Multiplication by a matrix as a linear map

Definition

Let $F$ be a field and let $A$ be an $m \times n$ matrix over $F$. The linear map of multiplication by $A$ is $$L_A \colon F^n \to F^m, \qquad L_A(x) = Ax,$$ where $Ax$ is the matrix–vector product.

Remarks

The map $L_A$ is linear, because $A(x + y) = Ax + Ay$ and $A(cx) = c\, Ax$. Its matrix in the standard bases is $A$, because $L_A(e_j)$ is column $j$ of $A$. Each linear map $F^n \to F^m$ equals $L_A$ for exactly one matrix $A$. The kernel of $L_A$ is the null space of $A$, and its image is the column space of $A$.

Explanations

L1 human

Multiplying vectors by a fixed matrix is a linear map. The matrix $\begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}$ stretches the $x$-direction by $2$ and the $y$-direction by $3$. Every linear map between coordinate spaces is of this kind.