In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the dimension of a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
''V'' is the
cardinality (i.e., the number of vectors) of a
basis of ''V'' over its base
field.
[ p. 44, §2.36] It is sometimes called Hamel dimension (after
Georg Hamel) or algebraic dimension to distinguish it from other types of
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
.
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. We say
is 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,
is typically written.
Examples
The vector space
has
as a
standard basis
In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as \mathbb^n or \mathbb^n) is the set of vectors whose components are all zero, except one that equals 1. For example, in the ...
, and therefore
More generally,
and even more generally,
for any
field
The
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s
are both a real and complex vector space; we have
and
So the dimension depends on the base field.
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
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 map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
s.
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
then 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
The dimension of a vector space may alternatively be characterized as the
trace of the
identity operator. For instance,
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
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
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 (
), 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
Character or Characters may refer to:
Arts, entertainment, and media Literature
* ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk
* ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
of a representation is the trace of the representation, hence a scalar-valued function on a
group whose value on the identity
is the dimension of the representation, as a representation sends the identity in the group to the identity matrix:
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
Notes
References
Sources
*
External links
MIT Linear Algebra Lecture on Independence, Basis, and Dimension by Gilbert Strangat MIT OpenCourseWare
{{DEFAULTSORT:Dimension (Vector Space)
Dimension
Linear algebra
Vectors (mathematics and physics)