---
kind: statement
msc:
- 15A03
name: Basis of R^m as columns row equivalent to the identity
role: proposition
sources:
- locator: src/section3-2.xml:390
  url: https://github.com/davidaustinm/ula/blob/b6ea263ef23703072b7007485bb7a2d89c5f399b/src/section3-2.xml#L390
tag: JXTGZPZP47
uses:
- 0KCQ9Y3QKA
- 095DDGZF16
- 06DW9BAWB1
- 0N1MY1T0W9
---

## Statement

Let $v_1, \dots, v_n$ be vectors in $\mathbb{R}^m$. Let $A$ be the [[matrix|0N1MY1T0W9]] whose columns are $v_1, \dots, v_n$.

The vectors $v_1, \dots, v_n$ form a [[basis|0KCQ9Y3QKA]] of $\mathbb{R}^m$ if and only if $A$ is [[row equivalent|095DDGZF16]] to the [[identity matrix|06DW9BAWB1]].

## Remarks

The identity matrix is square, so row equivalence to it forces $A$ to be an $m \times m$ matrix. Hence $n = m$. Every basis of $\mathbb{R}^m$ has exactly $m$ vectors.