---
kind: statement
name: A linearly independent set in R^m has at most m vectors
msc:
- 15A03
role: proposition
sources:
- locator: src/section2-4.xml:777
  url: https://github.com/davidaustinm/ula/blob/b6ea263ef23703072b7007485bb7a2d89c5f399b/src/section2-4.xml#L777
tag: XCG7M9ZHP2
uses:
- 0CRMGPAH83
- 08ZDRN6H41
---

## Statement

Let $m$ be a positive integer.
Suppose a set of vectors in $\mathbb{R}^m$ is [[linearly independent|0CRMGPAH83]].
Then the set has at most $m$ elements.

## Remarks

Equivalently, any set of more than $m$ vectors in $\mathbb{R}^m$ is linearly dependent.
This is a special case of the [[Steinitz exchange lemma|08ZDRN6H41]], since the $m$ standard unit vectors span $\mathbb{R}^m$.