basis
A (Hamel) basis of a vector space is a linearly independent
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 dimension of a vector space, which is precisely the cardinality of the basis, and is denoted by , where is the vector space.
The fact that every vector space has a Hamel basis (http://planetmath.org/EveryVectorSpaceHasABasis) is an important consequence of the axiom of choice
(in fact, that proposition
is equivalent
to the axiom of choice.)
Examples.
- •
, , is a basis for (the -dimensional vector space over the reals). For ,
- •
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 matrices over a division ring, and assuming that the characteristic of the ring is not 2, then so is
- •
The empty set
is a basis for the trivial vector space which consists of the unique element .
Remark. More generally, for any (left) right module over a ring , one may define a (left) right basis for as a subset of such that spans 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 |