What is a null space?

A null space, or the kernel, is a linear subspaceSubspace of a larger vector space. that contains a set of vectors that transform to the $zero^{th}$ vector under a given linear transformation: multiplication with the matrix $A$.

We represent it as follows:

Here:

• $A$ is an $m \times n$ matrix.

• Vector $\vec{x}$ is the vector that transforms to the $zero^{th}$ vector after multiplication with the matrix $A$ .

We can also define the null space of the matrix $A$ in the following notation:

Example 1

Let's determine the null space of the following matrix:

We find the null space of $A$ using the following steps:

We can write the equation above in the form of the matrix below:

We transform the matrix into a reduced row echelon form:

This gives us the following null space:

All the linear combinations of the vector $\{ -9 ,-5, -4, 1 \}$ belong to the null space of $A$. In simpler words, the spanThe set of all linear combinations of the vectors of the solution vector $\{ -9 ,-5, -4, 1 \}$ equals the null space of the matrix $A$.

Example 2

Let's determine the null space of the following matrix:

We finding the null space of $A$ using the following steps:

We can write the equation above in the form of the matrix below:

We transform the matrix into reduced row echelon form:

This gives us the following null space:

In simpler words, the null space of the matrix $A$ contains the $zero^{th}$ vector.

Note: If $A$ is an $n \times n$ matrix and has linearly independent column vectors, the null space of $A$ always contains a $zero$ vector.

Quiz

To understand the topic better, let's solve the following question:

Applications

Null spaces have numerous applications in linear algebra, and the real world too. Some of their applications are listed below:

• They determine the uniqueness of solutions.

• They are used in the calculation of Eigenvectors and Eigenspaces.

• They help in diagonalizing matrices.

• They aid the singular value decomposition of a matrix.

• They hold importance in solving differential equations.