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.
We say V is finitedimensional if the dimension of V is finite.
Contents
Examples
The vector space R^{3} has
as a basis, and therefore we have dim_{R}(R^{3}) = 3. More generally, dim_{R}(R^{n}) = n, and even more generally, dim_{F}(F^{n}) = 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 finitedimensional vector spaces are equal, one often uses the following criterion: if V is a finitedimensional vector space and W is a linear subspace of V with dim(W) = dim(V), then W = V.
R^{n} has the standard basis {e_{1}, ..., e_{n}}, where e_{i} is the ith column of the corresponding identity matrix. Therefore R^{n} 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 Fvector space.
An important result about dimensions is given by the ranknullity theorem for linear maps.
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