---
tag: 0M4SXE5WW3
kind: statement
role: theorem
name: Every linearly independent set extends to a basis
msc: [15A03]
uses: [0VTQTPQGAV, 0EVHJFTG9Z]
sources:
  - url: https://en.wikipedia.org/wiki/Basis_(linear_algebra)
    locator: Properties
---

## Statement

Let $F$ be a field, let $V$ be a vector space over $F$, and let $L \subseteq V$ be a [[linearly independent|0CRMGPAH83]] set.
Then there is a [[basis|0KCQ9Y3QKA]] $B$ of $V$ with $L \subseteq B$.

## Remarks

For finite-dimensional $V$, add to $L$, one at a time, a vector that is not in the span of the current set.
This stops after finitely many steps.
In general, the proof uses Zorn's lemma.
A stronger form: if $L \subseteq S$, with $L$ independent and $S$ spanning, then a basis $B$ exists with $L \subseteq B \subseteq S$.
For example, $\{(1, 1, 0)\}$ extends to the basis $\{(1, 1, 0), (0, 1, 0), (0, 0, 1)\}$ of $\R^3$.

## Explanations

### L1 human

We can always complete an independent set of vectors to a basis.
We keep adding vectors that point in new directions until the set reaches the whole space.
This shows that an independent set can never be larger than a basis.