Question

What is the null space of a matrix?

Answer 1

Null space of a matrix:

• The null space of a matrix $A$, is the set of all solutions to the homogeneous equation $A\overrightarrow{x}=\overrightarrow{0}$.
• It is denoted by $NulA$.
• The zero vector is always in the null space of a matrix.

Example:

The nullspace of the matrix $A=\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right]$

By definition, the null space of $A$ consists of all vectors $x$ such that $A\overrightarrow{x}=\overrightarrow{0}$.

Perform elementary row operations on a matrix $A$.

$\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]$

$R_1\leftrightarrow R_2$

$\left[\begin{array}{cc}1& 2\\ 2& 1\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]$

$R_2\to R_2-2R_1$

$\left[\begin{array}{cc}1& 2\\ 0& -3\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]$

From the second row, we get

$y=0$

Back‐substituting this into the first row, we get

$x=0$

Since, the only solution of $A\overrightarrow{x}=\overrightarrow{0}$ is $\overrightarrow{x}=\overrightarrow{0}$,

Hence, the null space of $A$ consists of the zero vector alone.