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Wikipedia / 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. 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, and all bases of a vector space have equal cardinality; is said to be is finite, and if its dimension is infinite.

instead.

Examples

as a standard basis for any field

The complex numbers. For example, 1 as the basis and the complex numbers as scalars (elements of the field for the vector space as the basis and real numbers as scalars form a 2-dimensional vector space.

the vector space consisting only of its zero element.

Properties

is a linear subspace

-th column of the corresponding identity matrix

with the same dimension are isomorphic. Any bijective-vector space.

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

is a field extension

Some formulae relate the dimension of a vector space with the cardinality then:

Generalizations

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 (18991971), 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 This appears to be a circular definition, but it allows useful generalizations.

Firstly, it allows for a definition of a notion of dimension when one has a trace but no natural sense of basis. For example, one may have an algebra (the inclusion of scalars, called the unit (corresponding to trace, called the counit 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), 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. 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 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 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.

See also

  • , also called Lebesgue covering dimension

Notes

References

Sources