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.)
, , is a basis for (the -dimensional vector space over the reals). For ,
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
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.
|Date of creation||2013-03-22 12:01:57|
|Last modified on||2013-03-22 12:01:57|
|Last modified by||mathcam (2727)|