Linear and multilinear algebra; matrix theory Proposition

Basis of R^m as columns row equivalent to the identity

Builds on Basis Row equivalence Identity matrix Matrix

Statement

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

The vectors $v_1, \dots, v_n$ form a basis of $\mathbb{R}^m$ if and only if $A$ is row equivalent to the identity matrix.

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.

Reading path

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Basis concept
Identity matrix object
Matrix definition
Elementary row operations definition
Row equivalence definition
Basis of R^m as columns row equivalent to the identity statement

Tag	`JXTGZPZP47`
URI	`m2://JXTGZPZP47`
Kind	statement, proposition
Writing	W0 stub
MSC	15A03
Status	This copy of the site has no build data. Run `m2 status JXTGZPZP47`.
Uses	Basis, Row equivalence, Identity matrix, Matrix
Changed	4 November 2026

Basis of R^m as columns row equivalent to the identity

Statement

Remarks

Reading path

Further reading

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