# 32 Null Spaces

The null space of an matrix can be thought of as the set of all matrices that can be multiplied by to get the zero matrix. That is, the null space of , denoted as , is the set of all solutions x of the homogeneous equation

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

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

which can be rewritten as

Using matrix multiplication, we have the following system of equations

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 , where is a scalar. It follows from the second equation that . Finally, it follows from equation one that . Therefore,

As we can see, every solution to

can be written as a multiple of the vector

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.