 7.6 & 9.8 & 3 & 2.6 & 6.8 \\ - 0.5 & 3.8 & - 4.9 & - 2.5 & - 2.1 & - 2.7 \end{array}\right) \sim \left(\begin{array}{rrrrrr}1 & - 7.6 & 9.8 & 3 & 2.6 & 6.8 \\ 0 & 0 & 0 & 2 & 1.6 & - 1.4 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 7.6 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 9.8 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 0.2 \\ 0 \\ 0 \\ - 0.8 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}- 8.9 \\ 0 \\ 0 \\ 0.7 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.5 \\ - 1.5 \\ -1 \\ 1.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ -1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 7.6 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 9.8 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 0.2 \\ 0 \\ 0 \\ - 0.8 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 8.9 \\ 0 \\ 0 \\ 0.7 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-1 \\ -2 \\ 0 \\ 1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ -1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 7.6 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 9.8 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 0.2 \\ 0 \\ 0 \\ - 0.8 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 8.9 \\ 0 \\ 0 \\ 0.7 \\ 0 \\ 1 \end{array}\right)

  v v v   v
* * * * * *
0 0 0 0 * *
0 0 0 0 0 0
0 0 0 0 0 0

Example matrix

A = \left(\begin{array}{rrrrrr}-1 & 5 & 0 & 2 & -3 & 6 \\ - 0.5 & 2.5 & 0 & 1 & 0.1 & 6.2 \\ - 0.8 & 4 & 0 & 1.6 & - 6.4 & - 3.2 \\ - 0.2 & 1 & 0 & 0.4 & - 3.4 & - 4.4 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-1 & 5 & 0 & 2 & -3 & 6 \\ 0 & 0 & 0 & 0 & -4 & -8 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}12 u_{6} \\ 0 \\ 0 \\ 0 \\ -2 u_{6} \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-1 \\ - 2.1 \\ 3.2 \\ 2.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}12 u_{6} \\ 0 \\ 0 \\ 0 \\ -2 u_{6} \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}0 \\ - 1.6 \\ 4 \\ 2.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}12 u_{6} \\ 0 \\ 0 \\ 0 \\ -2 u_{6} \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr} 0.1 & 0.2 & - 0.2 & 0.4 & - 4.2 & - 2.1 \\ -1 & -2 & 2 & -4 & 7 & 0 \\ - 0.9 & - 1.8 & 1.8 & - 3.6 & 1.3 & -3 \\ 0.3 & 0.6 & - 0.6 & 1.2 & - 3.1 & - 0.6 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-1 & -2 & 2 & -4 & 7 & 0 \\ 0 & 0 & 0 & 0 & -5 & -3 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-2 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}2 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-4 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 4.2 \\ 0 \\ 0 \\ 0 \\ - 0.6 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 3.8 \\ 3 \\ - 2.3 \\ - 1.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-2 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}2 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-4 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 4.2 \\ 0 \\ 0 \\ 0 \\ - 0.6 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 3.9 \\ 4 \\ - 1.4 \\ - 2.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-2 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}2 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-4 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 4.2 \\ 0 \\ 0 \\ 0 \\ - 0.6 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}- 0.4 & - 5.2 & - 2.8 & 2 & 1.2 & 4 \\ 0.4 & 5.2 & 2.8 & -2 & - 2.4 & - 8.8 \\ - 0.5 & - 6.5 & - 3.5 & 2.5 & 0.5 & 1 \\ - 0.1 & - 1.3 & - 0.7 & 0.5 & - 0.7 & -3 \end{array}\right) \sim \left(\begin{array}{rrrrrr}- 0.5 & - 6.5 & - 3.5 & 2.5 & 0.5 & 1 \\ 0 & 0 & 0 & 0 & -2 & -8 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-13 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-7 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}5 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-2 \\ 0 \\ 0 \\ 0 \\ -4 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}0 \\ - 1.2 \\ -1 \\ -1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-13 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-7 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}5 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}-2 \\ 0 \\ 0 \\ 0 \\ -4 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 0.4 \\ - 1.6 \\ - 0.5 \\ - 0.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}-13 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-7 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}5 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}-2 \\ 0 \\ 0 \\ 0 \\ -4 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}- 0.4 & 4.4 & - 4.2 & - 1.8 & 3.6 & - 1.6 \\ 0.2 & - 2.2 & 2.1 & 0.9 & - 0.8 & 3.2 \\ 0.2 & - 2.2 & 2.1 & 0.9 & - 3.8 & -4 \\ 0.2 & - 2.2 & 2.1 & 0.9 & - 2.8 & - 1.6 \end{array}\right) \sim \left(\begin{array}{rrrrrr}- 0.4 & 4.4 & - 4.2 & - 1.8 & 3.6 & - 1.6 \\ 0 & 0 & 0 & 0 & -2 & - 4.8 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}11 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 10.5 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 4.5 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 25.6 \\ 0 \\ 0 \\ 0 \\ - 2.4 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}2 \\ 0 \\ -3 \\ -2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}11 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 10.5 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 4.5 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 25.6 \\ 0 \\ 0 \\ 0 \\ - 2.4 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 2.4 \\ - 0.2 \\ - 3.2 \\ - 2.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}11 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 10.5 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 4.5 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 25.6 \\ 0 \\ 0 \\ 0 \\ - 2.4 \\ 1 \end{array}\right)

  v v v v  
* * * * * *
0 0 0 0 0 *
0 0 0 0 0 0
0 0 0 0 0 0

Example matrix

A = \left(\begin{array}{rrrrrr} 0.8 & 0 & 4 & - 1.6 & 6.4 & 2.2 \\ 1 & 0 & 5 & -2 & 8 & 4 \\ - 0.4 & 0 & -2 & 0.8 & - 3.2 & - 1.8 \\ - 0.3 & 0 & - 1.5 & 0.6 & - 2.4 & - 2.1 \end{array}\right) \sim \left(\begin{array}{rrrrrr}1 & 0 & 5 & -2 & 8 & 4 \\ 0 & 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}-8 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}1 \\ 0 \\ 0.2 \\ 0.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}-8 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 0.2 \\ -1 \\ 0.6 \\ 1.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-5 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}-8 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr} 0.8 & 0 & - 1.6 & - 1.2 & - 2.4 & 1 \\ -2 & 0 & 4 & 3 & 6 & -6 \\ - 0.4 & 0 & 0.8 & 0.6 & 1.2 & 0.8 \\ 0.6 & 0 & - 1.2 & - 0.9 & - 1.8 & 3.4 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-2 & 0 & 4 & 3 & 6 & -6 \\ 0 & 0 & 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 1.5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 1.4 \\ 0 \\ -2 \\ - 1.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 1.5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 0.6 \\ 2 \\ - 1.6 \\ - 2.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}2 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 1.5 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr} 0.3 & 2.1 & 1.2 & 5.4 & 3.9 & - 4.4 \\ - 0.4 & - 2.8 & - 1.6 & - 7.2 & - 5.2 & 14.1 \\ - 0.5 & - 3.5 & -2 & -9 & - 6.5 & 9.5 \\ 0.1 & 0.7 & 0.4 & 1.8 & 1.3 & 2 \end{array}\right) \sim \left(\begin{array}{rrrrrr}- 0.5 & - 3.5 & -2 & -9 & - 6.5 & 9.5 \\ 0 & 0 & 0 & 0 & 0 & 6.5 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-7 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-4 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-18 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-13 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 3.2 \\ 12.5 \\ 7.5 \\ 2.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}-7 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-4 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-18 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}-13 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 3.5 \\ 12.9 \\ 8 \\ 2.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}-7 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-4 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}-18 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}-13 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}1 & -2 & 0.2 & 0.4 & - 8.8 & - 6.2 \\ 0.5 & -1 & 0.1 & 0.2 & - 4.4 & 3.5 \\ - 0.5 & 1 & - 0.1 & - 0.2 & 4.4 & - 0.2 \\ 0.5 & -1 & 0.1 & 0.2 & - 4.4 & 0.2 \end{array}\right) \sim \left(\begin{array}{rrrrrr}1 & -2 & 0.2 & 0.4 & - 8.8 & - 6.2 \\ 0 & 0 & 0 & 0 & 0 & 6.6 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}2 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 0.2 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 0.4 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 8.8 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 4.2 \\ 4.5 \\ - 1.2 \\ 1.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}2 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 0.2 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 0.4 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 8.8 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 3.2 \\ 5 \\ - 1.7 \\ 1.7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}2 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 0.2 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 0.4 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 8.8 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

4 \times 6 with five free variables

  v v v v v
* * * * * *
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0

Example matrix

A = \left(\begin{array}{rrrrrr}- 0.6 & 0.3 & - 2.4 & - 2.1 & 0.6 & 2.4 \\ 2 & -1 & 8 & 7 & -2 & -8 \\ - 1.4 & 0.7 & - 5.6 & - 4.9 & 1.4 & 5.6 \\ 1 & - 0.5 & 4 & 3.5 & -1 & -4 \end{array}\right) \sim \left(\begin{array}{rrrrrr}2 & -1 & 8 & 7 & -2 & -8 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.5 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-4 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 3.5 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}1 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}4 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.6 \\ 2 \\ - 1.4 \\ 1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 0.5 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-4 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 3.5 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}1 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}4 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.2 \\ 4 \\ - 2.8 \\ 2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 0.5 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-4 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r}- 3.5 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}1 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}4 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}- 1.4 & 2.8 & - 5.6 & 0.7 & 4.2 & - 3.5 \\ 2 & -4 & 8 & -1 & -6 & 5 \\ - 0.8 & 1.6 & - 3.2 & 0.4 & 2.4 & -2 \\ - 1.6 & 3.2 & - 6.4 & 0.8 & 4.8 & -4 \end{array}\right) \sim \left(\begin{array}{rrrrrr}2 & -4 & 8 & -1 & -6 & 5 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}2 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-4 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 0.5 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}- 2.5 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.4 \\ 2 \\ - 0.8 \\ - 1.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}2 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-4 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 0.5 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 2.5 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 2.8 \\ 4 \\ - 1.6 \\ - 3.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}2 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-4 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 0.5 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}- 2.5 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}-1 & - 1.6 & 1.1 & 1.8 & - 0.5 & 0.1 \\ -2 & - 3.2 & 2.2 & 3.6 & -1 & 0.2 \\ -5 & -8 & 5.5 & 9 & - 2.5 & 0.5 \\ 1 & 1.6 & - 1.1 & - 1.8 & 0.5 & - 0.1 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-5 & -8 & 5.5 & 9 & - 2.5 & 0.5 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 1.6 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 1.1 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 1.8 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 0.5 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 0.1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-1 \\ -2 \\ -5 \\ 1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 1.6 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 1.1 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 1.8 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 0.5 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 0.1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-2 \\ -4 \\ -10 \\ 2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 1.6 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 1.1 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 1.8 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r}- 0.5 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r} 0.1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrrrrr}- 0.5 & 4 & - 0.4 & 0.6 & 1.6 & - 3.5 \\ 0.5 & -4 & 0.4 & - 0.6 & - 1.6 & 3.5 \\ -1 & 8 & - 0.8 & 1.2 & 3.2 & -7 \\ 0.5 & -4 & 0.4 & - 0.6 & - 1.6 & 3.5 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-1 & 8 & - 0.8 & 1.2 & 3.2 & -7 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}8 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 0.8 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 1.2 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 3.2 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}-7 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.5 \\ 0.5 \\ -1 \\ 0.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}8 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 0.8 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 1.2 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 3.2 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}-7 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-1 \\ 1 \\ -2 \\ 1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}8 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}- 0.8 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right) + u_4 \left(\begin{array}{r} 1.2 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array}\right) + u_5 \left(\begin{array}{r} 3.2 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right) + u_6 \left(\begin{array}{r}-7 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}\right)

5 \times 2

  1. 5 \times 2 with 0 free variables
  2. 5 \times 2 with 1 free variables

5 \times 2 with no free variables

   
* *
0 *
0 0
0 0
0 0

Example matrix

A = \left(\begin{array}{rr}6 & 5 \\ 5.4 & 2 \\ 2.4 & -3 \\ 0.6 & -1 \\ - 5.4 & -3 \end{array}\right) \sim \left(\begin{array}{rr}6 & 5 \\ 0 & -5 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}\right) , \textrm{null}(A) = \{\textbf{0}\}

Example solution

\textbf{b} = \left(\begin{array}{r}1 \\ 3.4 \\ 5.4 \\ 1.6 \\ - 2.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ -1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-4 \\ 1.4 \\ 8.4 \\ 2.6 \\ 0.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ -2 \end{array}\right)

Example matrix

A = \left(\begin{array}{rr}- 2.1 & - 1.6 \\ 2.8 & 1.9 \\ 7 & 6 \\ 3.5 & 4 \\ 5.6 & 4.2 \end{array}\right) \sim \left(\begin{array}{rr}7 & 6 \\ 0 & 1 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}\right) , \textrm{null}(A) = \{\textbf{0}\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.5 \\ 0.9 \\ 1 \\ - 0.5 \\ 1.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ -1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-1 \\ 1.8 \\ 2 \\ -1 \\ 2.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ -2 \end{array}\right)

Example matrix

A = \left(\begin{array}{rr} 1.8 & 2.1 \\ - 4.5 & - 8.5 \\ 3.6 & 0.3 \\ 0.9 & - 2.2 \\ - 1.8 & 0.5 \end{array}\right) \sim \left(\begin{array}{rr}- 4.5 & - 8.5 \\ 0 & - 6.5 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}\right) , \textrm{null}(A) = \{\textbf{0}\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.3 \\ 4 \\ 3.3 \\ 3.1 \\ - 2.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ -1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 1.5 \\ - 0.5 \\ 6.9 \\ 4 \\ - 4.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ -1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rr}- 1.7 & - 2.1 \\ 3.4 & - 1.8 \\ - 1.7 & 3.9 \\ 1.7 & 5.1 \\ - 1.7 & - 2.1 \end{array}\right) \sim \left(\begin{array}{rr} 3.4 & - 1.8 \\ 0 & 6 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}\right) , \textrm{null}(A) = \{\textbf{0}\}

Example solution

\textbf{b} = \left(\begin{array}{r} 0.4 \\ 5.2 \\ - 5.6 \\ - 3.4 \\ 0.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ -1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.3 \\ 8.6 \\ - 7.3 \\ - 1.7 \\ - 1.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ -1 \end{array}\right)

5 \times 2 with one free variable

  v
* *
0 0
0 0
0 0
0 0

Example matrix

A = \left(\begin{array}{rr}- 4.5 & - 0.9 \\ 5 & 1 \\ 3 & 0.6 \\ 0.5 & 0.1 \\ -2 & - 0.4 \end{array}\right) \sim \left(\begin{array}{rr}5 & 1 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 0.2 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 4.5 \\ 5 \\ 3 \\ 0.5 \\ -2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 0.2 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-9 \\ 10 \\ 6 \\ 1 \\ -4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 0.2 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rr}-5 & 3 \\ - 3.5 & 2.1 \\ 1 & - 0.6 \\ 3 & - 1.8 \\ 4 & - 2.4 \end{array}\right) \sim \left(\begin{array}{rr}-5 & 3 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.6 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-5 \\ - 3.5 \\ 1 \\ 3 \\ 4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 0.6 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-10 \\ -7 \\ 2 \\ 6 \\ 8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 0.6 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rr}- 0.5 & - 1.4 \\ - 2.5 & -7 \\ 2 & 5.6 \\ - 1.5 & - 4.2 \\ -1 & - 2.8 \end{array}\right) \sim \left(\begin{array}{rr}- 2.5 & -7 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 2.8 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.5 \\ - 2.5 \\ 2 \\ - 1.5 \\ -1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 2.8 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}-1 \\ -5 \\ 4 \\ -3 \\ -2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r}- 2.8 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rr} 0.5 & - 1.1 \\ 1 & - 2.2 \\ - 0.5 & 1.1 \\ 0.5 & - 1.1 \\ - 0.5 & 1.1 \end{array}\right) \sim \left(\begin{array}{rr}1 & - 2.2 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 2.2 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 0.5 \\ 1 \\ - 0.5 \\ 0.5 \\ - 0.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 2.2 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}1 \\ 2 \\ -1 \\ 1 \\ -1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{r} 2.2 \\ 1 \end{array}\right)

5 \times 3

  1. 5 \times 3 with 0 free variables
  2. 5 \times 3 with 1 free variables
  3. 5 \times 3 with 2 free variables

5 \times 3 with no free variables

     
* * *
0 * *
0 0 *
0 0 0
0 0 0

Example matrix

A = \left(\begin{array}{rrr}-1 & -4 & - 8.5 \\ 2 & -2 & 3 \\ 0.8 & - 0.3 & - 4.1 \\ - 0.4 & 4.4 & - 0.4 \\ 1.4 & 0.6 & 0.1 \end{array}\right) \sim \left(\begin{array}{rrr}2 & -2 & 3 \\ 0 & -5 & -7 \\ 0 & 0 & -6 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \{\textbf{0}\}

Example solution

\textbf{b} = \left(\begin{array}{r} 2.5 \\ -1 \\ 5.4 \\ 4 \\ 3.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 1 \\ -1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 1.5 \\ 1 \\ 6.2 \\ 3.6 \\ 4.7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 1 \\ -1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrr}- 1.8 & - 1.2 & 7.4 \\ 6.3 & - 4.5 & - 7.5 \\ - 6.3 & 3.9 & 8.9 \\ 9 & -9 & -7 \\ 5.4 & -6 & - 3.3 \end{array}\right) \sim \left(\begin{array}{rrr}9 & -9 & -7 \\ 0 & -3 & 6 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \{\textbf{0}\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.4 \\ 2.4 \\ - 2.2 \\ 2 \\ 0.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 2 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 3.4 \\ 4.2 \\ - 4.6 \\ 2 \\ 0.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 3 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrr}- 1.2 & - 0.8 & 9.2 \\ 2.4 & 4.4 & 3.6 \\ - 3.6 & - 5.6 & 6.1 \\ -6 & -9 & 8.5 \\ 2.4 & 3 & - 9.9 \end{array}\right) \sim \left(\begin{array}{rrr}-6 & -9 & 8.5 \\ 0 & 1 & 7.5 \\ 0 & 0 & 2.5 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \{\textbf{0}\}

Example solution

\textbf{b} = \left(\begin{array}{r}6 \\ 4.4 \\ 2.9 \\ 2.5 \\ - 6.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -2 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 6.4 \\ 6.4 \\ 0.9 \\ - 0.5 \\ - 5.7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ -1 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrr}- 2.8 & - 0.4 & 1.7 \\ 2.8 & 5.2 & - 5.5 \\ 5.6 & 4 & 5.8 \\ - 2.8 & - 0.4 & - 11.5 \\ 2.8 & 3.6 & 3.1 \end{array}\right) \sim \left(\begin{array}{rrr} 5.6 & 4 & 5.8 \\ 0 & 3.2 & - 8.4 \\ 0 & 0 & 8.8 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \{\textbf{0}\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 6.5 \\ 0.7 \\ - 2.6 \\ 6.7 \\ - 4.7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ -2 \\ -1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 9.3 \\ 3.5 \\ 3 \\ 3.9 \\ - 1.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ -2 \\ -1 \end{array}\right)

5 \times 3 with one free variable

    v
* * *
0 * *
0 0 0
0 0 0
0 0 0

Example matrix

A = \left(\begin{array}{rrr} 0.8 & - 3.3 & - 3.6 \\ 1.4 & - 0.4 & -2 \\ - 0.6 & 7.1 & 6.4 \\ -2 & 7 & 8 \\ - 0.2 & - 0.3 & 0 \end{array}\right) \sim \left(\begin{array}{rrr}-2 & 7 & 8 \\ 0 & 5 & 4 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 1.2 \\ - 0.8 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.9 \\ 3.8 \\ 5.3 \\ 1 \\ - 0.9 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 1.2 \\ - 0.8 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.1 \\ 5.2 \\ 4.7 \\ -1 \\ - 1.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 1.2 \\ - 0.8 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrr}-2 & 6 & -4 \\ - 1.8 & 9.4 & - 3.6 \\ 1 & -1 & 2 \\ 0.8 & - 0.8 & 1.6 \\ 0.4 & -4 & 0.8 \end{array}\right) \sim \left(\begin{array}{rrr}-2 & 6 & -4 \\ 0 & 4 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-2 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-2 \\ 2.2 \\ 3 \\ 2.4 \\ - 2.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-2 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}0 \\ 4 \\ 2 \\ 1.6 \\ - 2.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-2 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrr}- 0.8 & -5 & 3.8 \\ 2 & 0 & 8 \\ - 1.6 & -1 & -5 \\ - 0.4 & -4 & 4 \\ 0.8 & 3 & -1 \end{array}\right) \sim \left(\begin{array}{rrr}2 & 0 & 8 \\ 0 & -5 & 7 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-4 \\ 1.4 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 4.2 \\ 2 \\ - 0.6 \\ 3.6 \\ - 2.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}1 \\ -1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-4 \\ 1.4 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 3.4 \\ 4 \\ - 2.2 \\ 3.2 \\ - 1.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ -1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r}-4 \\ 1.4 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrr} 0.4 & 1.4 & 0 \\ 0.2 & 1.2 & 2.9 \\ - 0.2 & - 1.7 & - 5.8 \\ 0.2 & 0.2 & - 2.9 \\ - 0.2 & - 1.2 & - 2.9 \end{array}\right) \sim \left(\begin{array}{rrr} 0.4 & 1.4 & 0 \\ 0 & -1 & - 5.8 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 20.3 \\ - 5.8 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 0.2 \\ - 0.4 \\ 0.9 \\ 0.6 \\ 0.4 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ -1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 20.3 \\ - 5.8 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.2 \\ - 0.6 \\ 1.1 \\ 0.4 \\ 0.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ -1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{r} 20.3 \\ - 5.8 \\ 1 \end{array}\right)

  v  
* * *
0 0 *
0 0 0
0 0 0
0 0 0

Example matrix

A = \left(\begin{array}{rrr} 1.4 & - 5.6 & 3.1 \\ 0.2 & - 0.8 & - 0.3 \\ 1 & -4 & 6.5 \\ 2 & -8 & 5 \\ 0.6 & - 2.4 & 3.1 \end{array}\right) \sim \left(\begin{array}{rrr}2 & -8 & 5 \\ 0 & 0 & 4 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}4 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 1.1 \\ 0.9 \\ - 3.5 \\ 1 \\ - 1.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}4 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 2.5 \\ 1.1 \\ - 2.5 \\ 3 \\ - 0.7 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}4 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrr} 0.5 & - 2.5 & 6 \\ 0.6 & -3 & - 0.8 \\ 1 & -5 & 2 \\ - 0.5 & 2.5 & - 5.5 \\ 0.3 & - 1.5 & 0.1 \end{array}\right) \sim \left(\begin{array}{rrr}1 & -5 & 2 \\ 0 & 0 & 5 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}5 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}-4 \\ 3.2 \\ 2 \\ 3.5 \\ 1.1 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}5 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 4.5 \\ 2.6 \\ 1 \\ 4 \\ 0.8 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ -1 \end{array}\right) + u_2 \left(\begin{array}{r}5 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrr} 0.8 & 0.4 & - 3.6 \\ - 1.2 & - 0.6 & 4.2 \\ -2 & -1 & 6 \\ 0.4 & 0.2 & 0.3 \\ - 1.6 & - 0.8 & 5.1 \end{array}\right) \sim \left(\begin{array}{rrr}-2 & -1 & 6 \\ 0 & 0 & 1.5 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 0.5 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r}- 1.2 \\ 0.6 \\ 0 \\ 1.5 \\ 0.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}- 0.5 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r}- 0.4 \\ - 0.6 \\ -2 \\ 1.9 \\ - 1.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}4 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}- 0.5 \\ 1 \\ 0 \end{array}\right)

Example matrix

A = \left(\begin{array}{rrr}-2 & - 2.2 & 10.7 \\ -2 & - 2.2 & - 3.4 \\ -2 & - 2.2 & 6 \\ -4 & - 4.4 & 2.6 \\ -2 & - 2.2 & 6 \end{array}\right) \sim \left(\begin{array}{rrr}-4 & - 4.4 & 2.6 \\ 0 & 0 & 9.4 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 1.1 \\ 1 \\ 0 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{r} 6.7 \\ - 7.4 \\ 2 \\ - 5.4 \\ 2 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}2 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}- 1.1 \\ 1 \\ 0 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{r} 4.7 \\ - 9.4 \\ 0 \\ - 9.4 \\ 0 \end{array}\right) and \textbf{u} = \left(\begin{array}{r}3 \\ 0 \\ 1 \end{array}\right) + u_2 \left(\begin{array}{r}- 1.1 \\ 1 \\ 0 \end{array}\right)

5 \times 3 with two free variables

  v v
* * *
0 0 0
0 0 0
0 0 0
0 0 0

Example matrix

A = \left(\begin{array}{ccc}- 1.2 & 0.6 & 3.6 \\ - 0.8 & 0.4 & 2.4 \\ -2 & 1 & 6 \\ 0.8 & - 0.4 & - 2.4 \\ - 0.6 & 0.3 & 1.8 \end{array}\right) \sim \left(\begin{array}{ccc}-2 & 1 & 6 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{c} 0.5 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{c}3 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{c}- 1.2 \\ - 0.8 \\ -2 \\ 0.8 \\ - 0.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{c}1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{c} 0.5 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{c}3 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{c}- 2.4 \\ - 1.6 \\ -4 \\ 1.6 \\ - 1.2 \end{array}\right) and \textbf{u} = \left(\begin{array}{c}2 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{c} 0.5 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{c}3 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{ccc} 0.1 & - 0.1 & 0.5 \\ 0.3 & - 0.3 & 1.5 \\ 0.9 & - 0.9 & 4.5 \\ 1 & -1 & 5 \\ - 0.5 & 0.5 & - 2.5 \end{array}\right) \sim \left(\begin{array}{ccc}1 & -1 & 5 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{c}1 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{c}-5 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{c} 0.1 \\ 0.3 \\ 0.9 \\ 1 \\ - 0.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{c}1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{c}1 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{c}-5 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{c} 0.2 \\ 0.6 \\ 1.8 \\ 2 \\ -1 \end{array}\right) and \textbf{u} = \left(\begin{array}{c}2 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{c}1 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{c}-5 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{ccc}- 0.2 & 3 & - 2.8 \\ 0.2 & -3 & 2.8 \\ 0.5 & - 7.5 & 7 \\ 0.1 & - 1.5 & 1.4 \\ - 0.3 & 4.5 & - 4.2 \end{array}\right) \sim \left(\begin{array}{ccc} 0.5 & - 7.5 & 7 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{c}15 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{c}-14 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{c}- 0.2 \\ 0.2 \\ 0.5 \\ 0.1 \\ - 0.3 \end{array}\right) and \textbf{u} = \left(\begin{array}{c}1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{c}15 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{c}-14 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{c}- 0.4 \\ 0.4 \\ 1 \\ 0.2 \\ - 0.6 \end{array}\right) and \textbf{u} = \left(\begin{array}{c}2 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{c}15 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{c}-14 \\ 0 \\ 1 \end{array}\right)

Example matrix

A = \left(\begin{array}{ccc}1 & - 0.4 & 3.4 \\ 0.5 & - 0.2 & 1.7 \\ - 0.5 & 0.2 & - 1.7 \\ 0.5 & - 0.2 & 1.7 \\ - 0.5 & 0.2 & - 1.7 \end{array}\right) \sim \left(\begin{array}{ccc}1 & - 0.4 & 3.4 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) , \textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{c} 0.4 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{c}- 3.4 \\ 0 \\ 1 \end{array}\right)\right\}

Example solution

\textbf{b} = \left(\begin{array}{c}1 \\ 0.5 \\ - 0.5 \\ 0.5 \\ - 0.5 \end{array}\right) and \textbf{u} = \left(\begin{array}{c}1 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{c} 0.4 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{c}- 3.4 \\ 0 \\ 1 \end{array}\right)

Example solution

\textbf{b} = \left(\begin{array}{c}2 \\ 1 \\ -1 \\ 1 \\ -1 \end{array}\right) and \textbf{u} = \left(\begin{array}{c}2 \\ 0 \\ 0 \end{array}\right) + u_2 \left(\begin{array}{c} 0.4 \\ 1 \\ 0 \end{array}\right) + u_3 \left(\begin{array}{c}- 3.4 \\ 0 \\ 1 \end{array}\right)

5 \times 4

  1. 5 \times 4 with 0 free variables
  2. 5 \times 4 with 1 free variables
  3. 5 \times 4 with 2 free variables
  4. 5 \times 4 with 3 free variables

5 \times 4 with no free variables

       
* * * *
0 * * *
0 0 * *
0 0 0 *
0 0 0 0

5 \times 4 with one free variable

      v
* * * *
0 * * *
0 0 * *
0 0 0 0
0 0 0 0
    v  
* * * *
0 * * *
0 0 0 *
0 0 0 0
0 0 0 0
  v    
* * * *
0 0 * *
0 0 0 *
0 0 0 0
0 0 0 0

5 \times 4 with two free variables

    v v
* * * *
0 * * *
0 0 0 0
0 0 0 0
0 0 0 0
  v   v
* * * *
0 0 * *
0 0 0 0
0 0 0 0
0 0 0 0
  v v  
* * * *
0 0 0 *
0 0 0 0
0 0 0 0
0 0 0 0

5 \times 4 with three free variables

  v v v
* * * *
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0

5 \times 5

  1. 5 \times 5 with 0 free variables
  2. 5 \times 5 with 1 free variables
  3. 5 \times 5 with 2 free variables
  4. 5 \times 5 with 3 free variables
  5. 5 \times 5 with 4 free variables

5 \times 5 with no free variables

         
* * * * *
0 * * * *
0 0 * * *
0 0 0 * *
0 0 0 0 *

5 \times 5 with one free variable

        v
* * * * *
0 * * * *
0 0 * * *
0 0 0 * *
0 0 0 0 0
      v  
* * * * *
0 * * * *
0 0 * * *
0 0 0 0 *
0 0 0 0 0
    v    
* * * * *
0 * * * *
0 0 0 * *
0 0 0 0 *
0 0 0 0 0
  v      
* * * * *
0 0 * * *
0 0 0 * *
0 0 0 0 *
0 0 0 0 0

5 \times 5 with two free variables

      v v
* * * * *
0 * * * *
0 0 * * *
0 0 0 0 0
0 0 0 0 0
    v   v
* * * * *
0 * * * *
0 0 0 * *
0 0 0 0 0
0 0 0 0 0
    v v  
* * * * *
0 * * * *
0 0 0 0 *
0 0 0 0 0
0 0 0 0 0
  v     v
* * * * *
0 0 * * *
0 0 0 * *
0 0 0 0 0
0 0 0 0 0
  v   v  
* * * * *
0 0 * * *
0 0 0 0 *
0 0 0 0 0
0 0 0 0 0
  v v    
* * * * *
0 0 0 * *
0 0 0 0 *
0 0 0 0 0
0 0 0 0 0

5 \times 5 with three free variables

    v v v
* * * * *
0 * * * *
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
  v   v v
* * * * *
0 0 * * *
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
  v v   v
* * * * *
0 0 0 * *
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
  v v v  
* * * * *
0 0 0 0 *
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0

5 \times 5 with four free variables

  v v v v
* * * * *
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0

5 \times 6

  1. 5 \times 6 with 1 free variables
  2. 5 \times 6 with 2 free variables
  3. 5 \times 6 with 3 free variables
  4. 5 \times 6 with 4 free variables
  5. 5 \times 6 with 5 free variables

5 \times 6 with one free variable

          v
* * * * * *
0 * * * * *
0 0 * * * *
0 0 0 * * *
0 0 0 0 * *
        v  
* * * * * *
0 * * * * *
0 0 * * * *
0 0 0 * * *
0 0 0 0 0 *
      v    
* * * * * *
0 * * * * *
0 0 * * * *
0 0 0 0 * *
0 0 0 0 0 *
    v      
* * * * * *
0 * * * * *
0 0 0 * * *
0 0 0 0 * *
0 0 0 0 0 *
  v        
* * * * * *
0 0 * * * *
0 0 0 * * *
0 0 0 0 * *
0 0 0 0 0 *

5 \times 6 with two free variables

        v v
* * * * * *
0 * * * * *
0 0 * * * *
0 0 0 * * *
0 0 0 0 0 0
      v   v
* * * * * *
0 * * * * *
0 0 * * * *
0 0 0 0 * *
0 0 0 0 0 0
      v v  
* * * * * *
0 * * * * *
0 0 * * * *
0 0 0 0 0 *
0 0 0 0 0 0
    v     v
* * * * * *
0 * * * * *
0 0 0 * * *
0 0 0 0 * *
0 0 0 0 0 0
    v   v  
* * * * * *
0 * * * * *
0 0 0 * * *
0 0 0 0 0 *
0 0 0 0 0 0
    v v    
* * * * * *
0 * * * * *
0 0 0 0 * *
0 0 0 0 0 *
0 0 0 0 0 0
  v       v
* * * * * *
0 0 * * * *
0 0 0 * * *
0 0 0 0 * *
0 0 0 0 0 0
  v     v  
* * * * * *
0 0 * * * *
0 0 0 * * *
0 0 0 0 0 *
0 0 0 0 0 0
  v   v    
* * * * * *
0 0 * * * *
0 0 0 0 * *
0 0 0 0 0 *
0 0 0 0 0 0
  v v      
* * * * * *
0 0 0 * * *
0 0 0 0 * *
0 0 0 0 0 *
0 0 0 0 0 0

5 \times 6 with three free variables

      v v v
* * * * * *
0 * * * * *
0 0 * * * *
0 0 0 0 0 0
0 0 0 0 0 0
    v   v v
* * * * * *
0 * * * * *
0 0 0 * * *
0 0 0 0 0 0
0 0 0 0 0 0
    v v   v
* * * * * *
0 * * * * *
0 0 0 0 * *
0 0 0 0 0 0
0 0 0 0 0 0
    v v v  
* * * * * *
0 * * * * *
0 0 0 0 0 *
0 0 0 0 0 0
0 0 0 0 0 0
  v     v v
* * * * * *
0 0 * * * *
0 0 0 * * *
0 0 0 0 0 0
0 0 0 0 0 0
  v   v   v
* * * * * *
0 0 * * * *
0 0 0 0 * *
0 0 0 0 0 0
0 0 0 0 0 0
  v   v v  
* * * * * *
0 0 * * * *
0 0 0 0 0 *
0 0 0 0 0 0
0 0 0 0 0 0
  v v     v
* * * * * *
0 0 0 * * *
0 0 0 0 * *
0 0 0 0 0 0
0 0 0 0 0 0
  v v   v  
* * * * * *
0 0 0 * * *
0 0 0 0 0 *
0 0 0 0 0 0
0 0 0 0 0 0
  v v v    
* * * * * *
0 0 0 0 * *
0 0 0 0 0 *
0 0 0 0 0 0
0 0 0 0 0 0

5 \times 6 with four free variables

    v v v v
* * * * * *
0 * * * * *
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
  v   v v v
* * * * * *
0 0 * * * *
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
  v v   v v
* * * * * *
0 0 0 * * *
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
  v v v   v
* * * * * *
0 0 0 0 * *
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
  v v v v  
* * * * * *
0 0 0 0 0 *
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0

5 \times 6 with five free variables

  v v v v v
* * * * * *
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0

6 \times 2

  1. 6 \times 2 with 0 free variables
  2. 6 \times 2 with 1 free variables

6 \times 2 with no free variables

   
* *
0 *
0 0
0 0
0 0
0 0

6 \times 2 with one free variable

  v
* *
0 0
0 0
0 0
0 0
0 0

6 \times 3

  1. 6 \times 3 with 0 free variables
  2. 6 \times 3 with 1 free variables
  3. 6 \times 3 with 2 free variables

6 \times 3 with no free variables

     
* * *
0 * *
0 0 *
0 0 0
0 0 0
0 0 0

6 \times 3 with one free variable

    v
* * *
0 * *
0 0 0
0 0 0
0 0 0
0 0 0
  v  
* * *
0 0 *
0 0 0
0 0 0
0 0 0
0 0 0

6 \times 3 with two free variables

  v v
* * *
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0

6 \times 4

  1. 6 \times 4 with 0 free variables
  2. 6 \times 4 with 1 free variables
  3. 6 \times 4 with 2 free variables
  4. 6 \times 4 with 3 free variables

6 \times 4 with no free variables

       
* * * *
0 * * *
0 0 * *
0 0 0 *
0 0 0 0
0 0 0 0

6 \times 4 with one free variable

      v
* * * *
0 * * *
0 0 * *
0 0 0 0
0 0 0 0
0 0 0 0
    v  
* * * *
0 * * *
0 0 0 *
0 0 0 0
0 0 0 0
0 0 0 0
  v    
* * * *
0 0 * *
0 0 0 *
0 0 0 0
0 0 0 0
0 0 0 0

6 \times 4 with two free variables

    v v
* * * *
0 * * *
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
  v   v
* * * *
0 0 * *
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
  v v  
* * * *
0 0 0 *
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0

6 \times 4 with three free variables

  v v v
* * * *
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0

6 \times 5

  1. 6 \times 5 with 0 free variables
  2. 6 \times 5 with 1 free variables
  3. 6 \times 5 with 2 free variables
  4. 6 \times 5 with 3 free variables
  5. 6 \times 5 with 4 free variables

6 \times 5 with no free variables

         
* * * * *
0 * * * *
0 0 * * *
0 0 0 * *
0 0 0 0 *
0 0 0 0 0

6 \times 5 with one free variable

        v
* * * * *
0 * * * *
0 0 * * *
0 0 0 * *
0 0 0 0 0
0 0 0 0 0
      v  
* * * * *
0 * * * *
0 0 * * *
0 0 0 0 *
0 0 0 0 0
0 0 0 0 0
    v    
* * * * *
0 * * * *
0 0 0 * *
0 0 0 0 *
0 0 0 0 0
0 0 0 0 0
  v      
* * * * *
0 0 * * *
0 0 0 * *
0 0 0 0 *
0 0 0 0 0
0 0 0 0 0

6 \times 5 with two free variables

      v v
* * * * *
0 * * * *
0 0 * * *
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
    v   v
* * * * *
0 * * * *
0 0 0 * *
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
    v v  
* * * * *
0 * * * *
0 0 0 0 *
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
  v     v
* * * * *
0 0 * * *
0 0 0 * *
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
  v   v  
* * * * *
0 0 * * *
0 0 0 0 *
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
  v v    
* * * * *
0 0 0 * *
0 0 0 0 *
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0

6 \times 5 with three free variables

    v v v
* * * * *
0 * * * *
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
  v   v v
* * * * *
0 0 * * *
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
  v v   v
* * * * *
0 0 0 * *
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
  v v v  
* * * * *
0 0 0 0 *
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0

6 \times 5 with four free variables

  v v v v
* * * * *
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0

6 \times 6

  1. 6 \times 6 with 0 free variables
  2. 6 \times 6 with 1 free variables
  3. 6 \times 6 with 2 free variables
  4. 6 \times 6 with 3 free variables
  5. 6 \times 6 with 4 free variables
  6. 6 \times 6 with 5 free variables

6 \times 6 with no free variables

           
* * * * * *
0 * * * * *
0 0 * * * *
0 0 0 * * *
0 0 0 0 * *
0 0 0 0 0 *

6 \times 6 with one free variable

          v
* * * * * *
0 * * * * *
0 0 * * * *
0 0 0 * * *
0 0 0 0 * *
0 0 0 0 0 0
        v  
* * * * * *
0 * * * * *
0 0 * * * *
0 0 0 * * *
0 0 0 0 0 *
0 0 0 0 0 0
      v    
* * * * * *
0 * * * * *
0 0 * * * *
0 0 0 0 * *
0 0 0 0 0 *
0 0 0 0 0 0
    v      
* * * * * *
0 * * * * *
0 0 0 * * *
0 0 0 0 * *
0 0 0 0 0 *
0 0 0 0 0 0
  v        
* * * * * *
0 0 * * * *
0 0 0 * * *
0 0 0 0 * *
0 0 0 0 0 *
0 0 0 0 0 0

6 \times 6 with two free variables

        v v
* * * * * *
0 * * * * *
0 0 * * * *
0 0 0 * * *
0 0 0 0 0 0
0 0 0 0 0 0
      v   v
* * * * * *
0 * * * * *
0 0 * * * *
0 0 0 0 * *
0 0 0 0 0 0
0 0 0 0 0 0
      v v  
* * * * * *
0 * * * * *
0 0 * * * *
0 0 0 0 0 *
0 0 0 0 0 0
0 0 0 0 0 0
    v     v
* * * * * *
0 * * * * *
0 0 0 * * *
0 0 0 0 * *
0 0 0 0 0 0
0 0 0 0 0 0
    v   v  
* * * * * *
0 * * * * *
0 0 0 * * *
0 0 0 0 0 *
0 0 0 0 0 0
0 0 0 0 0 0
    v v    
* * * * * *
0 * * * * *
0 0 0 0 * *
0 0 0 0 0 *
0 0 0 0 0 0
0 0 0 0 0 0
  v       v
* * * * * *
0 0 * * * *
0 0 0 * * *
0 0 0 0 * *
0 0 0 0 0 0
0 0 0 0 0 0
  v     v  
* * * * * *
0 0 * * * *
0 0 0 * * *
0 0 0 0 0 *
0 0 0 0 0 0
0 0 0 0 0 0
  v   v    
* * * * * *
0 0 * * * *
0 0 0 0 * *
0 0 0 0 0 *
0 0 0 0 0 0
0 0 0 0 0 0
  v v      
* * * * * *
0 0 0 * * *
0 0 0 0 * *
0 0 0 0 0 *
0 0 0 0 0 0
0 0 0 0 0 0

6 \times 6 with three free variables

      v v v
* * * * * *
0 * * * * *
0 0 * * * *
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
    v   v v
* * * * * *
0 * * * * *
0 0 0 * * *
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
    v v   v
* * * * * *
0 * * * * *
0 0 0 0 * *
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
    v v v  
* * * * * *
0 * * * * *
0 0 0 0 0 *
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
  v     v v
* * * * * *
0 0 * * * *
0 0 0 * * *
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
  v   v   v
* * * * * *
0 0 * * * *
0 0 0 0 * *
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
  v   v v  
* * * * * *
0 0 * * * *
0 0 0 0 0 *
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
  v v     v
* * * * * *
0 0 0 * * *
0 0 0 0 * *
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
  v v   v  
* * * * * *
0 0 0 * * *
0 0 0 0 0 *
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
  v v v    
* * * * * *
0 0 0 0 * *
0 0 0 0 0 *
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0

6 \times 6 with four free variables

    v v v v
* * * * * *
0 * * * * *
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
  v   v v v
* * * * * *
0 0 * * * *
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
  v v   v v
* * * * * *
0 0 0 * * *
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
  v v v   v
* * * * * *
0 0 0 0 * *
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
  v v v v  
* * * * * *
0 0 0 0 0 *
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0

6 \times 6 with five free variables

  v v v v v
* * * * * *
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0