Dimension (vector space)

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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 dimF(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 finite-dimensional if the dimension of V is finite.

Contents

Examples

The vector space R3 has

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

The complex numbers C are both a real and complex vector space; we have dimR(C) = 2 and dimC(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, ..., en}, where ei 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 : BF 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.

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