The Left and Right Nullspaces

The (right) nullspace is the vector space consisting of all n×1 vectors X satisfying AX = 0. You find a basis for the nullspace by choosing Nullspace Basis from the Matrices submenu.

$\blacktriangleright$ Matrices + Nullspace Basis

$\left[\vphantom{
\begin{array}{rrrr}
-85 & -55 & -37 & -35 \\
97 & 50 & 79 & 56 \\
49 & 63 & 57 & -59 \\
-36 & 8 & 20 & -94
\end{array}
}\right.$$\begin{array}{rrrr}
-85 & -55 & -37 & -35 \\
97 & 50 & 79 & 56 \\
49 & 63 & 57 & -59 \\
-36 & 8 & 20 & -94
\end{array}$$\left.\vphantom{
\begin{array}{rrrr}
-85 & -55 & -37 & -35 \\
97 & 50 & 79 & 56 \\
49 & 63 & 57 & -59 \\
-36 & 8 & 20 & -94
\end{array}
}\right]$, nullspace basis: $\left[\vphantom{
\begin{array}{c}
\stackunder{}{-%
\dfrac{133337}{68264}} \...
...}{\dfrac{74049}{34132}} \\
\dfrac{3085}{9752} \\
1
\end{array}
}\right.$$\begin{array}{c}
\stackunder{}{-%
\dfrac{133337}{68264}} \\
\stackunder{}{\dfrac{74049}{34132}} \\
\dfrac{3085}{9752} \\
1
\end{array}$$\left.\vphantom{
\begin{array}{c}
\stackunder{}{-%
\dfrac{133337}{68264}} \...
...}{\dfrac{74049}{34132}} \\
\dfrac{3085}{9752} \\
1
\end{array}
}\right]$

The left nullspace is the vector space consisting of all 1×m vectors Y satisfying YA = 0. You find a basis for the left nullspace by first taking the transpose of A and then choosing Nullspace Basis from the Matrices submenu.

$\blacktriangleright$ Evaluate

$\left[\vphantom{
\begin{array}{rrrr}
-85 & -55 & -37 & -35 \\
97 & 50 & 79 & 56 \\
49 & 63 & 57 & -59 \\
-36 & 8 & 20 & -94
\end{array}
}\right.$$\begin{array}{rrrr}
-85 & -55 & -37 & -35 \\
97 & 50 & 79 & 56 \\
49 & 63 & 57 & -59 \\
-36 & 8 & 20 & -94
\end{array}$$\left.\vphantom{
\begin{array}{rrrr}
-85 & -55 & -37 & -35 \\
97 & 50 & 7...
...
49 & 63 & 57 & -59 \\
-36 & 8 & 20 & -94
\end{array}
}\right]^{{T}}_{}$ = $\left[\vphantom{
\begin{array}{cccc}
-85 & 97 & 49 & -36 \\
-55 & 50 & 63 & 8 \\
-37 & 79 & 57 & 20 \\
-35 & 56 & -59 & -94
\end{array}
}\right.$$\begin{array}{cccc}
-85 & 97 & 49 & -36 \\
-55 & 50 & 63 & 8 \\
-37 & 79 & 57 & 20 \\
-35 & 56 & -59 & -94
\end{array}$$\left.\vphantom{
\begin{array}{cccc}
-85 & 97 & 49 & -36 \\
-55 & 50 & 63 & 8 \\
-37 & 79 & 57 & 20 \\
-35 & 56 & -59 & -94
\end{array}
}\right]$

$\blacktriangleright$ Matrices + Nullspace Basis

 $\left[\vphantom{
\begin{array}{rrrr}
-85 & 97 & 49 & -36 \\
-55 & 50 & 63 & 8 \\
-37 & 79 & 57 & 20 \\
-35 & 56 & -59 & -94
\end{array}
}\right.$$\begin{array}{rrrr}
-85 & 97 & 49 & -36 \\
-55 & 50 & 63 & 8 \\
-37 & 79 & 57 & 20 \\
-35 & 56 & -59 & -94
\end{array}$$\left.\vphantom{
\begin{array}{rrrr}
-85 & 97 & 49 & -36 \\
-55 & 50 & 63 & 8 \\
-37 & 79 & 57 & 20 \\
-35 & 56 & -59 & -94
\end{array}
}\right]$, nullspace basis: $\left\{\vphantom{ \left[
\begin{array}{r}
1 \\
0 \\
1 \\
-1
\end{array}
\right] }\right.$$\left[\vphantom{
\begin{array}{r}
1 \\
0 \\
1 \\
-1
\end{array}
}\right.$$\begin{array}{r}
1 \\
0 \\
1 \\
-1
\end{array}$$\left.\vphantom{
\begin{array}{r}
1 \\
0 \\
1 \\
-1
\end{array}
}\right]$$\left.\vphantom{ \left[
\begin{array}{r}
1 \\
0 \\
1 \\
-1
\end{array}
\right] }\right\}$

To check that this vector is in the left nullspace, take the transpose of the vector and check the product.

$\blacktriangleright$ Evaluate

 $\left[\vphantom{
\begin{array}{r}
1 \\
0 \\
1 \\
-1
\end{array}
}\right.$$\begin{array}{r}
1 \\
0 \\
1 \\
-1
\end{array}$$\left.\vphantom{
\begin{array}{r}
1 \\
0 \\
1 \\
-1
\end{array}
}\right]^{{T}}_{}$$\left[\vphantom{
\begin{array}{rrrr}
-85 & -55 & -37 & -35 \\
97 & 50 & 79 & 56 \\
49 & 63 & 57 & -59 \\
-36 & 8 & 20 & -94
\end{array}
}\right.$$\begin{array}{rrrr}
-85 & -55 & -37 & -35 \\
97 & 50 & 79 & 56 \\
49 & 63 & 57 & -59 \\
-36 & 8 & 20 & -94
\end{array}$$\left.\vphantom{
\begin{array}{rrrr}
-85 & -55 & -37 & -35 \\
97 & 50 & 79 & 56 \\
49 & 63 & 57 & -59 \\
-36 & 8 & 20 & -94
\end{array}
}\right]$ =  $\left[\vphantom{
\begin{array}{cccc}
0 & 0 & 0 & 0
\end{array}
}\right.$$\begin{array}{cccc}
0 & 0 & 0 & 0
\end{array}$$\left.\vphantom{
\begin{array}{cccc}
0 & 0 & 0 & 0
\end{array}
}\right]$