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:
- Students face O(mn) additional opportunities to make
errors that do not actually reflect their understanding
of the algorithm.
- These additional errors make it exceedingly difficult to
equitably grade a solution when an insignificant error
early on leads to significantly divergent subsequent
calculations.
- The runtime increases by O(mn).
- 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,
- 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.
- If rank(A)=m, then two examples of either one or infinitely many solutions will be given.
- 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.
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
- 2×2 with 0 free variables
- 2×2 with 1 free variables
2×2
2×2 with no free variables
* *
0 *
Example matrix
A=(96−7.2−7.8)∼(9.06.00−3.0)
, null(A)={0}
Example solution
b=(30.6) and u=(1−1)
Example solution
b=(61.2) and u=(2−2)
Example matrix
A=(−6−11.25.2)∼(−6.0−1.005.0)
, null(A)={0}
Example solution
b=(−5−4) and u=(1−1)
Example solution
b=(−4−9.2) and u=(1−2)
Example matrix
A=(−8.57.53.4−10)∼(−8.57.50−7.0)
, null(A)={0}
Example solution
b=(−9.5−3.2) and u=(21)
Example solution
b=(−10.5−9.8) and u=(32)
Example matrix
A=(−0.8−5−0.4−2.1)∼(−0.8−5.000.4)
, null(A)={0}
Example solution
b=(1.80.5) and u=(4−1)
Example solution
b=(2.60.9) and u=(3−1)
2×2 with one free variable
v
* *
0 0
Example matrix
A=(−221.4−1.4)∼(−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=(−14−0.72.8)∼(−1400)
, null(A)=span{(41)}
Example solution
b=(−1−0.7) and u=(10)+u2(41)
Example solution
b=(−2−1.4) and u=(20)+u2(41)
Example matrix
A=(−10.50.2−0.1)∼(−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=(−22.6−11.3)∼(−22.600)
, null(A)=span{(1.31)}
Example solution
b=(−2−1) and u=(10)+u2(1.31)
Example solution
b=(−4−2) and u=(20)+u2(1.31)
2×3
- 2×3 with 1 free variables
- 2×3 with 2 free variables
2×3 with one free variable
v
* * *
0 * *
Example matrix
A=(−2−8−91.85.24.1)∼(−2−8−90−2−4)
, null(A)=span{(3.5−21)}
Example solution
b=(02) and u=(4−10)+u3(3.5−21)
Example solution
b=(20.2) and u=(3−10)+u3(3.5−21)
Example matrix
A=(−440−21−2)∼(−4400−1−2)
, null(A)=span{(−2−21)}
Example solution
b=(40) and u=(120)+u3(−2−21)
Example solution
b=(4−1) and u=(230)+u3(−2−21)
Example matrix
A=(0.59.53−0.2−5.80.8)∼(0.59.530−22)
, null(A)=span{(−2511)}
Example solution
b=(−7.55) and u=(4−10)+u3(−2511)
Example solution
b=(−85.2) and u=(3−10)+u3(−2511)
Example matrix
A=(−2−8−2.413.63)∼(−2−8−2.40−0.41.8)
, null(A)=span{(−19.24.51)}
Example solution
b=(00.4) and u=(4−10)+u3(−19.24.51)
Example solution
b=(2−0.6) and u=(3−10)+u3(−19.24.51)
v
* * *
0 0 *
Example matrix
A=(−2−49−1−22.5)∼(−2.0−4.09.000−2.0)
, null(A)=span{(−210)}
Example solution
b=(1−1.5) and u=(401)+u2(−210)
Example solution
b=(3−0.5) and u=(301)+u2(−210)
Example matrix
A=(5−173.5−0.78.9)∼(5.0−1.07.0004.0)
, null(A)=span{(0.210)}
Example solution
b=(3−1.9) and u=(20−1)+u2(0.210)
Example solution
b=(−2−5.4) and u=(10−1)+u2(0.210)
Example matrix
A=(2.5−39.50.5−0.60.9)∼(2.5−3.09.500−1.0)
, null(A)=span{(1.210)}
Example solution
b=(0.51.1) and u=(40−1)+u2(1.210)
Example solution
b=(−20.6) and u=(30−1)+u2(1.210)
Example matrix
A=(−2−3.4−2.611.7−3.9)∼(−2.0−3.4−2.600−5.2)
, null(A)=span{(−1.710)}
Example solution
b=(0.64.9) and u=(10−1)+u2(−1.710)
Example solution
b=(−1.45.9) and u=(20−1)+u2(−1.710)
2×3 with two free variables
v v
* * *
0 0 0
Example matrix
A=(−1−5−6−0.1−0.5−0.6)∼(−1−5−6000)
, null(A)=span{(−510),(−601)}
Example solution
b=(−1−0.1) and u=(100)+u2(−510)+u3(−601)
Example solution
b=(−2−0.2) and u=(200)+u2(−510)+u3(−601)
Example matrix
A=(1−24−0.91.8−3.6)∼(1−24000)
, null(A)=span{(210),(−401)}
Example solution
b=(1−0.9) and u=(100)+u2(210)+u3(−401)
Example solution
b=(2−1.8) and u=(200)+u2(210)+u3(−401)
Example matrix
A=(−0.58.5−7.5−0.35.1−4.5)∼(−0.58.5−7.5000)
, null(A)=span{(1710),(−1501)}
Example solution
b=(−0.5−0.3) and u=(100)+u2(1710)+u3(−1501)
Example solution
b=(−1−0.6) and u=(200)+u2(1710)+u3(−1501)
Example matrix
A=(0.21.2−2.40.10.6−1.2)∼(0.21.2−2.4000)
, 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
- 2×4 with 2 free variables
- 2×4 with 3 free variables
2×4 with two free variables
v v
* * * *
0 * * *
Example matrix
A=(1−5−4−30.8−3−0.2−4.4)∼(1−5−4−3013−2)
, null(A)=span{(−11−310),(13201)}
Example solution
b=(−10.2) and u=(4100)+u3(−11−310)+u4(13201)
Example solution
b=(−2−0.6) and u=(3100)+u3(−11−310)+u4(13201)
Example matrix
A=(−11−3−60.64.410.811.6)∼(−11−3−60598)
, null(A)=span{(−4.8−1.810),(−7.6−1.601)}
Example solution
b=(−2−3.8) and u=(1−100)+u3(−4.8−1.810)+u4(−7.6−1.601)
Example solution
b=(−3−3.2) and u=(2−100)+u3(−4.8−1.810)+u4(−7.6−1.601)
Example matrix
A=(−0.58.5−95−0.47.3−4.78)∼(−0.58.5−9500.52.54)
, null(A)=span{(−103−510),(−126−801)}
Example solution
b=(6.55.7) and u=(4100)+u3(−103−510)+u4(−126−801)
Example solution
b=(76.1) and u=(3100)+u3(−103−510)+u4(−126−801)
Example matrix
A=(−0.29.6−69−0.15.2−0.83.3)∼(−0.29.6−6900.42.2−1.2)
, null(A)=span{(−294−5.510),(189301)}
Example solution
b=(8.84.8) and u=(4100)+u3(−294−5.510)+u4(189301)
Example solution
b=(94.9) and u=(3100)+u3(−294−5.510)+u4(189301)
v v
* * * *
0 0 * *
Example matrix
A=(2−886−0.41.60.47.8)∼(2−8860029)
, null(A)=span{(4100),(150−4.51)}
Example solution
b=(0−2) and u=(40−10)+u2(4100)+u4(150−4.51)
Example solution
b=(−2−1.6) and u=(30−10)+u2(4100)+u4(150−4.51)
Example matrix
A=(−2−309−0.2−0.3−1−1.1)∼(−2−30900−1−2)
, null(A)=span{(−1.5100),(4.50−21)}
Example solution
b=(−20.8) and u=(10−10)+u2(−1.5100)+u4(4.50−21)
Example solution
b=(−21.8) and u=(10−20)+u2(−1.5100)+u4(4.50−21)
Example matrix
A=(0.58.5−5.52−0.4−6.85.4−4.6)∼(0.58.5−5.52001−3)
, 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=(−0.2−2.8−9.2−7.80.11.43.65.5)∼(−0.2−2.8−9.2−7.800−11.6)
, null(A)=span{(−14100),(−112.601.61)}
Example solution
b=(8.4−3.2) and u=(40−10)+u2(−14100)+u4(−112.601.61)
Example solution
b=(8.6−3.3) and u=(30−10)+u2(−14100)+u4(−112.601.61)
v v
* * * *
0 0 0 *
Example matrix
A=(−267−70.4−1.2−1.4−6.6)∼(−267−7000−8)
, null(A)=span{(3100),(3.5010)}
Example solution
b=(17.8) and u=(300−1)+u2(3100)+u3(3.5010)
Example solution
b=(37.4) and u=(200−1)+u2(3100)+u3(3.5010)
Example matrix
A=(−59−7−13−5.44.26.6)∼(−59−7−10006)
, null(A)=span{(1.8100),(−1.4010)}
Example solution
b=(−4−3.6) and u=(100−1)+u2(1.8100)+u3(−1.4010)
Example solution
b=(−9−0.6) and u=(200−1)+u2(1.8100)+u3(−1.4010)
Example matrix
A=(−5−5.54.5−933.3−2.713.4)∼(−5−5.54.5−90008)
, null(A)=span{(−1.1100),(0.9010)}
Example solution
b=(−6−4.4) and u=(300−1)+u2(−1.1100)+u3(0.9010)
Example solution
b=(−1−7.4) and u=(200−1)+u2(−1.1100)+u3(0.9010)
Example matrix
A=(21.40.2−1.4−1−0.7−0.1−9.1)∼(21.40.2−1.4000−9.8)
, null(A)=span{(−0.7100),(−0.1010)}
Example solution
b=(3.48.1) and u=(100−1)+u2(−0.7100)+u3(−0.1010)
Example solution
b=(5.47.1) and u=(200−1)+u2(−0.7100)+u3(−0.1010)
2×4 with three free variables
v v v
* * * *
0 0 0 0
Example matrix
A=(−17−730.9−6.36.3−2.7)∼(−17−730000)
, 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=(2−9−7−11.8−8.1−6.3−0.9)∼(2−9−7−10000)
, 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=(0.58.5−970.23.4−3.62.8)∼(0.58.5−970000)
, 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=(2−1.4−0.64.6−10.70.3−2.3)∼(2−1.4−0.64.60000)
, null(A)=span{(0.7100),(0.3010),(−2.3001)}
Example solution
b=(2−1) and u=(1000)+u2(0.7100)+u3(0.3010)+u4(−2.3001)
Example solution
b=(4−2) and u=(2000)+u2(0.7100)+u3(0.3010)+u4(−2.3001)
3×2
- 3×2 with 0 free variables
- 3×2 with 1 free variables
3×2 with no free variables
* *
0 *
0 0
Example matrix
A=(−8−4−464−5.2)∼(−8−40800)
, null(A)={0}
Example solution
b=(−4−109.2) and u=(1−1)
Example solution
b=(−168−6.4) and u=(12)
Example matrix
A=(−44−2.4−0.6−22.6)∼(−440−300)
, null(A)={0}
Example solution
b=(4−3.63.2) and u=(12)
Example solution
b=(−4−5.4−1.4) and u=(21)
Example matrix
A=(23.5−1.2−4.6−1.6−3.3)∼(23.50−2.500)
, null(A)={0}
Example solution
b=(0.52.20.1) and u=(2−1)
Example solution
b=(2.51−1.5) and u=(3−1)
Example matrix
A=(−1.312.12.6−9.8−1.38.5)∼(2.6−9.807.200)
, null(A)={0}
Example solution
b=(6.90.63.3) and u=(41)
Example solution
b=(8.2−24.6) and u=(31)
3×2 with one free variable
v
* *
0 0
0 0
Example matrix
A=(−1.60.84−2−3.21.6)∼(4−20000)
, null(A)=span{(0.51)}
Example solution
b=(−1.64−3.2) and u=(10)+u2(0.51)
Example solution
b=(−3.28−6.4) and u=(20)+u2(0.51)
Example matrix
A=(−4−0.851−4.5−0.9)∼(510000)
, null(A)=span{(−0.21)}
Example solution
b=(−45−4.5) and u=(10)+u2(−0.21)
Example solution
b=(−810−9) and u=(20)+u2(−0.21)
Example matrix
A=(−2.55.51.5−3.31−2.2)∼(−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=(−224−42−2)∼(4−40000)
, 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=(−1.60.84−2−3.21.6)∼(4−20000)
, null(A)=span{(0.51)}
Example solution
b=(−1.64−3.2) and u=(10)+u2(0.51)
Example solution
b=(−3.28−6.4) and u=(20)+u2(0.51)
Example matrix
A=(−4−0.851−4.5−0.9)∼(510000)
, null(A)=span{(−0.21)}
Example solution
b=(−45−4.5) and u=(10)+u2(−0.21)
Example solution
b=(−810−9) and u=(20)+u2(−0.21)
Example matrix
A=(−2.55.51.5−3.31−2.2)∼(−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=(−224−42−2)∼(4−40000)
, null(A)=span{(11)}
Example solution
b=(−242) and u=(10)+u2(11)
Example solution
b=(−484) and u=(20)+u2(11)
3×3
- 3×3 with 2 free variables
- 3×3 with 1 free variables
- 3×3 with 2 free variables
3×3 with no free variables
* * *
0 * *
0 0 *
Example matrix
A=(−4.25.39.46−9−2−1.83.2−2.4)∼(6−9−20−18001)
, null(A)={0}
Example solution
b=(3.24−3.2) and u=(421)
Example solution
b=(7.4−2−1.4) and u=(321)
Example matrix
A=(6−91−1.29.86.8−3.611.84)∼(6−9108700−1)
, null(A)={0}
Example solution
b=(20.60.6) and u=(21−1)
Example solution
b=(41.21.2) and u=(42−2)
Example matrix
A=(−1−4−70.20.3−4.1−0.2−1.23.7)∼(−1−4−70−0.5−5.5009.5)
, null(A)={0}
Example solution
b=(−2−4.15.5) and u=(3−21)
Example solution
b=(−1−4.35.7) and u=(2−21)
Example matrix
A=(−3.81.22.67.6−6.84.4−3.82.3−5.8)∼(7.6−6.84.40−2.24.800−6)
, null(A)={0}
Example solution
b=(−1.4−0.8−6.5) and u=(231)
Example solution
b=(−0.2−7.6−4.2) and u=(241)
3×3 with one free variable
v
* * *
0 * *
0 0 0
Example matrix
A=(−2−86−0.6−4.4−1.2−0.2−0.60.9)∼(−2−860−2−3000)
, null(A)=span{(9−1.51)}
Example solution
b=(02−0.2) and u=(4−10)+u3(9−1.51)
Example solution
b=(22.60) and u=(3−10)+u3(9−1.51)
Example matrix
A=(0.82−0.2−1−59−0.1−0.92.3)∼(−1−590−27000)
, null(A)=span{(−8.53.51)}
Example solution
b=(1.210.5) and u=(4−10)+u3(−8.53.51)
Example solution
b=(0.420.6) and u=(3−10)+u3(−8.53.51)
Example matrix
A=(−5−14.5−40.27.6−20.24.2)∼(−5−14.5014000)
, null(A)=span{(1.7−41)}
Example solution
b=(−1−4.8−2.8) and u=(1−40)+u3(1.7−41)
Example solution
b=(−2−4.6−2.6) and u=(1−30)+u3(1.7−41)
Example matrix
A=(0.2310.3−0.4−4−2.6−0.2−1.53.2)∼(−0.4−4−2.6019000)
, null(A)=span{(83.5−91)}
Example solution
b=(−2.22.40.7) and u=(4−10)+u3(83.5−91)
Example solution
b=(−2.42.80.9) and u=(3−10)+u3(83.5−91)
v
* * *
0 0 *
0 0 0
Example matrix
A=(−4.55.47.15−612−2.42.8)∼(5−61008000)
, null(A)=span{(1.210)}
Example solution
b=(−11.64−0.8) and u=(10−1)+u2(1.210)
Example solution
b=(−1.9116.8) and u=(201)+u2(1.210)
Example matrix
A=(2791.86.35.1−1.6−5.6−8.7)∼(27900−3000)
, null(A)=span{(−3.510)}
Example solution
b=(−12.12.3) and u=(40−1)+u2(−3.510)
Example solution
b=(−30.33.9) and u=(30−1)+u2(−3.510)
Example matrix
A=(−21.4−2.3−53.5−9.52−1.45)∼(−53.5−9.5001.5000)
, null(A)=span{(0.710)}
Example solution
b=(−1.7−0.5−1) and u=(20−1)+u2(0.710)
Example solution
b=(−3.4−1−2) and u=(40−2)+u2(0.710)
Example matrix
A=(0.4−1.83.80.2−0.96.3−0.20.9−4.1)∼(0.4−1.83.8004.4000)
, null(A)=span{(4.510)}
Example solution
b=(−2.2−5.53.3) and u=(40−1)+u2(4.510)
Example solution
b=(−2.6−5.73.5) and u=(30−1)+u2(4.510)
3×3 with two free variables
v v
* * *
0 0 0
0 0 0
Example matrix
A=(−2−3−4−1.4−2.1−2.81.62.43.2)∼(−2−3−4000000)
, null(A)=span{(−1.510),(−201)}
Example solution
b=(−2−1.41.6) and u=(100)+u2(−1.510)+u3(−201)
Example solution
b=(−4−2.83.2) and u=(200)+u2(−1.510)+u3(−201)
Example matrix
A=(−2−941.46.3−2.80.20.9−0.4)∼(−2−94000000)
, 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=(2−1.415−3.52.5−42.8−2)∼(5−3.52.5000000)
, null(A)=span{(0.710),(−0.501)}
Example solution
b=(25−4) and u=(100)+u2(0.710)+u3(−0.501)
Example solution
b=(410−8) and u=(200)+u2(0.710)+u3(−0.501)
Example matrix
A=(−16.8−6.6−0.53.4−3.30.5−3.43.3)∼(−16.8−6.6000000)
, null(A)=span{(6.810),(−6.601)}
Example solution
b=(−1−0.50.5) and u=(100)+u2(6.810)+u3(−6.601)
Example solution
b=(−2−11) and u=(200)+u2(6.810)+u3(−6.601)
Example matrix
A=(−2−3−4−1.4−2.1−2.81.62.43.2)∼(−2−3−4000000)
, null(A)=span{(−1.510),(−201)}
Example solution
b=(−2−1.41.6) and u=(100)+u2(−1.510)+u3(−201)
Example solution
b=(−4−2.83.2) and u=(200)+u2(−1.510)+u3(−201)
Example matrix
A=(−2−941.46.3−2.80.20.9−0.4)∼(−2−94000000)
, 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=(2−1.415−3.52.5−42.8−2)∼(5−3.52.5000000)
, null(A)=span{(0.710),(−0.501)}
Example solution
b=(25−4) and u=(100)+u2(0.710)+u3(−0.501)
Example solution
b=(410−8) and u=(200)+u2(0.710)+u3(−0.501)
Example matrix
A=(−16.8−6.6−0.53.4−3.30.5−3.43.3)∼(−16.8−6.6000000)
, null(A)=span{(6.810),(−6.601)}
Example solution
b=(−1−0.50.5) and u=(100)+u2(6.810)+u3(−6.601)
Example solution
b=(−2−11) and u=(200)+u2(6.810)+u3(−6.601)
3×4
- 3×4 with 1 free variables
- 3×4 with 2 free variables
- 3×4 with 3 free variables
3×4 with one free variable
v
* * * *
0 * * *
0 0 * *
Example matrix
A=(−0.8−55.8−7.6−40−621.24.5−5.52.6)∼(−40−620−57−800−1−4)
, null(A)=span{(6.5−7.2−41)}
Example solution
b=(101.1) and u=(3−3−20)+u4(6.5−7.2−41)
Example solution
b=(2.6−2−1.1) and u=(2−2−10)+u4(6.5−7.2−41)
Example matrix
A=(−0.42.47.47−27−35−1.23.3−5−2.4)∼(−27−3501860040)
, null(A)=span{(−18.5−601)}
Example solution
b=(−6.623.5) and u=(41−10)+u4(−18.5−601)
Example solution
b=(−6.244.7) and u=(31−10)+u4(−18.5−601)
Example matrix
A=(2.5−7.5−9.541−8−7.82.6−1.50.51.5−4.1)∼(2.5−7.5−9.540−5−4100−1−2.5)
, null(A)=span{(−4.52.2−2.51)}
Example solution
b=(−1.51.40.5) and u=(1−220)+u4(−4.52.2−2.51)
Example solution
b=(−12.60) and u=(2−330)+u4(−4.52.2−2.51)
Example matrix
A=(57.65.2−2.42.539.6−0.62.54.2−0.7−9.7)∼(57.65.2−2.40−0.870.6000.2−8.2)
, null(A)=span{(−588.6359.5411)}
Example solution
b=(−0.4−5.62.3) and u=(4−2−10)+u4(−588.6359.5411)
Example solution
b=(08.6−4.1) and u=(2−210)+u4(−588.6359.5411)
v
* * * *
0 * * *
0 0 0 *
Example matrix
A=(17−72−0.4−7.8−6.2−0.80.62.2−7.80.2)∼(17−720−5−90000−1)
, null(A)=span{(19.6−1.810)}
Example solution
b=(24.2−0.8) and u=(1−104)+u3(19.6−1.810)
Example solution
b=(14.6−0.4) and u=(2−103)+u3(19.6−1.810)
Example matrix
A=(2−6.84.86.45−776−1.50.1−1.1−1.8)∼(5−7760−424000−2)
, null(A)=span{(−0.70.510)}
Example solution
b=(−402) and u=(1−10−2)+u3(−0.70.510)
Example solution
b=(−3.613.7) and u=(1−20−3)+u3(−0.70.510)
Example matrix
A=(5−2.5−7−8.5107.14.33−1.9−11−17.9)∼(5−2.5−7−8.500.58.56000−8)
, null(A)=span{(−7.1−1710)}
Example solution
b=(6.56.3−8.1) and u=(2−201)+u3(−7.1−1710)
Example solution
b=(46.3−10) and u=(2−101)+u3(−7.1−1710)
Example matrix
A=(0.4−6.4−89.8−0.245.6−13.7−0.23.64.80.3)∼(0.4−6.4−89.800.81.6−8.80009.6)
, null(A)=span{(−12−210)}
Example solution
b=(−1.4−6.56.7) and u=(4201)+u3(−12−210)
Example solution
b=(−1.8−6.36.9) and u=(3201)+u3(−12−210)
v
* * * *
0 0 * *
0 0 0 *
Example matrix
A=(1.4−1.48.3−8.12−29−3−0.80.8−5.414.6)∼(2−29−3002−60008)
, null(A)=span{(1100)}
Example solution
b=(−2.9−7−7) and u=(40−2−1)+u2(1100)
Example solution
b=(−4.3−9−6.2) and u=(30−2−1)+u2(1100)
Example matrix
A=(4−3.25.20.8−541−61.5−1.2−1.5−0.4)∼(−541−6006−4000−3)
, null(A)=span{(0.8100)}
Example solution
b=(1.215.3) and u=(20−1−2)+u2(0.8100)
Example solution
b=(−2.863.8) and u=(10−1−2)+u2(0.8100)
Example matrix
A=(−530−9.51−0.6−84.4−42.43.2−13.1)∼(−530−9.500−82.5000−4.5)
, null(A)=span{(0.6100)}
Example solution
b=(−13.27) and u=(40−1−2)+u2(0.6100)
Example solution
b=(−5.5−9.44.3) and u=(301−1)+u2(0.6100)
Example matrix
A=(−1−2.76.1−14.125.43.49.812.75.6−4.3)∼(25.43.49.8007.8−9.2000−4.6)
, null(A)=span{(−2.7100)}
Example solution
b=(−2.1−8.6−2.9) and u=(40−2−1)+u2(−2.7100)
Example solution
b=(−1.1−10.6−3.9) and u=(30−2−1)+u2(−2.7100)
Example matrix
A=(1.4−1.48.3−8.12−29−3−0.80.8−5.414.6)∼(2−29−3002−60008)
, null(A)=span{(1100)}
Example solution
b=(−2.9−7−7) and u=(40−2−1)+u2(1100)
Example solution
b=(−4.3−9−6.2) and u=(30−2−1)+u2(1100)
Example matrix
A=(4−3.25.20.8−541−61.5−1.2−1.5−0.4)∼(−541−6006−4000−3)
, null(A)=span{(0.8100)}
Example solution
b=(1.215.3) and u=(20−1−2)+u2(0.8100)
Example solution
b=(−2.863.8) and u=(10−1−2)+u2(0.8100)
Example matrix
A=(−530−9.51−0.6−84.4−42.43.2−13.1)∼(−530−9.500−82.5000−4.5)
, null(A)=span{(0.6100)}
Example solution
b=(−13.27) and u=(40−1−2)+u2(0.6100)
Example solution
b=(−5.5−9.44.3) and u=(301−1)+u2(0.6100)
Example matrix
A=(−1−2.76.1−14.125.43.49.812.75.6−4.3)∼(25.43.49.8007.8−9.2000−4.6)
, null(A)=span{(−2.7100)}
Example solution
b=(−2.1−8.6−2.9) and u=(40−2−1)+u2(−2.7100)
Example solution
b=(−1.1−10.6−3.9) and u=(30−2−1)+u2(−2.7100)
3×4 with two free variables
v v
* * * *
0 * * *
0 0 0 0
Example matrix
A=(1.43.60.53.1−2−8570.81.80.82.8)∼(−2−8570−2480000)
, null(A)=span{(−5.5210),(−12.5401)}
Example solution
b=(0.620.6) and u=(3−100)+u3(−5.5210)+u4(−12.5401)
Example solution
b=(201.4) and u=(4−100)+u3(−5.5210)+u4(−12.5401)
Example matrix
A=(0.86.24.4−2.4−1−92−2−0.5−4−21)∼(−1−92−20−16−40000)
, null(A)=span{(−52610),(34−401)}
Example solution
b=(−352) and u=(4−100)+u3(−52610)+u4(34−401)
Example solution
b=(−3.862.5) and u=(3−100)+u3(−52610)+u4(34−401)
Example matrix
A=(10380.25−3.40.60.8−34.87)∼(103805−4−10000)
, null(A)=span{(−30.810),(−80.201)}
Example solution
b=(25.4−1.4) and u=(2100)+u3(−30.810)+u4(−80.201)
Example solution
b=(1−4.83.8) and u=(1−100)+u3(−30.810)+u4(−80.201)
Example matrix
A=(2.54.26.1−12.158.89.4−5.4−2.5−4.3−5.47.4)∼(58.89.4−5.40−0.21.4−9.40000)
, null(A)=span{(−14.2710),(83.8−4701)}
Example solution
b=(0.81.2−0.7) and u=(2−100)+u3(−14.2710)+u4(83.8−4701)
Example solution
b=(−0.9−2.61.1) and u=(3−200)+u3(−14.2710)+u4(83.8−4701)
v v
* * * *
0 0 * *
0 0 0 0
Example matrix
A=(1.21.8−1.610.4−2−31−90.60.90.20.2)∼(−2−31−900−150000)
, null(A)=span{(−1.5100),(−2051)}
Example solution
b=(−201) and u=(1020)+u2(−1.5100)+u4(−2051)
Example solution
b=(−2.4−11.8) and u=(2030)+u2(−1.5100)+u4(−2051)
Example matrix
A=(3.67.2−10.1−4−4−89024−3.91.2)∼(−4−89000−2−40000)
, null(A)=span{(−2100),(−4.50−21)}
Example solution
b=(−2.910.1) and u=(2010)+u2(−2100)+u4(−4.50−21)
Example solution
b=(0.7−32.1) and u=(3010)+u2(−2100)+u4(−4.50−21)
Example matrix
A=(−1−88.5−7.5−0.4−3.23.9−3.5−0.8−6.46.7−5.9)∼(−1−88.5−7.5000.5−0.50000)
, 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=(0.25.87.6−4.4−0.1−2.9−2.8−7.60.12.94.3−7.1)∼(0.25.87.6−4.4001−9.80000)
, null(A)=span{(−29100),(−350.409.81)}
Example solution
b=(−6.82.4−3.9) and u=(40−10)+u2(−29100)+u4(−350.409.81)
Example solution
b=(−72.5−4) and u=(30−10)+u2(−29100)+u4(−350.409.81)
v v
* * * *
0 0 0 *
0 0 0 0
Example matrix
A=(5−49−7−32.4−5.41.23−2.45.4−3.9)∼(5−49−7000−30000)
, null(A)=span{(0.8100),(−1.8010)}
Example solution
b=(3−4.82.1) and u=(2001)+u2(0.8100)+u3(−1.8010)
Example solution
b=(1−6.61.2) and u=(3002)+u2(0.8100)+u3(−1.8010)
Example matrix
A=(3.2−6.44.8−9.64−86−72−43−3.1)∼(4−86−7000−40000)
, \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
- 3 \times 5 with 2 free variables
- 3 \times 5 with 3 free variables
- 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
- 4 \times 2 with 0 free variables
- 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
- 4 \times 3 with 0 free variables
- 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
- 4 \times 4 with 0 free variables
- 4 \times 4 with 1 free variables
- 4 \times 4 with 2 free variables
- 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
- 4 \times 5 with 1 free variables
- 4 \times 5 with 2 free variables
- 4 \times 5 with 3 free variables
- 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
- 4 \times 6 with 2 free variables
- 4 \times 6 with 3 free variables
- 4 \times 6 with 4 free variables
- 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
- 5 \times 2 with 0 free variables
- 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
- 5 \times 3 with 0 free variables
- 5 \times 3 with 1 free variables
- 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
- 5 \times 4 with 0 free variables
- 5 \times 4 with 1 free variables
- 5 \times 4 with 2 free variables
- 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
- 5 \times 5 with 0 free variables
- 5 \times 5 with 1 free variables
- 5 \times 5 with 2 free variables
- 5 \times 5 with 3 free variables
- 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
- 5 \times 6 with 1 free variables
- 5 \times 6 with 2 free variables
- 5 \times 6 with 3 free variables
- 5 \times 6 with 4 free variables
- 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
- 6 \times 2 with 0 free variables
- 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
- 6 \times 3 with 0 free variables
- 6 \times 3 with 1 free variables
- 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
- 6 \times 4 with 0 free variables
- 6 \times 4 with 1 free variables
- 6 \times 4 with 2 free variables
- 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
- 6 \times 5 with 0 free variables
- 6 \times 5 with 1 free variables
- 6 \times 5 with 2 free variables
- 6 \times 5 with 3 free variables
- 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
- 6 \times 6 with 0 free variables
- 6 \times 6 with 1 free variables
- 6 \times 6 with 2 free variables
- 6 \times 6 with 3 free variables
- 6 \times 6 with 4 free variables
- 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