---
tag: 00MS9VMQ9Z
kind: statement
role: theorem
name: Every vector space has a basis
msc: [15A03, "03E25"]
uses: [0VTQTPQGAV]
sources:
  - url: https://en.wikipedia.org/wiki/Basis_(linear_algebra)
    locator: Proof that every vector space has a basis
---

## Statement

Let $F$ be a field and let $V$ be a vector space over $F$.
Then $V$ has a [[basis|0KCQ9Y3QKA]].

## Remarks

For a space with a finite spanning set, the proof is elementary: remove vectors from the spanning set until it is linearly independent.
In general, the proof uses Zorn's lemma: a maximal linearly independent subset of $V$ is a basis.
The statement for all vector spaces is equivalent to the axiom of choice, over set theory without choice.
For example, the axiom of choice gives a basis of $\R$ as a vector space over $\Q$, a Hamel basis.

## Explanations

### L1 human

Every vector space has a set of building blocks that is independent and reaches every vector.
For finite-dimensional spaces, we can find one by hand.
For very large spaces, we must use the axiom of choice, and we may not be able to write the basis down.