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.

For every vector space there exists a basis,[a] and all bases of a vector space have equal cardinality;[b] as a result, the dimension of a vector space is uniquely defined.

are both a real and complex vector space; we have

So the dimension depends on the base field.

the vector space consisting only of its zero element.

To show that two finite-dimensional vector spaces are equal, the following criterion can be used: if

-th column of the corresponding identity matrix.

Any two finite dimensional vector spaces over

Any bijective map between their bases can be uniquely extended to a bijective linear map between the vector spaces.

is some set, a vector space with dimension

These functions can be added and multiplied with elements of

An important result about dimensions is given by the rank–nullity theorem for linear maps.

In particular, every complex vector space of dimension

is a real vector space of dimension

Some formulae relate the dimension of a vector space with the cardinality of the base field and the cardinality of the space itself.

then: A vector space can be seen 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.

The dimension of a vector space may alternatively be characterized as the trace of the identity operator.

Firstly, it allows for a definition of a notion of dimension when one has a trace but no natural sense of basis.

(the inclusion of scalars, called the unit) and a map

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, this map is required to be the identity, which can be obtained by normalizing the counit by dividing by dimension (

), so in these cases the normalizing constant corresponds to dimension.

Alternatively, it may be possible 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.

of the character can be viewed 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

-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.