Null Spaces

Abigail Huettig

32 Null Spaces

The null space of an $m\times n$ matrix $A$ can be thought of as the set of all matrices that can be multiplied by $A$ to get the zero matrix. That is, the null space of $A$ , denoted as $N(A)$ , is the set of all solutions x of the homogeneous equation

$A\text{\textbf{x}=\textbf{0}}$

according to Larson and Falvo [16].

Let’s now look at an example. This problem is one I completed as homework for Elementary Linear Algebra and comes from Larson and Falvo [16].

Example 66
Find a basis for the null space of the matrix

$A=\begin{bmatrix} 1&2&3\\ 0&1&0 \end{bmatrix}$

Solution
To find the null space, we need to find the solutions x to

$\begin{bmatrix} 1&2&3\\ 0&1&0 \end{bmatrix}\textbf{x}=\textbf{0},$

which can be rewritten as

$\begin{bmatrix} 1&2&3\\ 0&1&0 \end{bmatrix}\begin{bmatrix} x_1\\ x_2\\ x_3 \end{bmatrix}=\begin{bmatrix} 0\\ 0 \end{bmatrix}.$

Using matrix multiplication, we have the following system of equations

$\begin{align*} 1x_1+2x_2+3x_3&=0\\ 0x_1+1x_2+0x_3&=0 \end{align*}$

Because the coefficient matrix of this system is already in row-echelon form, we do not need to perform any elementary row operations before solving this system. So, we can let $x_1=s$ , where $s$ is a scalar. It follows from the second equation that $x_2=0$ . Finally, it follows from equation one that $x_3=-\displaystyle\frac{s}{3}$ . Therefore,

$\begin{align*} \textbf{x}&=\left[ \begin{array}{@{}*{7}{r}@{}} s\\ 0 \\ -\displaystyle\frac{s}{3} \\ \end{array} \right] \\ &=s\left[ \begin{array}{@{}*{7}{r}@{}} 1\\ 0 \\ -\displaystyle\frac{1}{3} \\ \end{array} \right] &&\text{where $s$ is a real number} \end{align*}$

As we can see, every solution to

$A\textbf{x}=\textbf{0}$

can be written as a multiple of the vector

$\left[ \begin{array}{@{}*{7}{r}@{}} 1\\ 0 \\ -\displaystyle\frac{1}{3} \\ \end{array} \right]$

so this vector forms a basis for the null space.

We have seen how matrices and other vectors work together regarding row, column, and null spaces along with their bases. We are now going to look at a topic that utilizes the ideas of square matrices, their determinants, and vectors known as eigenvectors.

32 Null Spaces

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