Dimension (vector space)

In mathematics, the dimension of a vector space V is the cardinality (i.e. the number of vectors) of a basis of V over its base field.[1][2] It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension.

An important result about dimensions is given by the rank–nullity theorem for linear maps.

A vector space can be seen as a particular case of a matroid, and in the latter there is a well-defined notion of dimension. The length of a module and the rank of an abelian group both have several properties similar to the dimension of vector spaces.

The Krull dimension of a commutative ring, named after Wolfgang Krull (1899–1971), is defined to be the maximal number of strict inclusions in an increasing chain of prime ideals in the ring.

Alternatively, it may be possible to take the trace of operators on an infinite-dimensional space; in this case a (finite) trace is defined, even though no (finite) dimension exists, and gives a notion of "dimension of the operator". These fall under the rubric of "trace class operators" on a Hilbert space, or more generally nuclear operators on a Banach space.