\begin{definition}[Definition:Column Space]
Let $R$ be a ring.
Let:
:$\mathbf A_{m \times n} = \begin{bmatrix}
a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\
a_{2 1} & a_{2 2} & \cdots & a_{2 n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m 1} & a_{m 2} & \cdots & a_{m n} \\
\end{bmatrix}$
be a matrix over $R$ such that every column is defined as a vector:
:$\forall i: 1 \le i \le m: \begin {bmatrix} a_{1 i} \\ a_{2 i} \\ \vdots \\ a_{m i} \end {bmatrix} \in \mathbf V$
where $\mathbf V$ is some vector space.
Then the '''column space of $\mathbf A$''' is the linear span of all such column vectors:
:$\map {\mathrm C} {\mathbf A} = \map \span {\begin {bmatrix} a_{1 1} \\ a_{2 1} \\ \vdots \\ a_{m 1} \end {bmatrix}, \begin {bmatrix} a_{1 2} \\ a_{2 2} \\ \vdots \\ a_{m 2} \end {bmatrix}, \cdots, \begin {bmatrix} a_{1 n} \\ a_{2 n} \\ \vdots \\ a_{ mn} \end {bmatrix} }$
\end{definition}