# Definition:Null Space

## Definition

Let:

$\mathbf A_{m \times n} = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \\ \end{bmatrix}$, $\mathbf x_{n \times 1} = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}$, $\mathbf 0_{m \times 1} = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix}$

be matrices where each column is a member of a real vector space.

The set of all solutions to $\mathbf A \mathbf x = \mathbf 0$:

$\map {\operatorname N} {\mathbf A} = \set {\mathbf x \in \R^n : \mathbf {A x} = \mathbf 0}$

is called the null space of $\mathbf A$.

## Also known as

The null space of $\mathbf A$ is also known as the nullspace of $\mathbf A$.

## Also see

• Results about null spaces can be found here.

## Sources

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