properties of linear independence

Let V be a vector spaceMathworldPlanetmath over a field k. Below are some basic properties of linear independence.

  1. 1.

    SV is never linearly independentMathworldPlanetmath if 0S.


    Since 10=0. ∎

  2. 2.

    If S is linearly independent, so is any subset of S. As a result, if S and T are linearly independent, so is ST. In addition, is linearly independent, its spanning setPlanetmathPlanetmath being the singleton consisting of the zero vector 0.


    If r1v1+rnvn=0, where viT, then viS, so ri=0 for all i=1,,n. ∎

  3. 3.

    If S1S2 is a chain of linearly independent subsets of V, so is their union.


    Let S be the union. If r1v1+rnvn=0, then viSa(i), for each i. Pick the largest Sa(i) so that all vi’s are in it. Since this set is linearly independent, ri=0 for all i. ∎

  4. 4.

    S is a basis for V iff S is a maximal linear independent subset of V. Here, maximal means that any proper supersetMathworldPlanetmath of S is linearly dependent.


    If S is a basis for V, then it is linearly independent and spans V. If we take any vector vS, then v can be expressed as a linear combinationMathworldPlanetmath of elements in S, so that S{v} is no longer linearly independent, for the coefficient in front of v is non-zero. Therefore, S is maximal.

    Conversely, suppose S is a maximal linearly independent set in V. Let W be the span of S. If WV, pick an element vV-W. Suppose 0=r1v1+rnvn+rv, where viS, then -rv=r1v1++rnvn. If r0, then v would be in the span of S, contradicting the assumptionPlanetmathPlanetmath. So r=0, and as a result, ri=0, since S is linearly independent. This shows that S{v} is linearly independent, which is impossible since S is assumed to be maximal. Therefore, W=V. ∎

Remark. All of the properties above can be generalized to modules over rings, except the last one, where the implicationMathworldPlanetmath is only one-sided: basis implying maximal linear independence.

Title properties of linear independence
Canonical name PropertiesOfLinearIndependence
Date of creation 2013-03-22 18:05:37
Last modified on 2013-03-22 18:05:37
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 5
Author CWoo (3771)
Entry type Result
Classification msc 15A03