vector subspace

Definition Let V be a vector spaceMathworldPlanetmath over a field F, and let W be a subset of V. If W is itself a vector space, then W is said to be a vector subspace of V. If in addtition VW, then W is a proper vector subspace of V.

If W is a nonempty subset of V, then a necessary and sufficient condition for W to be a subspacePlanetmathPlanetmath is that a+γbW for all a,bW and all γF.

0.0.1 Examples

  1. 1.

    Every vector space is a vector subspace of itself.

  2. 2.

    In every vector space, {0} is a vector subspace.

  3. 3.

    If S and T are vector subspaces of a vector space V, then the vector sum


    and the intersectionMathworldPlanetmath


    are vector subspaces of V.

  4. 4.

    Suppose S and T are vector spaces, and suppose L is a linear mapping L:ST. Then ImL is a vector subspace of T, and KerL is a vector subspace of S.

  5. 5.

    If V is an inner product spaceMathworldPlanetmath, then the orthogonal complementMathworldPlanetmathPlanetmath of any subset of V is a vector subspace of V.

0.0.2 Results for vector subspaces

Theorem 1 [1] Let V be a finite dimensional vector space. If W is a vector subspace of V and dimW=dimV, then W=V.

Theorem 2 [2] (Dimension theorem for subspaces) Let V be a vector space with subspaces S and T. Then

dim(S+T)+dim(ST) = dimS+dimT.


  • 1 S. Lang, Linear Algebra, Addison-Wesley, 1966.
  • 2 W.E. Deskins, Abstract Algebra, Dover publications, 1995.
Title vector subspace
Canonical name VectorSubspace
Date of creation 2013-03-22 11:55:24
Last modified on 2013-03-22 11:55:24
Owner yark (2760)
Last modified by yark (2760)
Numerical id 20
Author yark (2760)
Entry type Definition
Classification msc 15-00
Synonym subspace
Synonym linear subspace
Related topic VectorSpace
Related topic LinearManifold
Defines dimension theorem for subspaces
Defines proper vector subspace