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Hamel dimension)

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 Hamel dimension of a vector space is uniquely defined. The dimension of the vectorspace *V* over the field *F* is written as dim_{F}(*V*).

We say *V* is **finite-dimensional** if the dimension of *V* is finite.

## Examples

E.g. The vector space **R**^{3} has {(1,0,0), (0,1,0), (0,0,1)} as a basis, and therefore we have dim_{R}(**R**^{3}) = 3. More generally, dim_{R}(**R**^{n}) = *n*. And more generally still, dim_{F}(*F*^{n}) = *n*.

The complex numbers **C** are a real vector space; we have dim_{R}(**C**) = 2 and dim_{C}(**C**) = 1. So the Hamel 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*.

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*-vectorspace.

An important result about dimensions related to a linear transformation is given by the rank-nullity theorem.

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 2*n*.

Some simple formulae relate the Hamel 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 Hamel dimension of *V* by dim*V*, we have:

- If dim
*V* is finite, then |*V*| = |*F*|^{dimV}.
- If dim
*V* is infinite, then |*V*| = max(|*F*|, dim*V*).

## Generalizations

One can see a vector space as a particular case of a pregeometry, 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 Hamel dimension of vector spaces.

## See also

Last updated: 05-07-2005 17:57:12

Last updated: 05-13-2005 07:56:04