---
tag: 0R8X9XA83A
kind: statement
role: proposition
name: A linear map is determined by its values on a basis
msc: [15A04]
uses: [0VTQTPQGAV, 0RJF668P1T]
sources:
  - url: https://en.wikipedia.org/wiki/Basis_(linear_algebra)
    locator: Properties
---

## Statement

Let $F$ be a field, let $V$ and $W$ be vector spaces over $F$, and let $(b_i)_{i \in I}$ be a [[basis|0KCQ9Y3QKA]] of $V$.
If $S, T \colon V \to W$ are [[linear maps|0NRYHJPQG2]] with $S(b_i) = T(b_i)$ for all $i \in I$, then $S = T$.
Moreover, for each family $(w_i)_{i \in I}$ of vectors of $W$, there is exactly one linear map $T \colon V \to W$ with $T(b_i) = w_i$ for all $i \in I$.

## Remarks

The map is $T(\sum_i a_i b_i) = \sum_i a_i w_i$.
The values on a basis can be chosen freely; the values on a merely spanning family cannot.
This statement is the reason why a linear map between finite-dimensional spaces is described by a matrix.

## Explanations

### L1 human

A linear map is known as soon as we know where it sends the basis vectors.
Every other vector is a combination of basis vectors, and linearity tells us where it goes.
Conversely, we may send the basis vectors anywhere we like.