---
tag: "D3PN42EP4X"
kind: statement
role: proposition
name: Vectors span R^m iff the matrix of columns has a pivot in every row
msc: [15A03, 15A06]
uses: [0DTY8B9HMS, 0BCMGDHZGR, 0XM8BZD76G]
about: [0DTY8B9HMS]
sources:
  - url: https://github.com/davidaustinm/ula/blob/b6ea263ef23703072b7007485bb7a2d89c5f399b/src/section2-3.xml#L1347
    locator: src/section2-3.xml:1347
---

## Statement

Let $v_1, \dots, v_n$ be vectors in [[the coordinate space|0XM8BZD76G]] $\R^m$.
Let $A$ be the $m \times n$ matrix whose columns are $v_1, \dots, v_n$.

Then the [[linear span|0DTY8B9HMS]] of $v_1, \dots, v_n$ is all of $\R^m$ if and only if every row of $A$ contains a [[pivot|0BCMGDHZGR]] 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.