\begin{definition}[Definition:Null Space]
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 $\mathbf A$.
{{expand|and it's also the solution set to a system of homogeneous equations<br/>No, add that on a separate page.}}
\end{definition}