linearly independent

Let V be a vector spaceMathworldPlanetmath over a field F. We say that v1,,vkV are linearly dependent if there exist scalars λ1,,λkF, not all zero, such that


If no such scalars exist, then we say that the vectors are linearly independent. More generally, we say that a (possibly infinite) subset SV is linearly independent if all finite subsets of S are linearly independent.

In the case of two vectors, linear dependence means that one of the vectors is a scalar multiple of the other. As an alternate characterization of dependence, we also have the following.

Proposition 1.

Let SV be a subset of a vector space. Then, S is linearly dependent if and only if there exists a vS such that v can be expressed as a linear combinationMathworldPlanetmath of the vectors in the set S\{v} (all the vectors in S other than v (

Remark. Linear independence can be defined more generally for modules over rings: if M is a (left) module over a ring R. A subset S of M is linearly independent if whenever r1m1++rnmn=0 for riR and miM, then r1==rn=0.

Title linearly independent
Canonical name LinearlyIndependent
Date of creation 2013-03-22 11:58:40
Last modified on 2013-03-22 11:58:40
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 30
Author rmilson (146)
Entry type Definition
Classification msc 15A03
Synonym linear independence
Defines linearly dependent
Defines linear dependence