---
kind: statement
msc:
- 15A03
name: Columns are independent iff every column has a pivot
role: proposition
sources:
- locator: src/section2-4.xml:739
  url: https://github.com/davidaustinm/ula/blob/b6ea263ef23703072b7007485bb7a2d89c5f399b/src/section2-4.xml#L739
tag: 55K7JBE13G
uses:
- 0EVHJFTG9Z
- 0BCMGDHZGR
- 0N1MY1T0W9
---

## Statement

Let $F$ be a field and let $A$ be a [[matrix|0N1MY1T0W9]] over $F$.
The columns of $A$ are [[linearly independent|0CRMGPAH83]] if and only if each column of $A$ contains a [[pivot|0J0BBCMJ5W]] position.

## Remarks

Equivalently, the columns are dependent exactly when some column has no pivot.
A column without a pivot gives a free variable of the system $Ax = 0$, and then this system has a nonzero solution.