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. It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension. For every vector space there exists a basis, and all bases of a vector space have equal cardinality; as a result, the dimension of a vector space is uniquely defined. We say is if the dimension of is finite, and if its dimension is infinite. The dimension of the vector space over the field can be written as or as read "dimension of over ". When can be inferred from context, is typically written. The vector space has as a standard basis, and therefore More generally, and even more generally, for any field The complex numbers are both a real and complex vector space; we have and So the dimension depends on the base field. The only vector space with dimension is the vector space consisting only of its zero element. If is a linear subspace of then To show that two finite-dimensional vector spaces are equal, the following criterion can be used: if is a finite-dimensional vector space and is a linear subspace of with then The space has the standard basis where is the -th column of the corresponding identity matrix. Therefore, has dimension Any two finite dimensional vector spaces over with the same dimension are isomorphic. Any bijective map between their bases can be uniquely extended to a bijective linear map between the vector spaces. If is some set, a vector space with dimension over can be constructed as follows: take the set of all functions such that for all but finitely many in These functions can be added and multiplied with elements of to obtain the desired -vector space. An important result about dimensions is given by the rank–nullity theorem for linear maps. If is a field extension, then is in particular a vector space over Furthermore, every -vector space is also a -vector space.

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