---
tag: 0WK0Y8RH1M
kind: statement
role: theorem
name: Dimension theorem for vector spaces
aliases: [invariance-of-dimension]
msc: [15A03]
uses: [0VTQTPQGAV]
sources:
  - url: https://en.wikipedia.org/wiki/Dimension_theorem_for_vector_spaces
    locator: lead section
same_as:
  wikidata: Q3527205
  wikipedia: Dimension theorem for vector spaces
---

## Statement

Let $F$ be a field and let $V$ be a vector space over $F$.
Any two [[bases|0KCQ9Y3QKA]] of $V$ have the same cardinality.

## Remarks

For finite bases, this follows from the [[Steinitz exchange lemma|08ZDRN6H41]].
For infinite bases, the proof uses cardinal arithmetic: each vector of one basis is a combination of finitely many vectors of the other basis.
The common cardinality is the [[dimension|03GSZ8RTZJ]] of $V$.
The theorem holds more generally for free modules over a commutative ring that is not the zero ring.

## Explanations

### L1 human

However we choose a basis of a space, it always has the same number of vectors.
So "the number of vectors in a basis" is a property of the space itself.
We call this number the dimension.