Linear and multilinear algebra; matrix theory Proposition

Vectors span R^m iff the matrix of columns has a pivot in every row

Builds on Linear span Pivot Coordinate space

Statement

Let $v_1, \dots, v_n$ be vectors in the coordinate space $\R^m$ . Let $A$ be the $m \times n$ matrix whose columns are $v_1, \dots, v_n$ .

Then the linear span of $v_1, \dots, v_n$ is all of $\R^m$ if and only if every row of $A$ contains a pivot position.

Remarks

In particular, if $n < m$ , then the vectors cannot span $\R^m$ , since $A$ has at most $n$ pivot positions. A row without a pivot gives a vector $b$ for which the system $Ax = b$ has no solution.

Reading path

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Coordinate space object
Linear span concept
Matrix definition
Row echelon form definition
Pivot definition
Vectors span R^m iff the matrix of columns has a pivot in every row statement

Tag	`D3PN42EP4X`
URI	`m2://D3PN42EP4X`
Kind	statement, proposition
Writing	W0 stub
MSC	15A03, 15A06
Status	This copy of the site has no build data. Run `m2 status D3PN42EP4X`.
About	Linear span
Uses	Linear span, Pivot, Coordinate space
Changed	4 November 2026

Vectors span R^m iff the matrix of columns has a pivot in every row

Statement

Remarks

Reading path

Further reading

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