Systems of linear equations

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

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

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

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

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

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

In each case,

1. If $\textrm{rank}(A) < m$, then two examples with no solutions will be given, and two examples with one or infinitely many solutions will be given.
2. If $\textrm{rank}(A) = m$, then two examples of either one or infinitely many solutions will be given.
3. If $\textrm{rank}(A) < n$, 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

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

$2 \times 2$

$2 \times 2$ with no free variables

   
* *
0 *

Example matrix

$A = \left(\begin{array}{rr}9 & 6 \\ - 7.2 & - 7.8 \end{array}\right) \sim \left(\begin{array}{rr} 9.0 & 6.0 \\ 0 & - 3.0 \end{array}\right)$ , $\textrm{null}(A) = \{\textbf{0}\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rr}-6 & -1 \\ 1.2 & 5.2 \end{array}\right) \sim \left(\begin{array}{rr}- 6.0 & - 1.0 \\ 0 & 5.0 \end{array}\right)$ , $\textrm{null}(A) = \{\textbf{0}\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rr}- 8.5 & 7.5 \\ 3.4 & -10 \end{array}\right) \sim \left(\begin{array}{rr}- 8.5 & 7.5 \\ 0 & - 7.0 \end{array}\right)$ , $\textrm{null}(A) = \{\textbf{0}\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rr}- 0.8 & -5 \\ - 0.4 & - 2.1 \end{array}\right) \sim \left(\begin{array}{rr}- 0.8 & - 5.0 \\ 0 & 0.4 \end{array}\right)$ , $\textrm{null}(A) = \{\textbf{0}\}$

Example solution

Example solution

$2 \times 2$ with one free variable

  v
* *
0 0

Example matrix

$A = \left(\begin{array}{rr}-2 & 2 \\ 1.4 & - 1.4 \end{array}\right) \sim \left(\begin{array}{rr}-2 & 2 \\ 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}1 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rr}-1 & 4 \\ - 0.7 & 2.8 \end{array}\right) \sim \left(\begin{array}{rr}-1 & 4 \\ 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}4 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rr}-1 & 0.5 \\ 0.2 & - 0.1 \end{array}\right) \sim \left(\begin{array}{rr}-1 & 0.5 \\ 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.5 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rr}-2 & 2.6 \\ -1 & 1.3 \end{array}\right) \sim \left(\begin{array}{rr}-2 & 2.6 \\ 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 1.3 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

$2 \times 3$

1. $2 \times 3$ with 1 free variables
2. $2 \times 3$ with 2 free variables

$2 \times 3$ with one free variable

    v
* * *
0 * *

Example matrix

$A = \left(\begin{array}{rrr}-2 & -8 & -9 \\ 1.8 & 5.2 & 4.1 \end{array}\right) \sim \left(\begin{array}{rrr}-2 & -8 & -9 \\ 0 & -2 & -4 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 3.5 \\ -2 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrr}-4 & 4 & 0 \\ -2 & 1 & -2 \end{array}\right) \sim \left(\begin{array}{rrr}-4 & 4 & 0 \\ 0 & -1 & -2 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-2 \\ -2 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrr} 0.5 & 9.5 & 3 \\ - 0.2 & - 5.8 & 0.8 \end{array}\right) \sim \left(\begin{array}{rrr} 0.5 & 9.5 & 3 \\ 0 & -2 & 2 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-25 \\ 1 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrr}-2 & -8 & - 2.4 \\ 1 & 3.6 & 3 \end{array}\right) \sim \left(\begin{array}{rrr}-2 & -8 & - 2.4 \\ 0 & - 0.4 & 1.8 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 19.2 \\ 4.5 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

  v  
* * *
0 0 *

Example matrix

$A = \left(\begin{array}{rrr}-2 & -4 & 9 \\ -1 & -2 & 2.5 \end{array}\right) \sim \left(\begin{array}{rrr}- 2.0 & - 4.0 & 9.0 \\ 0 & 0 & - 2.0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-2 \\ 1 \\ 0 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrr}5 & -1 & 7 \\ 3.5 & - 0.7 & 8.9 \end{array}\right) \sim \left(\begin{array}{rrr} 5.0 & - 1.0 & 7.0 \\ 0 & 0 & 4.0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.2 \\ 1 \\ 0 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrr} 2.5 & -3 & 9.5 \\ 0.5 & - 0.6 & 0.9 \end{array}\right) \sim \left(\begin{array}{rrr} 2.5 & - 3.0 & 9.5 \\ 0 & 0 & - 1.0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 1.2 \\ 1 \\ 0 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrr}-2 & - 3.4 & - 2.6 \\ 1 & 1.7 & - 3.9 \end{array}\right) \sim \left(\begin{array}{rrr}- 2.0 & - 3.4 & - 2.6 \\ 0 & 0 & - 5.2 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 1.7 \\ 1 \\ 0 \end{array}\right)\right\}$

Example solution

Example solution

$2 \times 3$ with two free variables

  v v
* * *
0 0 0

Example matrix

$A = \left(\begin{array}{rrr}-1 & -5 & -6 \\ - 0.1 & - 0.5 & - 0.6 \end{array}\right) \sim \left(\begin{array}{rrr}-1 & -5 & -6 \\ 0 & 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-5 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}-6 \\ 0 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrr}1 & -2 & 4 \\ - 0.9 & 1.8 & - 3.6 \end{array}\right) \sim \left(\begin{array}{rrr}1 & -2 & 4 \\ 0 & 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}2 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}-4 \\ 0 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrr}- 0.5 & 8.5 & - 7.5 \\ - 0.3 & 5.1 & - 4.5 \end{array}\right) \sim \left(\begin{array}{rrr}- 0.5 & 8.5 & - 7.5 \\ 0 & 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}17 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}-15 \\ 0 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrr} 0.2 & 1.2 & - 2.4 \\ 0.1 & 0.6 & - 1.2 \end{array}\right) \sim \left(\begin{array}{rrr} 0.2 & 1.2 & - 2.4 \\ 0 & 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-6 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}12 \\ 0 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

$2 \times 4$

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

$2 \times 4$ with two free variables

    v v
* * * *
0 * * *

Example matrix

$A = \left(\begin{array}{rrrr}1 & -5 & -4 & -3 \\ 0.8 & -3 & - 0.2 & - 4.4 \end{array}\right) \sim \left(\begin{array}{rrrr}1 & -5 & -4 & -3 \\ 0 & 1 & 3 & -2 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-11 \\ -3 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}13 \\ 2 \\ 0 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrrr}-1 & 1 & -3 & -6 \\ 0.6 & 4.4 & 10.8 & 11.6 \end{array}\right) \sim \left(\begin{array}{rrrr}-1 & 1 & -3 & -6 \\ 0 & 5 & 9 & 8 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 4.8 \\ - 1.8 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}- 7.6 \\ - 1.6 \\ 0 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrrr}- 0.5 & 8.5 & -9 & 5 \\ - 0.4 & 7.3 & - 4.7 & 8 \end{array}\right) \sim \left(\begin{array}{rrrr}- 0.5 & 8.5 & -9 & 5 \\ 0 & 0.5 & 2.5 & 4 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-103 \\ -5 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}-126 \\ -8 \\ 0 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrrr}- 0.2 & 9.6 & -6 & 9 \\ - 0.1 & 5.2 & - 0.8 & 3.3 \end{array}\right) \sim \left(\begin{array}{rrrr}- 0.2 & 9.6 & -6 & 9 \\ 0 & 0.4 & 2.2 & - 1.2 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-294 \\ - 5.5 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}189 \\ 3 \\ 0 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

  v   v
* * * *
0 0 * *

Example matrix

$A = \left(\begin{array}{rrrr}2 & -8 & 8 & 6 \\ - 0.4 & 1.6 & 0.4 & 7.8 \end{array}\right) \sim \left(\begin{array}{rrrr}2 & -8 & 8 & 6 \\ 0 & 0 & 2 & 9 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}4 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}15 \\ 0 \\ - 4.5 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrrr}-2 & -3 & 0 & 9 \\ - 0.2 & - 0.3 & -1 & - 1.1 \end{array}\right) \sim \left(\begin{array}{rrrr}-2 & -3 & 0 & 9 \\ 0 & 0 & -1 & -2 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 1.5 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 4.5 \\ 0 \\ -2 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrrr} 0.5 & 8.5 & - 5.5 & 2 \\ - 0.4 & - 6.8 & 5.4 & - 4.6 \end{array}\right) \sim \left(\begin{array}{rrrr} 0.5 & 8.5 & - 5.5 & 2 \\ 0 & 0 & 1 & -3 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-17 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}29 \\ 0 \\ 3 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrrr}- 0.2 & - 2.8 & - 9.2 & - 7.8 \\ 0.1 & 1.4 & 3.6 & 5.5 \end{array}\right) \sim \left(\begin{array}{rrrr}- 0.2 & - 2.8 & - 9.2 & - 7.8 \\ 0 & 0 & -1 & 1.6 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-14 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 112.6 \\ 0 \\ 1.6 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

  v v  
* * * *
0 0 0 *

Example matrix

$A = \left(\begin{array}{rrrr}-2 & 6 & 7 & -7 \\ 0.4 & - 1.2 & - 1.4 & - 6.6 \end{array}\right) \sim \left(\begin{array}{rrrr}-2 & 6 & 7 & -7 \\ 0 & 0 & 0 & -8 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}3 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 3.5 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrrr}-5 & 9 & -7 & -1 \\ 3 & - 5.4 & 4.2 & 6.6 \end{array}\right) \sim \left(\begin{array}{rrrr}-5 & 9 & -7 & -1 \\ 0 & 0 & 0 & 6 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 1.8 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 1.4 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrrr}-5 & - 5.5 & 4.5 & -9 \\ 3 & 3.3 & - 2.7 & 13.4 \end{array}\right) \sim \left(\begin{array}{rrrr}-5 & - 5.5 & 4.5 & -9 \\ 0 & 0 & 0 & 8 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 1.1 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 0.9 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrrr}2 & 1.4 & 0.2 & - 1.4 \\ -1 & - 0.7 & - 0.1 & - 9.1 \end{array}\right) \sim \left(\begin{array}{rrrr}2 & 1.4 & 0.2 & - 1.4 \\ 0 & 0 & 0 & - 9.8 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 0.7 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 0.1 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}$

Example solution

Example solution

$2 \times 4$ with three free variables

  v v v
* * * *
0 0 0 0

Example matrix

$A = \left(\begin{array}{rrrr}-1 & 7 & -7 & 3 \\ 0.9 & - 6.3 & 6.3 & - 2.7 \end{array}\right) \sim \left(\begin{array}{rrrr}-1 & 7 & -7 & 3 \\ 0 & 0 & 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}7 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-7 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}3 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrrr}2 & -9 & -7 & -1 \\ 1.8 & - 8.1 & - 6.3 & - 0.9 \end{array}\right) \sim \left(\begin{array}{rrrr}2 & -9 & -7 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 4.5 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 3.5 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 0.5 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrrr} 0.5 & 8.5 & -9 & 7 \\ 0.2 & 3.4 & - 3.6 & 2.8 \end{array}\right) \sim \left(\begin{array}{rrrr} 0.5 & 8.5 & -9 & 7 \\ 0 & 0 & 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-17 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}18 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}-14 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrrr}2 & - 1.4 & - 0.6 & 4.6 \\ -1 & 0.7 & 0.3 & - 2.3 \end{array}\right) \sim \left(\begin{array}{rrrr}2 & - 1.4 & - 0.6 & 4.6 \\ 0 & 0 & 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.7 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r} 0.3 \\ 0 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}- 2.3 \\ 0 \\ 0 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

$3 \times 2$

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

$3 \times 2$ with no free variables

   
* *
0 *
0 0

Example matrix

$A = \left(\begin{array}{rr}-8 & -4 \\ -4 & 6 \\ 4 & - 5.2 \end{array}\right) \sim \left(\begin{array}{rr}-8 & -4 \\ 0 & 8 \\ 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \{\textbf{0}\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rr}-4 & 4 \\ - 2.4 & - 0.6 \\ -2 & 2.6 \end{array}\right) \sim \left(\begin{array}{rr}-4 & 4 \\ 0 & -3 \\ 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \{\textbf{0}\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rr}2 & 3.5 \\ - 1.2 & - 4.6 \\ - 1.6 & - 3.3 \end{array}\right) \sim \left(\begin{array}{rr}2 & 3.5 \\ 0 & - 2.5 \\ 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \{\textbf{0}\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rr}- 1.3 & 12.1 \\ 2.6 & - 9.8 \\ - 1.3 & 8.5 \end{array}\right) \sim \left(\begin{array}{rr} 2.6 & - 9.8 \\ 0 & 7.2 \\ 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \{\textbf{0}\}$

Example solution

Example solution

$3 \times 2$ with one free variable

  v
* *
0 0
0 0

Example matrix

$A = \left(\begin{array}{rr}- 1.6 & 0.8 \\ 4 & -2 \\ - 3.2 & 1.6 \end{array}\right) \sim \left(\begin{array}{rr}4 & -2 \\ 0 & 0 \\ 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.5 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rr}-4 & - 0.8 \\ 5 & 1 \\ - 4.5 & - 0.9 \end{array}\right) \sim \left(\begin{array}{rr}5 & 1 \\ 0 & 0 \\ 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 0.2 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rr}- 2.5 & 5.5 \\ 1.5 & - 3.3 \\ 1 & - 2.2 \end{array}\right) \sim \left(\begin{array}{rr}- 2.5 & 5.5 \\ 0 & 0 \\ 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 2.2 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rr}-2 & 2 \\ 4 & -4 \\ 2 & -2 \end{array}\right) \sim \left(\begin{array}{rr}4 & -4 \\ 0 & 0 \\ 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}1 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rr}- 1.6 & 0.8 \\ 4 & -2 \\ - 3.2 & 1.6 \end{array}\right) \sim \left(\begin{array}{rr}4 & -2 \\ 0 & 0 \\ 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.5 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rr}-4 & - 0.8 \\ 5 & 1 \\ - 4.5 & - 0.9 \end{array}\right) \sim \left(\begin{array}{rr}5 & 1 \\ 0 & 0 \\ 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 0.2 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rr}- 2.5 & 5.5 \\ 1.5 & - 3.3 \\ 1 & - 2.2 \end{array}\right) \sim \left(\begin{array}{rr}- 2.5 & 5.5 \\ 0 & 0 \\ 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 2.2 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rr}-2 & 2 \\ 4 & -4 \\ 2 & -2 \end{array}\right) \sim \left(\begin{array}{rr}4 & -4 \\ 0 & 0 \\ 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}1 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

$3 \times 3$

1. $3 \times 3$ with 2 free variables
2. $3 \times 3$ with 1 free variables
3. $3 \times 3$ with 2 free variables

$3 \times 3$ with no free variables

     
* * *
0 * *
0 0 *

Example matrix

$A = \left(\begin{array}{rrr}- 4.2 & 5.3 & 9.4 \\ 6 & -9 & -2 \\ - 1.8 & 3.2 & - 2.4 \end{array}\right) \sim \left(\begin{array}{rrr}6 & -9 & -2 \\ 0 & -1 & 8 \\ 0 & 0 & 1 \end{array}\right)$ , $\textrm{null}(A) = \{\textbf{0}\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrr}6 & -9 & 1 \\ - 1.2 & 9.8 & 6.8 \\ - 3.6 & 11.8 & 4 \end{array}\right) \sim \left(\begin{array}{rrr}6 & -9 & 1 \\ 0 & 8 & 7 \\ 0 & 0 & -1 \end{array}\right)$ , $\textrm{null}(A) = \{\textbf{0}\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrr}-1 & -4 & -7 \\ 0.2 & 0.3 & - 4.1 \\ - 0.2 & - 1.2 & 3.7 \end{array}\right) \sim \left(\begin{array}{rrr}-1 & -4 & -7 \\ 0 & - 0.5 & - 5.5 \\ 0 & 0 & 9.5 \end{array}\right)$ , $\textrm{null}(A) = \{\textbf{0}\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrr}- 3.8 & 1.2 & 2.6 \\ 7.6 & - 6.8 & 4.4 \\ - 3.8 & 2.3 & - 5.8 \end{array}\right) \sim \left(\begin{array}{rrr} 7.6 & - 6.8 & 4.4 \\ 0 & - 2.2 & 4.8 \\ 0 & 0 & -6 \end{array}\right)$ , $\textrm{null}(A) = \{\textbf{0}\}$

Example solution

Example solution

$3 \times 3$ with one free variable

    v
* * *
0 * *
0 0 0

Example matrix

$A = \left(\begin{array}{rrr}-2 & -8 & 6 \\ - 0.6 & - 4.4 & - 1.2 \\ - 0.2 & - 0.6 & 0.9 \end{array}\right) \sim \left(\begin{array}{rrr}-2 & -8 & 6 \\ 0 & -2 & -3 \\ 0 & 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}9 \\ - 1.5 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrr} 0.8 & 2 & - 0.2 \\ -1 & -5 & 9 \\ - 0.1 & - 0.9 & 2.3 \end{array}\right) \sim \left(\begin{array}{rrr}-1 & -5 & 9 \\ 0 & -2 & 7 \\ 0 & 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 8.5 \\ 3.5 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrr}-5 & -1 & 4.5 \\ -4 & 0.2 & 7.6 \\ -2 & 0.2 & 4.2 \end{array}\right) \sim \left(\begin{array}{rrr}-5 & -1 & 4.5 \\ 0 & 1 & 4 \\ 0 & 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 1.7 \\ -4 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrr} 0.2 & 3 & 10.3 \\ - 0.4 & -4 & - 2.6 \\ - 0.2 & - 1.5 & 3.2 \end{array}\right) \sim \left(\begin{array}{rrr}- 0.4 & -4 & - 2.6 \\ 0 & 1 & 9 \\ 0 & 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 83.5 \\ -9 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

  v  
* * *
0 0 *
0 0 0

Example matrix

$A = \left(\begin{array}{rrr}- 4.5 & 5.4 & 7.1 \\ 5 & -6 & 1 \\ 2 & - 2.4 & 2.8 \end{array}\right) \sim \left(\begin{array}{rrr}5 & -6 & 1 \\ 0 & 0 & 8 \\ 0 & 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 1.2 \\ 1 \\ 0 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrr}2 & 7 & 9 \\ 1.8 & 6.3 & 5.1 \\ - 1.6 & - 5.6 & - 8.7 \end{array}\right) \sim \left(\begin{array}{rrr}2 & 7 & 9 \\ 0 & 0 & -3 \\ 0 & 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 3.5 \\ 1 \\ 0 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrr}-2 & 1.4 & - 2.3 \\ -5 & 3.5 & - 9.5 \\ 2 & - 1.4 & 5 \end{array}\right) \sim \left(\begin{array}{rrr}-5 & 3.5 & - 9.5 \\ 0 & 0 & 1.5 \\ 0 & 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.7 \\ 1 \\ 0 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrr} 0.4 & - 1.8 & 3.8 \\ 0.2 & - 0.9 & 6.3 \\ - 0.2 & 0.9 & - 4.1 \end{array}\right) \sim \left(\begin{array}{rrr} 0.4 & - 1.8 & 3.8 \\ 0 & 0 & 4.4 \\ 0 & 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 4.5 \\ 1 \\ 0 \end{array}\right)\right\}$

Example solution

Example solution

$3 \times 3$ with two free variables

  v v
* * *
0 0 0
0 0 0

Example matrix

$A = \left(\begin{array}{rrr}-2 & -3 & -4 \\ - 1.4 & - 2.1 & - 2.8 \\ 1.6 & 2.4 & 3.2 \end{array}\right) \sim \left(\begin{array}{rrr}-2 & -3 & -4 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 1.5 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}-2 \\ 0 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrr}-2 & -9 & 4 \\ 1.4 & 6.3 & - 2.8 \\ 0.2 & 0.9 & - 0.4 \end{array}\right) \sim \left(\begin{array}{rrr}-2 & -9 & 4 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 4.5 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}2 \\ 0 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrr}2 & - 1.4 & 1 \\ 5 & - 3.5 & 2.5 \\ -4 & 2.8 & -2 \end{array}\right) \sim \left(\begin{array}{rrr}5 & - 3.5 & 2.5 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.7 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}- 0.5 \\ 0 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrr}-1 & 6.8 & - 6.6 \\ - 0.5 & 3.4 & - 3.3 \\ 0.5 & - 3.4 & 3.3 \end{array}\right) \sim \left(\begin{array}{rrr}-1 & 6.8 & - 6.6 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 6.8 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}- 6.6 \\ 0 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrr}-2 & -3 & -4 \\ - 1.4 & - 2.1 & - 2.8 \\ 1.6 & 2.4 & 3.2 \end{array}\right) \sim \left(\begin{array}{rrr}-2 & -3 & -4 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 1.5 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}-2 \\ 0 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrr}-2 & -9 & 4 \\ 1.4 & 6.3 & - 2.8 \\ 0.2 & 0.9 & - 0.4 \end{array}\right) \sim \left(\begin{array}{rrr}-2 & -9 & 4 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 4.5 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}2 \\ 0 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrr}2 & - 1.4 & 1 \\ 5 & - 3.5 & 2.5 \\ -4 & 2.8 & -2 \end{array}\right) \sim \left(\begin{array}{rrr}5 & - 3.5 & 2.5 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.7 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}- 0.5 \\ 0 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrr}-1 & 6.8 & - 6.6 \\ - 0.5 & 3.4 & - 3.3 \\ 0.5 & - 3.4 & 3.3 \end{array}\right) \sim \left(\begin{array}{rrr}-1 & 6.8 & - 6.6 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 6.8 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}- 6.6 \\ 0 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

$3 \times 4$

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

$3 \times 4$ with one free variable

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

Example matrix

$A = \left(\begin{array}{rrrr}- 0.8 & -5 & 5.8 & - 7.6 \\ -4 & 0 & -6 & 2 \\ 1.2 & 4.5 & - 5.5 & 2.6 \end{array}\right) \sim \left(\begin{array}{rrrr}-4 & 0 & -6 & 2 \\ 0 & -5 & 7 & -8 \\ 0 & 0 & -1 & -4 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 6.5 \\ - 7.2 \\ -4 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrrr}- 0.4 & 2.4 & 7.4 & 7 \\ -2 & 7 & -3 & 5 \\ - 1.2 & 3.3 & -5 & - 2.4 \end{array}\right) \sim \left(\begin{array}{rrrr}-2 & 7 & -3 & 5 \\ 0 & 1 & 8 & 6 \\ 0 & 0 & 4 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 18.5 \\ -6 \\ 0 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrrr} 2.5 & - 7.5 & - 9.5 & 4 \\ 1 & -8 & - 7.8 & 2.6 \\ - 1.5 & 0.5 & 1.5 & - 4.1 \end{array}\right) \sim \left(\begin{array}{rrrr} 2.5 & - 7.5 & - 9.5 & 4 \\ 0 & -5 & -4 & 1 \\ 0 & 0 & -1 & - 2.5 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 4.5 \\ 2.2 \\ - 2.5 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrrr}5 & 7.6 & 5.2 & - 2.4 \\ 2.5 & 3 & 9.6 & - 0.6 \\ 2.5 & 4.2 & - 0.7 & - 9.7 \end{array}\right) \sim \left(\begin{array}{rrrr}5 & 7.6 & 5.2 & - 2.4 \\ 0 & - 0.8 & 7 & 0.6 \\ 0 & 0 & 0.2 & - 8.2 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 588.6 \\ 359.5 \\ 41 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

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

Example matrix

$A = \left(\begin{array}{rrrr}1 & 7 & -7 & 2 \\ - 0.4 & - 7.8 & - 6.2 & - 0.8 \\ 0.6 & 2.2 & - 7.8 & 0.2 \end{array}\right) \sim \left(\begin{array}{rrrr}1 & 7 & -7 & 2 \\ 0 & -5 & -9 & 0 \\ 0 & 0 & 0 & -1 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 19.6 \\ - 1.8 \\ 1 \\ 0 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrrr}2 & - 6.8 & 4.8 & 6.4 \\ 5 & -7 & 7 & 6 \\ - 1.5 & 0.1 & - 1.1 & - 1.8 \end{array}\right) \sim \left(\begin{array}{rrrr}5 & -7 & 7 & 6 \\ 0 & -4 & 2 & 4 \\ 0 & 0 & 0 & -2 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 0.7 \\ 0.5 \\ 1 \\ 0 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrrr}5 & - 2.5 & -7 & - 8.5 \\ 1 & 0 & 7.1 & 4.3 \\ 3 & - 1.9 & -11 & - 17.9 \end{array}\right) \sim \left(\begin{array}{rrrr}5 & - 2.5 & -7 & - 8.5 \\ 0 & 0.5 & 8.5 & 6 \\ 0 & 0 & 0 & -8 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 7.1 \\ -17 \\ 1 \\ 0 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrrr} 0.4 & - 6.4 & -8 & 9.8 \\ - 0.2 & 4 & 5.6 & - 13.7 \\ - 0.2 & 3.6 & 4.8 & 0.3 \end{array}\right) \sim \left(\begin{array}{rrrr} 0.4 & - 6.4 & -8 & 9.8 \\ 0 & 0.8 & 1.6 & - 8.8 \\ 0 & 0 & 0 & 9.6 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-12 \\ -2 \\ 1 \\ 0 \end{array}\right)\right\}$

Example solution

Example solution

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

Example matrix

$A = \left(\begin{array}{rrrr} 1.4 & - 1.4 & 8.3 & - 8.1 \\ 2 & -2 & 9 & -3 \\ - 0.8 & 0.8 & - 5.4 & 14.6 \end{array}\right) \sim \left(\begin{array}{rrrr}2 & -2 & 9 & -3 \\ 0 & 0 & 2 & -6 \\ 0 & 0 & 0 & 8 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}1 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrrr}4 & - 3.2 & 5.2 & 0.8 \\ -5 & 4 & 1 & -6 \\ 1.5 & - 1.2 & - 1.5 & - 0.4 \end{array}\right) \sim \left(\begin{array}{rrrr}-5 & 4 & 1 & -6 \\ 0 & 0 & 6 & -4 \\ 0 & 0 & 0 & -3 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.8 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrrr}-5 & 3 & 0 & - 9.5 \\ 1 & - 0.6 & -8 & 4.4 \\ -4 & 2.4 & 3.2 & - 13.1 \end{array}\right) \sim \left(\begin{array}{rrrr}-5 & 3 & 0 & - 9.5 \\ 0 & 0 & -8 & 2.5 \\ 0 & 0 & 0 & - 4.5 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.6 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrrr}-1 & - 2.7 & 6.1 & - 14.1 \\ 2 & 5.4 & 3.4 & 9.8 \\ 1 & 2.7 & 5.6 & - 4.3 \end{array}\right) \sim \left(\begin{array}{rrrr}2 & 5.4 & 3.4 & 9.8 \\ 0 & 0 & 7.8 & - 9.2 \\ 0 & 0 & 0 & - 4.6 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 2.7 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrrr} 1.4 & - 1.4 & 8.3 & - 8.1 \\ 2 & -2 & 9 & -3 \\ - 0.8 & 0.8 & - 5.4 & 14.6 \end{array}\right) \sim \left(\begin{array}{rrrr}2 & -2 & 9 & -3 \\ 0 & 0 & 2 & -6 \\ 0 & 0 & 0 & 8 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}1 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrrr}4 & - 3.2 & 5.2 & 0.8 \\ -5 & 4 & 1 & -6 \\ 1.5 & - 1.2 & - 1.5 & - 0.4 \end{array}\right) \sim \left(\begin{array}{rrrr}-5 & 4 & 1 & -6 \\ 0 & 0 & 6 & -4 \\ 0 & 0 & 0 & -3 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.8 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrrr}-5 & 3 & 0 & - 9.5 \\ 1 & - 0.6 & -8 & 4.4 \\ -4 & 2.4 & 3.2 & - 13.1 \end{array}\right) \sim \left(\begin{array}{rrrr}-5 & 3 & 0 & - 9.5 \\ 0 & 0 & -8 & 2.5 \\ 0 & 0 & 0 & - 4.5 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.6 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrrr}-1 & - 2.7 & 6.1 & - 14.1 \\ 2 & 5.4 & 3.4 & 9.8 \\ 1 & 2.7 & 5.6 & - 4.3 \end{array}\right) \sim \left(\begin{array}{rrrr}2 & 5.4 & 3.4 & 9.8 \\ 0 & 0 & 7.8 & - 9.2 \\ 0 & 0 & 0 & - 4.6 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 2.7 \\ 1 \\ 0 \\ 0 \end{array}\right)\right\}$

Example solution

Example solution

$3 \times 4$ with two free variables

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

Example matrix

$A = \left(\begin{array}{rrrr} 1.4 & 3.6 & 0.5 & 3.1 \\ -2 & -8 & 5 & 7 \\ 0.8 & 1.8 & 0.8 & 2.8 \end{array}\right) \sim \left(\begin{array}{rrrr}-2 & -8 & 5 & 7 \\ 0 & -2 & 4 & 8 \\ 0 & 0 & 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 5.5 \\ 2 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}- 12.5 \\ 4 \\ 0 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrrr} 0.8 & 6.2 & 4.4 & - 2.4 \\ -1 & -9 & 2 & -2 \\ - 0.5 & -4 & -2 & 1 \end{array}\right) \sim \left(\begin{array}{rrrr}-1 & -9 & 2 & -2 \\ 0 & -1 & 6 & -4 \\ 0 & 0 & 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-52 \\ 6 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}34 \\ -4 \\ 0 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrrr}1 & 0 & 3 & 8 \\ 0.2 & 5 & - 3.4 & 0.6 \\ 0.8 & -3 & 4.8 & 7 \end{array}\right) \sim \left(\begin{array}{rrrr}1 & 0 & 3 & 8 \\ 0 & 5 & -4 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-3 \\ 0.8 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r}-8 \\ 0.2 \\ 0 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrrr} 2.5 & 4.2 & 6.1 & - 12.1 \\ 5 & 8.8 & 9.4 & - 5.4 \\ - 2.5 & - 4.3 & - 5.4 & 7.4 \end{array}\right) \sim \left(\begin{array}{rrrr}5 & 8.8 & 9.4 & - 5.4 \\ 0 & - 0.2 & 1.4 & - 9.4 \\ 0 & 0 & 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 14.2 \\ 7 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 83.8 \\ -47 \\ 0 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

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

Example matrix

$A = \left(\begin{array}{rrrr} 1.2 & 1.8 & - 1.6 & 10.4 \\ -2 & -3 & 1 & -9 \\ 0.6 & 0.9 & 0.2 & 0.2 \end{array}\right) \sim \left(\begin{array}{rrrr}-2 & -3 & 1 & -9 \\ 0 & 0 & -1 & 5 \\ 0 & 0 & 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}- 1.5 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}-2 \\ 0 \\ 5 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrrr} 3.6 & 7.2 & - 10.1 & -4 \\ -4 & -8 & 9 & 0 \\ 2 & 4 & - 3.9 & 1.2 \end{array}\right) \sim \left(\begin{array}{rrrr}-4 & -8 & 9 & 0 \\ 0 & 0 & -2 & -4 \\ 0 & 0 & 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-2 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 4.5 \\ 0 \\ -2 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrrr}-1 & -8 & 8.5 & - 7.5 \\ - 0.4 & - 3.2 & 3.9 & - 3.5 \\ - 0.8 & - 6.4 & 6.7 & - 5.9 \end{array}\right) \sim \left(\begin{array}{rrrr}-1 & -8 & 8.5 & - 7.5 \\ 0 & 0 & 0.5 & - 0.5 \\ 0 & 0 & 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-8 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}1 \\ 0 \\ 1 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrrr} 0.2 & 5.8 & 7.6 & - 4.4 \\ - 0.1 & - 2.9 & - 2.8 & - 7.6 \\ 0.1 & 2.9 & 4.3 & - 7.1 \end{array}\right) \sim \left(\begin{array}{rrrr} 0.2 & 5.8 & 7.6 & - 4.4 \\ 0 & 0 & 1 & - 9.8 \\ 0 & 0 & 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}-29 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 350.4 \\ 0 \\ 9.8 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

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

Example matrix

$A = \left(\begin{array}{rrrr}5 & -4 & 9 & -7 \\ -3 & 2.4 & - 5.4 & 1.2 \\ 3 & - 2.4 & 5.4 & - 3.9 \end{array}\right) \sim \left(\begin{array}{rrrr}5 & -4 & 9 & -7 \\ 0 & 0 & 0 & -3 \\ 0 & 0 & 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 0.8 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 1.8 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}$

Example solution

Example solution

Example matrix

$A = \left(\begin{array}{rrrr} 3.2 & - 6.4 & 4.8 & - 9.6 \\ 4 & -8 & 6 & -7 \\ 2 & -4 & 3 & - 3.1 \end{array}\right) \sim \left(\begin{array}{rrrr}4 & -8 & 6 & -7 \\ 0 & 0 & 0 & -4 \\ 0 & 0 & 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r}2 \\ 1 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{r}- 1.5 \\ 0 \\ 1 \\ 0 \end{array}\right)\right\}$

Example solution

Example solution

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

Example solution

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

Example solution

$3 \times 4$ 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

Example solution

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

Example solution

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

Example solution

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

Example solution

$3 \times 5$

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

$3 \times 5$ with two free variables

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

Example matrix

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

Example solution

Example solution

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

Example solution

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

Example solution

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

Example solution

    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

Example solution

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

Example solution

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

Example solution

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

Example solution

    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

Example solution

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

Example solution

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

Example solution

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

Example solution

  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

Example solution

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

Example solution

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

Example solution

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

Example solution

  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

Example solution

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

Example solution

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

Example solution

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

Example solution

  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

Example solution

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

Example solution

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

Example solution

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

Example solution

$3 \times 5$ 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

Example solution

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

Example solution

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

Example solution

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

Example solution

  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

Example solution

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

Example solution

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

Example solution

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

Example solution

  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

Example solution

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

Example solution

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

Example solution

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

Example solution

  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

Example solution

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

Example solution

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

Example solution

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

Example solution

$3 \times 5$ 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

Example solution

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

Example solution

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

Example solution

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

Example solution

$4 \times 2$

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

$4 \times 2$ with no free variables

   
* *
0 *
0 0
0 0

Example matrix

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

Example solution

Example solution

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

Example solution

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

Example solution

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

Example solution

$4 \times 2$ 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

Example solution

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

Example solution

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

Example solution

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

Example solution

$4 \times 3$

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

$4 \times 3$ with no free variables

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

Example matrix

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

Example solution

Example solution

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

Example solution

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

Example solution

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

Example solution

$4 \times 3$ 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

Example solution

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

Example solution

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

Example solution

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

Example solution

  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

Example solution

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

Example solution

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

Example solution

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

Example solution

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

Example solution

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

Example solution

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

Example solution

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

Example solution

$4 \times 3$ 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

Example solution

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

Example solution

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

Example solution

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

Example solution

$4 \times 4$

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

$4 \times 4$ with no free variables

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

Example matrix

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

Example solution

Example solution

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

Example solution

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

Example solution

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

Example solution

$4 \times 4$ 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

Example solution

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

Example solution

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

Example solution

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

Example solution

    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

Example solution

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

Example solution

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

Example solution

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

Example solution

  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

Example solution

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

Example solution

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

Example solution

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

Example solution

$4 \times 4$ 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

Example solution

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

Example solution

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

Example solution

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

Example solution

  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

Example solution

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

Example solution

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

Example solution

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

Example solution

  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

Example solution

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

Example solution

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

Example solution

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

Example solution

$4 \times 4$ 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

Example solution

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

Example solution

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

Example solution

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

Example solution

$4 \times 5$

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

$4 \times 5$ with one free variable

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

Example matrix

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

Example solution

Example solution

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

Example solution

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

Example solution

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

Example solution

      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

Example solution

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

Example solution

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

Example solution

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

Example solution

    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

Example solution

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

Example solution

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

Example solution

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

Example solution

  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

Example solution

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

Example solution

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

Example solution

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

Example solution

$4 \times 5$ 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

Example solution

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

Example solution

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

Example solution

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

Example solution

    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

Example solution

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

Example solution

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

Example solution

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

Example solution

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

Example solution

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

Example solution

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

Example solution

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

Example solution

    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

Example solution

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

Example solution

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

Example solution

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

Example solution

  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

Example solution

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

Example solution

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

Example solution

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

Example solution

  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

Example solution

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

Example solution

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

Example solution

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

Example solution

  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

Example solution

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

Example solution

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

Example solution

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

Example solution

$4 \times 5$ 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

Example solution

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

Example solution

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

Example solution

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

Example solution

  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

Example solution

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

Example solution

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

Example solution

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

Example solution

  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

Example solution

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

Example solution

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

Example solution

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

Example solution

  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

Example solution

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

Example solution

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

Example solution

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

Example solution

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

Example solution

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

Example solution

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

Example solution

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

Example solution

$4 \times 5$ 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

Example solution

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

Example solution

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

Example solution

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

Example solution

$4 \times 6$

1. $4 \times 6$ with 2 free variables
2. $4 \times 6$ with 3 free variables
3. $4 \times 6$ with 4 free variables
4. $4 \times 6$ with 5 free variables

$4 \times 6$ with two free variables

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

Example matrix

$A = \left(\begin{array}{rrrrrr}-5 & 7 & 8 & -4 & -1 & 0 \\ -3 & 3.2 & 3.8 & - 6.9 & 8.4 & -6 \\ 3.5 & - 2.9 & - 1.6 & 5.8 & - 5.3 & -2 \\ - 3.5 & 6.5 & 8 & -3 & - 5.3 & -4 \end{array}\right) \sim \left(\begin{array}{rrrrrr}-5 & 7 & 8 & -4 & -1 & 0 \\ 0 & 2 & 4 & 3 & -6 & -2 \\ 0 & 0 & 1 & -3 & 6 & -7 \\ 0 & 0 & 0 & -5 & 5 & -8 \end{array}\right)$ , $\textrm{null}(A) = \textrm{span}\left\{\left(\begin{array}{r} 4.7 \\ 7.5 \\ -3 \\ 1 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{r} 3.4 \\ -1 \\ 2.2 \\ - 1.6 \\ 0 \\ 1 \end{array}\right)\right\}$

Example solution

Example solution

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

Example solution

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

Example solution

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

Example solution

      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

Example solution

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

Example solution

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

Example solution

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

Example solution

      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

Example solution

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

Example solution

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

Example solution

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

Example solution

    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

Example solution

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

Example solution

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

Example solution

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

Example solution

    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

Example solution

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

Example solution

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

Example solution

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

Example solution

    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

Example solution

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

Example solution

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

Example solution

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

Example solution

  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

Example solution

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

Example solution

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

Example solution

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

Example solution

  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

Example solution

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

Example solution

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

Example solution

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

Example solution

  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

Example solution

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

Example solution

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

Example solution

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

Example solution

  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

Example solution

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

Example solution

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

Example solution

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

Example solution

$4 \times 6$ 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

Example solution

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

Example solution

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

Example solution

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

Example solution

    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

Example solution

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

Example solution

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

Example solution

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

Example solution

    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

Example solution

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

Example solution

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

Example solution

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

Example solution

    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

Example solution

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

Example solution

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

Example solution

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

Example solution

  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

Example solution

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

Example solution

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

Example solution

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

Example solution

  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

Example solution

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

Example solution

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

Example solution

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

Example solution

  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

Example solution

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

Example solution

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

Example solution

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

Example solution

  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

Example solution

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

Example solution

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

Example solution

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

Example solution

  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

Example solution

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

Example solution

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

Example solution

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

Example solution

  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

Example solution

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

Example solution

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

Example solution

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

Example solution

$4 \times 6$ 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

Example solution

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

Example solution

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

Example solution

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

Example solution

  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

Example solution

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

Example solution

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

Example solution

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

Example solution

  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

Example solution

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

Example solution

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

Example solution

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

Example solution

  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

Example solution

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

Example solution

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

Example solution

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

Example solution

  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

Example solution

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

Example solution

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

Example solution

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

Example solution

$4 \times 6$ 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

Example solution

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

Example solution

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

Example solution

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

Example solution

$5 \times 2$

1. $5 \times 2$ with 0 free variables
2. $5 \times 2$ with 1 free variables

$5 \times 2$ with no free variables

   
* *
0 *
0 0
0 0
0 0

Example matrix

$A = \left(\begin{array}{rr}6 & 5 \\ 5.4 & 2 \\ 2.4 & -3 \\ 0.6 & -1 \\ - 5.4 & -3 \end{array}\right) \sim \left(\begin{array}{rr}6 & 5 \\ 0 & -5 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \{\textbf{0}\}$

Example solution

Example solution

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

Example solution

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

Example solution

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

Example solution

$5 \times 2$ 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

Example solution

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

Example solution

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

Example solution

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

Example solution

$5 \times 3$

1. $5 \times 3$ with 0 free variables
2. $5 \times 3$ with 1 free variables
3. $5 \times 3$ with 2 free variables

$5 \times 3$ with no free variables

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

Example matrix

$A = \left(\begin{array}{rrr}-1 & -4 & - 8.5 \\ 2 & -2 & 3 \\ 0.8 & - 0.3 & - 4.1 \\ - 0.4 & 4.4 & - 0.4 \\ 1.4 & 0.6 & 0.1 \end{array}\right) \sim \left(\begin{array}{rrr}2 & -2 & 3 \\ 0 & -5 & -7 \\ 0 & 0 & -6 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$ , $\textrm{null}(A) = \{\textbf{0}\}$

Example solution

Example solution

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

Example solution

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

Example solution

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

Example solution

$5 \times 3$ 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

Example solution

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

Example solution

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

Example solution

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

Example solution

  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

Example solution

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

Example solution

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

Example solution

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

Example solution

$5 \times 3$ 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

Example solution

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

Example solution

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

Example solution

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

Example solution

$5 \times 4$

1. $5 \times 4$ with 0 free variables
2. $5 \times 4$ with 1 free variables
3. $5 \times 4$ with 2 free variables
4. $5 \times 4$ with 3 free variables

$5 \times 4$ with no free variables

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

$5 \times 4$ with one free variable

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

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

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

$5 \times 4$ with two free variables

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

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

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

$5 \times 4$ with three free variables

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

$5 \times 5$

1. $5 \times 5$ with 0 free variables
2. $5 \times 5$ with 1 free variables
3. $5 \times 5$ with 2 free variables
4. $5 \times 5$ with 3 free variables
5. $5 \times 5$ with 4 free variables

$5 \times 5$ with no free variables

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

$5 \times 5$ with one free variable

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

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

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

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

$5 \times 5$ with two free variables

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

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

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

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

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

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

$5 \times 5$ with three free variables

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

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

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

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

$5 \times 5$ with four free variables

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

$5 \times 6$

1. $5 \times 6$ with 1 free variables
2. $5 \times 6$ with 2 free variables
3. $5 \times 6$ with 3 free variables
4. $5 \times 6$ with 4 free variables
5. $5 \times 6$ with 5 free variables

$5 \times 6$ with one free variable

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

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

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

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

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

$5 \times 6$ with two free variables

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

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

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

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

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

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

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

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

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

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

$5 \times 6$ with three free variables

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

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

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

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

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

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

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

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

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

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

$5 \times 6$ with four free variables

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

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

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

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

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

$5 \times 6$ with five free variables

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

$6 \times 2$

1. $6 \times 2$ with 0 free variables
2. $6 \times 2$ with 1 free variables

$6 \times 2$ with no free variables

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

$6 \times 2$ with one free variable

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

$6 \times 3$

1. $6 \times 3$ with 0 free variables
2. $6 \times 3$ with 1 free variables
3. $6 \times 3$ with 2 free variables

$6 \times 3$ with no free variables

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

$6 \times 3$ with one free variable

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

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

$6 \times 3$ with two free variables

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

$6 \times 4$

1. $6 \times 4$ with 0 free variables
2. $6 \times 4$ with 1 free variables
3. $6 \times 4$ with 2 free variables
4. $6 \times 4$ with 3 free variables

$6 \times 4$ with no free variables

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

$6 \times 4$ with one free variable

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

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

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

$6 \times 4$ with two free variables

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

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

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

$6 \times 4$ with three free variables

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

$6 \times 5$

1. $6 \times 5$ with 0 free variables
2. $6 \times 5$ with 1 free variables
3. $6 \times 5$ with 2 free variables
4. $6 \times 5$ with 3 free variables
5. $6 \times 5$ with 4 free variables

$6 \times 5$ with no free variables

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

$6 \times 5$ with one free variable

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

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

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

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

$6 \times 5$ with two free variables

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

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

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

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

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

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

$6 \times 5$ with three free variables

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

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

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

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

$6 \times 5$ with four free variables

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

$6 \times 6$

1. $6 \times 6$ with 0 free variables
2. $6 \times 6$ with 1 free variables
3. $6 \times 6$ with 2 free variables
4. $6 \times 6$ with 3 free variables
5. $6 \times 6$ with 4 free variables
6. $6 \times 6$ with 5 free variables

$6 \times 6$ with no free variables

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

$6 \times 6$ with one free variable

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

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

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

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

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

$6 \times 6$ with two free variables

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

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

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

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

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

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

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

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

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

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

$6 \times 6$ with three free variables

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

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

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

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

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

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

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

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

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

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

$6 \times 6$ with four free variables

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

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

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

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

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

$6 \times 6$ with five free variables

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