Hamel dimension : Dimension (linear algebra) 

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

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