---
tag: 0AJ0KP1B31
kind: definition
name: Multiplication by a matrix as a linear map
msc: [15A04]
defines: 094V2VBJXJ
uses: [0CCQV3884B, 0RJF668P1T]
sources:
  - url: https://en.wikipedia.org/wiki/Transformation_matrix
    locator: Uses
---

## 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|0PDVQEP0ZC]].

## 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|0PHDYNR24D]] of $A$, and its image is the [[column space|03AJKXH1AV]] 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.