Definition:Null Space

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In particular: Put this on a more abstract footing: into a more general vector space, according to Kelley.
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Definition

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

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 $\mathbf A$.



This page or section has statements made on it that ought to be extracted and proved in a Theorem page.
In particular: and it's also the solution set to a system of homogeneous equations
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Also known as

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


Also see

  • Results about null spaces can be found here.


Sources

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