Chemistry - 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 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.

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Examples

E.g. The vector space R³ has {(1,0,0), (0,1,0), (0,0,1)} as a basis, and therefore we have dim_R(R³) = 3. More generally, dim_R(Rⁿ) = n. And more generally still, dim_F(Fⁿ) = 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 2n.

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 dimV, we have:

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

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

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.