Definition:Column Space
(Redirected from Definition:Image of Matrix)
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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:
- $\map {\mathrm C} {\mathbf A} = \map \span {\begin {bmatrix} a_{1 1} \\ a_{2 1} \\ \vdots \\ a_{m 1} \end {bmatrix}, \begin {bmatrix} a_{1 2} \\ a_{2 2} \\ \vdots \\ a_{m 2} \end {bmatrix}, \cdots, \begin {bmatrix} a_{1 n} \\ a_{2 n} \\ \vdots \\ a_{m n} \end {bmatrix} }$
Also known as
The column space of a matrix is also known as its image.
Also see
- Results about column space can be found here.
Sources
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): column space
- For a video presentation of the contents of this page, visit the Khan Academy.