Column space

Definition

Let $F$ be a field and let $A$ be an $m \times n$ matrix over $F$ with columns $a_1, \dots, a_n \in F^m$. The column space of $A$ is $$\operatorname{Col}(A) = \operatorname{span}(a_1, \dots, a_n) \subseteq F^m,$$ the linear span of the columns.

Remarks

The column space is $\{Ax : x \in F^n\}$, the image of the map $x \mapsto Ax$. The system $Ax = b$ is consistent if and only if $b \in \operatorname{Col}(A)$. The dimension of the column space is the rank of $A$. The pivot columns of $A$ form a basis of $\operatorname{Col}(A)$.

Explanations

L1 human

The column space collects all the combinations of the columns of a matrix. These are exactly the vectors $b$ for which $Ax = b$ has a solution. For the matrix $\begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix}$, the column space is the line spanned by $(1, 2)$.