Encyclopedia > Linear algebra Dimension of a vector space

  Article Content

Hamel dimension

Redirected from Linear algebra/Dimension of a vector space

The dimension of a vector space V is the cardinality (i.e. the number of vectors) of a basis of V. It is sometimes called Hamel dimension when it is necessary to distinguish it from other types of dimension. Every basis of a vector space has equal cardinality and so the Hamel dimension of a vector space is uniquely defined.

The Hamel dimension is a natural generalization of the dimension of Euclidean space, since E n is a vector space of dimension n over R (the reals). However, the Hamel dimension depends on the base field, so while R has dimension 1 when considered as a vector space over itself, it has dimension c (the cardinality of the continuum) when considered as a vector space over Q (the rationals).

Some simple formulae relate the Hamel dimension of a vector space with the cardinality of the base field and the cardinality of the space itself. If V is a vector space over a field K then, denoting the Hamel dimension of V by dimV, we have:

If dimV is finite, then |V| = |K|dimV.
If dimV is infinite, then |V| = max(|K|, dimV).

See also: Dimension

All Wikipedia text is available under the terms of the GNU Free Documentation License

  Search Encyclopedia

Search over one million articles, find something about almost anything!
  Featured Article
Wheatley Heights, New York

... 3.41% Asian, 0.00% Pacific Islander, 4.15% from other races, and 3.73% from two or more races. 11.67% of the population are Hispanic or Latino of any race. There ...

This page was created in 85 ms