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 over its base field.^[1] It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension.

For every vector space there exists a basis,^{[lower-alpha 1]} and all bases of a vector space have equal cardinality;^{[lower-alpha 2]} as a result, the dimension of a vector space is uniquely defined. We say V is finite-dimensional if the dimension of V is finite, and infinite-dimensional if its dimension is infinite.

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, dim(V) is typically written.

Examples

The vector space R³ has

\left\{{\begin{pmatrix}1\\0\\0\end{pmatrix}},{\begin{pmatrix}0\\1\\0\end{pmatrix}},{\begin{pmatrix}0\\0\\1\end{pmatrix}}\right\}

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 dim V, we have:

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

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

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.

Trace

The dimension of a vector space may alternatively be characterized as the trace of the identity operator. For instance, $\operatorname {tr} \ \operatorname {id} _{\mathbf {R} ^{2}}=\operatorname {tr} \left({\begin{smallmatrix}1&0\\0&1\end{smallmatrix}}\right)=1+1=2.$ This appears to be a circular definition, but it allows useful generalizations.

Firstly, it allows one to define a notion of dimension when one has a trace but no natural sense of basis. For example, one may have an algebra A with maps $\eta \colon K\to A$ (the inclusion of scalars, called the unit) and a map $\epsilon \colon A\to K$ (corresponding to trace, called the counit). The composition $\epsilon \circ \eta \colon K\to K$ is a scalar (being a linear operator on a 1-dimensional space) corresponds to "trace of identity", and gives a notion of dimension for an abstract algebra. In practice, in bialgebras one requires that this map be the identity, which can be obtained by normalizing the counit by dividing by dimension ( $\epsilon :=\textstyle {\frac {1}{n}}\operatorname {tr}$ ), so in these cases the normalizing constant corresponds to dimension.

Alternatively, one may be able to take the trace of operators on an infinite-dimensional space; in this case a (finite) trace is defined, even though no (finite) dimension exists, and gives a notion of "dimension of the operator". These fall under the rubric of "trace class operators" on a Hilbert space, or more generally nuclear operators on a Banach space.

A subtler generalization is to consider the trace of a family of operators as a kind of "twisted" dimension. This occurs significantly in representation theory, where the character of a representation is the trace of the representation, hence a scalar-valued function on a group $\chi \colon G\to K,$ whose value on the identity $1\in G$ is the dimension of the representation, as a representation sends the identity in the group to the identity matrix: $\chi (1_{G})=\operatorname {tr} \ I_{V}=\dim V.$ One can view the other values $\chi (g)$ of the character as "twisted" dimensions, and find analogs or generalizations of statements about dimensions to statements about characters or representations. A sophisticated example of this occurs in the theory of monstrous moonshine: the j-invariant is the graded dimension of an infinite-dimensional graded representation of the Monster group, and replacing the dimension with the character gives the McKay–Thompson series for each element of the Monster group.^[2]

Notes

↑ if one assumes the axiom of choice
↑ see dimension theorem for vector spaces

References

↑ Itzkov, Mikhail (2009). Tensor Algebra and Tensor Analysis for Engineers: With Applications to Continuum Mechanics. Springer. p. 4. ISBN 978-3-540-93906-1.
↑ Gannon, Terry (2006), Moonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics, ISBN 0-521-83531-3

External links

MIT Linear Algebra Lecture on Independence, Basis, and Dimension by Gilbert Strang at MIT OpenCourseWare

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