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. It is sometimes called Hamel dimension or algebraic dimension to distinguish it from other types of dimension. All bases of a vector space have equal cardinality (see dimension theorem for vector spaces) and so the dimension of a vector space is uniquely defined. The dimension of the vector space V over the field F can be written as dim_F(V) or as [V : F], read "dimension of V over F". When F can be inferred from context, often just dim(V) is written.

Examples

The vector space R³ has

as a basis, and therefore we have dim_R(R³) = 3. More generally, dim_R(Rⁿ) = n, and even more generally, dim_F(Fⁿ) = n for any field F.

The complex numbers C are both a real and complex vector space; we have dim_R(C) = 2 and dim_C(C) = 1. So the dimension depends on the base field.

The only vector space with dimension 0 is {0}, the vector space consisting only of its zero element.

Facts

If W is a linear subspace of V, then dim(W) ≤ dim(V).

To show that two finite-dimensional vector spaces are equal, one often uses the following criterion: if V is a finite-dimensional vector space and W is a linear subspace of V with dim(W) = dim(V), then W = V.

Rⁿ has the standard basis {e₁, ..., e_n}, where e_i is the i-th column of the corresponding identity matrix. Therefore Rⁿ has dimension n.

Any two vector spaces over F having 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 B is some set, a vector space with dimension |B| over F can be constructed as follows: take the set F^(B) of all functions f : B → F such that f(b) = 0 for all but finitely many b in B. These functions can be added and multiplied with elements of F, and we obtain the desired F-vector space.

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

If F/K is a field extension, then F is in particular a vector space over K. Furthermore, every F-vector space V is also a K-vector space. The dimensions are related by the formula

dim_K(V) = dim_K(F) dim_F(V).

In particular, every complex vector space of dimension n is a real vector space of dimension 2n.

Some simple formulae relate the 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 F then, denoting the dimension of V by dimV, we have:

If dim V is finite, then |V| = |F|^dimV.

If dim V is infinite, then |V| = max(|F|, dimV).

Generalizations

One can see a vector space 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.