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
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
(i.e., the number of vectors) of a
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
of ''V'' over its base
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
.
[ p. 44, §2.36] It is sometimes called Hamel dimension (after
Georg Hamel
Georg Karl Wilhelm Hamel (12 September 1877 – 4 October 1954) was a German mathematician with interests in mechanics, the foundations of mathematics and function theory.
Biography
Hamel was born in Düren, Rhenish Prussia. He studied at A ...
) or algebraic dimension to distinguish it from other types of
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
.
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
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marke ...
, and if its dimension is
infinite
Infinite may refer to:
Mathematics
* Infinite set, a set that is not a finite set
*Infinity, an abstract concept describing something without any limit
Music
*Infinite (group), a South Korean boy band
*''Infinite'' (EP), debut EP of American m ...
.
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 c ...
, and therefore
More generally,
and even more generally,
for any
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
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 form ...
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
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
. Any
bijective
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
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
The rank–nullity theorem is a theorem in linear algebra, which asserts that the dimension of the domain of a linear map is the sum of its rank (the dimension of its image) and its ''nullity'' (the dimension of its kernel). p. 70, §2.1, Theo ...
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 Map (mathematics), mapping V \to W between two vect ...
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
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
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
In combinatorics, a branch of mathematics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid axiomatically, the most significant being ...
, and in the latter there is a well-defined notion of dimension. The
length of a module In abstract algebra, the length of a module is a generalization of the dimension of a vector space which measures its size. page 153 In particular, as in the case of vector spaces, the only modules of finite length are finitely generated modules. It ...
and the
rank of an abelian group both have several properties similar to the dimension of vector spaces.
The
Krull dimension
In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally th ...
of a commutative
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
, named after
Wolfgang Krull
Wolfgang Krull (26 August 1899 – 12 April 1971) was a German mathematician who made fundamental contributions to commutative algebra, introducing concepts that are now central to the subject.
Krull was born and went to school in Baden-Baden. H ...
(1899–1971), is defined to be the maximal number of strict inclusions in an increasing chain of
prime ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...
s in the ring.
Trace
The dimension of a vector space may alternatively be characterized as the
trace
Trace may refer to:
Arts and entertainment Music
* ''Trace'' (Son Volt album), 1995
* ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* ''The Trace'' (album)
Other uses in arts and entertainment
* ''Trace'' ...
of the
identity operator
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film) ...
. 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 a ...
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
bialgebra
In mathematics, a bialgebra over a field ''K'' is a vector space over ''K'' which is both a unital associative algebra and a counital coassociative coalgebra. The algebraic and coalgebraic structures are made compatible with a few more axioms. ...
s, 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 In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of trace ...
operators" on a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
, or more generally
nuclear operator
In mathematics, nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately tied to the projective tensor product of two topological vector spac ...
s on a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
.
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
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
, 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
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
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
In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group ''M'' and modular functions, in particular, the ''j'' function. The term was coined by John Conway and Simon P. Norton in 1979. ...
: the
-invariant is the
graded dimension of an infinite-dimensional graded representation of the
monster group
In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order
246320597611213317192329314147 ...
, 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)