Dimension (vector space) explained
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] [2] 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; as a result, the dimension of a vector space is uniquely defined.
is said to be
if the dimension of
is finite, and
if its dimension is
infinite.
The dimension of the vector space
over the field
can be written as
or as
read "dimension of
over
". When
can be inferred from context, the dimension is generally written as
instead.
Examples
The vector space
has
as a
standard basis, and therefore
More generally,
and even more generally,
for any
field
The complex numbers
are both a real and complex vector space. Their dimension depends on the base field, as
and
.
The only vector space with dimension
is
the vector space consisting only of its zero element.
Properties
If
is a
linear subspace of
, then
To show that two finite-dimensional vector spaces are equal, the following criterion can be used: if
is a finite-dimensional vector space and
is a linear subspace of
with
then
The space
has the standard basis
\left\{e1,\ldots,en\right\},
where
is the
-th column of the corresponding
identity matrix. Therefore,
has dimension
Any two finite dimensional vector spaces over
with 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
is some set, a vector space with dimension
over
can be constructed as follows: take the set
of all functions
such that
for all but finitely many
in
These functions can be added and multiplied with elements of
to obtain the desired
-vector space.
An important result about dimensions is given by the rank–nullity theorem for linear maps.
If
is a
field extension, then
is in particular a vector space over
Furthermore, every
-vector space
is also a
-vector space. The dimensions are related by the formula
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.If
is a vector space over a field
and if the dimension of
is denoted by
then:
If dim
is finite then
If dim
is infinite 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 (1899 - 1971), is defined to be the maximal number of strict inclusions in an increasing chain of prime ideals in the ring.
Trace
See also: Trace (linear algebra).
The dimension of a vector space may alternatively be characterized as the trace of the identity operator. For instance,
| \operatorname{tr} \operatorname{id} | |
| \R2 |
=\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.
with maps
(the inclusion of scalars, called the
unit) and a map
(corresponding to trace, called the
counit). The composition
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 (
} \operatorname), 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.
whose value on the identity
is the dimension of the representation, as a representation sends the identity in the group to the identity matrix:
\chi(1G)=\operatorname{tr} IV=\dimV.
The other values
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.
See also
- , also called Lebesgue covering dimension
Sources
External links
Notes and References
- Book: Itzkov, Mikhail. Tensor Algebra and Tensor Analysis for Engineers: With Applications to Continuum Mechanics. Springer. 2009. 978-3-540-93906-1. 4.
- p. 44, §2.36
