---
tag: 009892PZ30
kind: definition
name: Linear map of a matrix
msc: [15A04]
defines: 06HMCTAQH3
uses: [0YYMY95FWS]
sources:
  - url: https://en.wikipedia.org/wiki/Transformation_matrix
    locator: lead section
---

## Definition

Let $F$ be a field, and let $V$ and $W$ be vector spaces over $F$ with bases $B = (b_1, \dots, b_n)$ and $C = (c_1, \dots, c_m)$.
Let $A = (a_{ij})$ be an $m \times n$ matrix over $F$.
The *linear map of $A$* in the bases $B$ and $C$ is the linear map $T_A \colon V \to W$ with
$$T_A(b_j) = \sum_{i=1}^{m} a_{ij} c_i \qquad (1 \le j \le n).$$

## Remarks

The map $A \mapsto T_A$ is the inverse of the map $T \mapsto [T]_{C,B}$ of the [[matrix of a linear map|0YQSGNYE3K]].
For $V = F^n$ and $W = F^m$ with the standard bases, $T_A$ is the map $x \mapsto Ax$, see [[multiplication by a matrix as a linear map|094V2VBJXJ]].
In Mathlib, this map is `Matrix.toLin`.

## Explanations

### L1 human

A matrix and two bases together define a linear map.
Column $j$ tells us where the map sends basis vector $j$.
This undoes the step from a linear map to its matrix.