A (Hamel) basis of a vector spaceMathworldPlanetmath is a linearly independentMathworldPlanetmath spanning set.

It can be proved that any two bases of the same vector space must have the same cardinality. This introduces the notion of dimensionPlanetmathPlanetmathPlanetmath of a vector space, which is precisely the cardinality of the basis, and is denoted by dim(V), where V is the vector space.

The fact that every vector space has a Hamel basisMathworldPlanetmath ( is an important consequence of the axiom of choiceMathworldPlanetmath (in fact, that propositionPlanetmathPlanetmath is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to the axiom of choice.)


  • β={ei}, 1in, is a basis for n (the n-dimensional vector space over the reals). For n=4,

  • β={1,x,x2} is a basis for the vector space of polynomials with degree at most 2, over a division ring.

  • The set


    is a basis for the vector space of 2×2 matrices over a division ring, and assuming that the characteristic of the ring is not 2, then so is

  • The empty setMathworldPlanetmath is a basis for the trivial vector space which consists of the unique element 0.

Remark. More generally, for any (left) right module M over a ring R, one may define a (left) right basis for M as a subset B of M such that B spans M and is linearly independent. However, unlike bases for a vector space, bases for a module may not have the same cardinality.

Title basis
Canonical name Basis
Date of creation 2013-03-22 12:01:57
Last modified on 2013-03-22 12:01:57
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 22
Author mathcam (2727)
Entry type Definition
Classification msc 15A03
Synonym Hamel basis
Related topic Span
Related topic IntegralBasis
Related topic BasicTensor
Related topic Aliasing
Related topic Subbasis
Related topic Blade
Related topic ProofOfGramSchmidtOrthogonalizationProcedure
Related topic LinearExtension