# Definition:Null Space

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## 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 $\mathbf A \mathbf x = \mathbf 0$:

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

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

- Definition:Homogeneous Linear Equations
- Definition:Kernel of Linear Transformation
- Kernel of Linear Transformation is Null Space of Matrix Representation

- Results about
**null spaces**can be found here.

## Sources

- 1955: John L. Kelley:
*General Topology*... (previous) ... (next): Chapter $0$: Algebraic Concepts - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**kernel** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**kernel**

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