# Definition:Column Space

## Definition

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:

$\map {\mathrm C} {\mathbf A} = \map \span {\begin {bmatrix} a_{1 1} \\ a_{2 1} \\ \vdots \\ a_{m 1} \end {bmatrix}, \begin {bmatrix} a_{1 2} \\ a_{2 2} \\ \vdots \\ a_{m 2} \end {bmatrix}, \cdots, \begin {bmatrix} a_{1 n} \\ a_{2 n} \\ \vdots \\ a_{ mn} \end {bmatrix} }$

## Also known as

The column space is also known as the image.