Linear and multilinear algebra; matrix theory Proposition

Equivalent conditions for linearly dependent columns of a matrix

Builds on Linear independence Linear combination Pivot Homogeneous linear equation

Statement

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

The columns of $A$ are linearly dependent.
At least one of the vectors $v_1, \dots, v_n$ is a linear combination of the other ones.
Some column of $A$ has no pivot position.
The homogeneous equation $Ax = 0$ has infinitely many solutions, so it has a nonzero solution.
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.

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Homogeneous linear equation concept
Linear combination concept
Linear independence concept
Matrix definition
Row echelon form definition
Pivot definition
Equivalent conditions for linearly dependent columns of a matrix statement

Tag	`76GG9B25A1`
URI	`m2://76GG9B25A1`
Kind	statement, proposition
Writing	W0 stub
MSC	15A03
Status	This copy of the site has no build data. Run `m2 status 76GG9B25A1`.
Uses	Linear independence, Linear combination, Pivot, Homogeneous linear equation
Changed	4 November 2026

Equivalent conditions for linearly dependent columns of a matrix

Statement

Remarks

Reading path

Further reading

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