---
tag: 0XSJ25Z9M4
kind: definition
name: Linear isomorphism
msc: [15A04]
defines: 0B8T10CE7N
uses: [0RJF668P1T]
sources:
  - url: https://en.wikipedia.org/wiki/Linear_map
    locator: Definition and first consequences
---

## Definition

Let $F$ be a field and let $V$ and $W$ be vector spaces over $F$.
A *linear isomorphism* from $V$ to $W$ is a bijective [[linear map|0NRYHJPQG2]] $T \colon V \to W$.
The spaces $V$ and $W$ are *isomorphic*, written $V \cong W$, when a linear isomorphism $V \to W$ exists.

## Remarks

The inverse map $T^{-1} \colon W \to V$ of a linear isomorphism is linear.
A linear map is an isomorphism if and only if it sends some basis, or each basis, of $V$ to a basis of $W$.
For an $n \times n$ matrix $A$ over $F$, the map $x \mapsto Ax$ is an isomorphism of $F^n$ if and only if $A$ is [[invertible|0EB9N0S33C]].
[[Finite-dimensional spaces are isomorphic if and only if they have equal dimension|0VJBSK15SN]].

## Explanations

### L1 human

A linear isomorphism is a perfect translation between two vector spaces.
It matches the vectors one to one and respects addition and scaling.
Isomorphic spaces have the same linear algebra, even when their vectors look different.