Null Space

A null space, or kernel, of a $\mathbf{A}$ is the set of solutions to

$$\mathbf{A}\vec{x}=\vec{0}$$

For example, for

$$\mathbf{A}=\left(\begin{array}{ccccc}-3& 6& -1& 1& -7\\ 1& -2& 2& 3& -1\\ 2& -4& 5& 8& -4\end{array}\right)$$

Solving

$$\mathbf{A}\vec{x}=\vec{0}⟹\text{rref}\left(\begin{array}{cccccc}-3& 6& -1& 1& -7& 0\\ 1& -2& 2& 3& -1& 0\\ 2& -4& 5& 8& -4& 0\end{array}\right)$$

gives us

$$\{\begin{array}{l}{x}_1-2x_2-x_4+3x_5=0\\ x_3+2x_4-2x_5=0\end{array}$$

The general form of $\vec{x}$ must be

$$\vec{x}=x_2\left(\begin{array}{c}2\\ 1\\ 0\\ 0\\ 0\end{array}\right)+x_4\left(\begin{array}{c}1\\ 0\\ -2\\ 0\\ 0\end{array}\right)+x_5\left(\begin{array}{c}-3\\ 0\\ 2\\ 0\\ 0\end{array}\right)$$

So

$$\text{Nul}\mathbf{A}=\text{span}\{\overrightarrow{x_2}\overrightarrow{x_3},\overrightarrow{x_4}\}$$

Linear Transformations for Vector Spaces §

1. $T(\vec{u}+\vec{v})=T(\vec{u})+T(\vec{v})$
2. $T(c\vec{u})=cT(\vec{u})$

The kernel of $T$ is the set of all $\vec{u}\in V:T(\vec{u})=\vec{0}$.