---
tag: 76GG9B25A1
kind: statement
role: proposition
name: Equivalent conditions for linearly dependent columns of a matrix
msc: [15A03]
uses: [0CRMGPAH83, 0V8F3E48E6, 0BCMGDHZGR, 0KP0QNXPRT]
sources:
  - url: https://github.com/davidaustinm/ula/blob/b6ea263ef23703072b7007485bb7a2d89c5f399b/src/section2-4.xml#L1036
    locator: src/section2-4.xml:1036
---

## Statement

Let $A$ be a matrix with columns $v_1, v_2, \dots, v_n$.
Then the following five conditions are equivalent.

1. The columns of $A$ are [[linearly dependent|0CRMGPAH83]].
2. At least one of the vectors $v_1, \dots, v_n$ is a [[linear combination|0V8F3E48E6]] of the other ones.
3. Some column of $A$ has no [[pivot|0BCMGDHZGR]] position.
4. The [[homogeneous equation|0KP0QNXPRT]] $Ax = 0$ has infinitely many solutions, so it has a nonzero solution.
5. There are scalars $c_1, \dots, c_n$, not all zero, with $c_1 v_1 + \cdots + c_n v_n = 0$.

## Remarks

Condition 5 is the definition of dependence written out for the columns of $A$.
It says the same as condition 4, because $Ax$ is the combination of the columns with weights taken from $x$.
Condition 3 lets one test dependence by row reduction.