Definition:Column Space

Definition
Let $R$ be a ring.

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}$

be a matrix over $R$ such that every column is defined as a vector:


 * $\forall i: 1 \le i \le m: \begin{bmatrix} a_{1i} \\ a_{2i} \\ \vdots \\ a_{mi} \end{bmatrix} \in \mathbf V$

where $\mathbf V$ is some vector space.

Then the column space of $\mathbf A$ is the linear span of all such column vectors:


 * $\operatorname{C}\left({\mathbf A}\right) = \operatorname{span}\left({\begin{bmatrix} a_{11} \\ a_{21} \\ \vdots \\ a_{m1} \end{bmatrix},\begin{bmatrix} a_{12} \\ a_{22} \\ \vdots \\ a_{m2} \end{bmatrix},\cdots,\begin{bmatrix} a_{1n} \\ a_{2n} \\ \vdots \\ a_{mn} \end{bmatrix}}\right)$

Also known as
The column space is also known as the image.

Also see

 * Definition:Rank
 * Definition:Row Space