---
tag: 0VJBSK15SN
kind: statement
role: theorem
name: Finite-dimensional spaces are isomorphic iff they have equal dimension
msc: [15A03]
uses: [0XSJ25Z9M4, 05G2G0P224]
sources:
  - url: https://en.wikipedia.org/wiki/Dimension_(vector_space)
    locator: Properties
---

## Statement

Let $F$ be a field and let $V$ and $W$ be finite-dimensional vector spaces over $F$.
Then $V$ and $W$ are [[isomorphic|0B8T10CE7N]] if and only if $\dim V = \dim W$, where $\dim$ is the [[dimension|03GSZ8RTZJ]].

## Remarks

An isomorphism sends a basis to a basis, so it keeps the dimension.
Conversely, a bijection between bases of $V$ and $W$ extends to a linear isomorphism.
The statement holds for all vector spaces, with dimension as a cardinal number.
For example, $M_{2 \times 3}(F)$, $F^6$ and $F[x]_{<6}$ are pairwise isomorphic.

## Explanations

### L1 human

Up to isomorphism, a finite-dimensional vector space is determined by one number: its dimension.
All three-dimensional real spaces behave like $\R^3$.
Only the names of the vectors differ.