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:
See also: Dimension
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