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

λ1v1++λkvk=0.

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 (http://planetmath.org/SetDifference)).

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