View source for Dimension (vector space)

{{short description|Number of vectors in any basis of the vector space}}
[[File:Dimension levels.svg|thumb|A diagram of dimensions 1, 2, 3, and 4]]
In [[mathematics]], the '''dimension''' of a [[vector space]] ''V'' is the [[cardinality]] (i.e., the number of vectors) of a [[Basis (linear algebra)|basis]] of ''V'' over its base [[Field (mathematics)|field]].<ref>{{cite book|last=Itzkov|first=Mikhail|title=Tensor Algebra and Tensor Analysis for Engineers: With Applications to Continuum Mechanics|publisher=Springer|year=2009|isbn=978-3-540-93906-1|page=4|url=https://books.google.com/books?id=8FVk_KRY7zwC&pg=PA4}}</ref><ref>{{Harvard citation text|Axler|2015}} p. 44, §2.36</ref> 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,{{efn|if one assumes the [[axiom of choice]]}} and all bases of a vector space have equal cardinality;{{efn|see [[dimension theorem for vector spaces]]}} as a result, the dimension of a vector space is uniquely defined. <math>V</math> is said to be '''{{visible anchor|finite-dimensional}}''' if the dimension of <math>V</math> is [[wiktionary:finite|finite]], and '''{{visible anchor|infinite-dimensional}}''' if its dimension is [[infinity|infinite]].

The dimension of the vector space <math>V</math> over the field <math>F</math> can be written as <math>\dim_F(V)</math> or as <math>[V : F],</math> read "dimension of <math>V</math> over <math>F</math>". When <math>F</math> can be inferred from context, the dimension is generally written as <math>\dim(V)</math> instead.

== Examples ==

The vector space <math>\R^3</math> has
<math display=block>\left\{\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}  , \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} , \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}\right\}</math>
as a [[standard basis]], and therefore <math>\dim_{\R}(\R^3) = 3.</math> More generally, <math>\dim_{\R}(\R^n) = n,</math> and even more generally, <math>\dim_{F}(F^n) = n</math> for any [[Field (mathematics)|field]] <math>F.</math> 

The [[complex number]]s <math>\Complex</math> are both a real and complex vector space. Their dimension depends on the base field, as <math>\dim_{\R}(\Complex) = 2</math> and <math>\dim_{\Complex}(\Complex) = 1</math>.

The only vector space with dimension <math>0</math> is <math>\{0\},</math> the vector space consisting only of its zero element.

== Properties ==

If <math>W</math> is a [[linear subspace]] of <math>V</math>, then <math>\dim (W) \leq \dim (V).</math>

To show that two finite-dimensional vector spaces are equal, the following criterion can be used: if <math>V</math> is a finite-dimensional vector space and <math>W</math> is a linear subspace of <math>V</math> with <math>\dim (W) = \dim (V),</math> then <math>W = V.</math> 

The space <math>\R^n</math> has the standard basis <math>\left\{e_1, \ldots, e_n\right\},</math> where <math>e_i</math> is the <math>i</math>-th column of the corresponding [[identity matrix]]. Therefore, <math>\R^n</math> has dimension <math>n.</math>

Any two finite-dimensional vector spaces over <math>F</math> 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 <math>B</math> is some set, a vector space with dimension <math>|B|</math> over <math>F</math> can be constructed as follows: take the set <math>F(B)</math> of all functions <math>f : B \to F</math> such that <math>f(b) = 0</math> for all but finitely many <math>b</math> in <math>B.</math> These functions can be added and multiplied with elements of <math>F</math> to obtain the desired <math>F</math>-vector space.

An important result about dimensions is given by the [[rank–nullity theorem]] for [[linear map]]s.

If <math>F / K</math> is a [[field extension]], then <math>F</math> is in particular a vector space over <math>K.</math> Furthermore, every <math>F</math>-vector space <math>V</math> is also a <math>K</math>-vector space. The dimensions are related by the formula
<math display=block>\dim_K(V) = \dim_K(F) \dim_F(V).</math>
In particular, every complex vector space of dimension <math>n</math> is a real vector space of dimension <math>2n.</math>

Some formulae relate the dimension of a vector space with the [[cardinality]] of the base field and the cardinality of the space itself.
If <math>V</math> is a vector space over a field <math>F</math> and if the dimension of <math>V</math> is denoted by <math>\dim V,</math> then:

:If dim <math>V</math> is finite then <math>|V| = |F|^{\dim V}.</math>
:If dim <math>V</math> is infinite then <math>|V| = \max (|F|, \dim V).</math>

== 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 (algebra)|ring]], named after [[Wolfgang Krull]] (1899&ndash;1971), is defined to be the maximal number of strict inclusions in an increasing chain of [[prime ideal]]s in the ring.

=== Trace ===
{{see also|Trace (linear algebra)}}

The dimension of a vector space may alternatively be characterized as the [[Trace (linear algebra)|trace]] of the [[identity operator]]. For instance, 
<math>\operatorname{tr}\ \operatorname{id}_{\R^2} = \operatorname{tr} \left(\begin{smallmatrix} 1 & 0 \\ 0 & 1 \end{smallmatrix}\right) = 1 + 1 = 2.</math> 
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 over a field|algebra]] <math>A</math> with maps <math>\eta : K \to A</math> (the inclusion of scalars, called the ''unit'') and a map <math>\epsilon : A \to K</math> (corresponding to trace, called the ''[[counit]]''). The composition <math>\epsilon \circ \eta : K \to K</math> 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]]s, this map is required to be the identity, which can be obtained by normalizing the counit by dividing by dimension (<math>\epsilon := \textstyle{\frac{1}{n}} \operatorname{tr}</math>), 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 operator]]s 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 (mathematics)|character]] of a representation is the trace of the representation, hence a scalar-valued function on a [[Group (mathematics)|group]] <math>\chi : G \to K,</math> whose value on the identity <math>1 \in G</math> is the dimension of the representation, as a representation sends the identity in the group to the identity matrix: <math>\chi(1_G) = \operatorname{tr}\ I_V = \dim V.</math> The other values <math>\chi(g)</math> 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|<math>j</math>-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.<ref name="gannon">{{Citation|last=Gannon|first=Terry|title=Moonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics|year=2006|publisher=Cambridge University Press |isbn=978-0-521-83531-2}}</ref>

== See also ==

* {{annotated link|Fractal dimension}}
* {{annotated link|Krull dimension}}
* {{annotated link|Matroid rank}}
* {{annotated link|Rank (linear algebra)}}
* {{annotated link|Topological dimension}}, also called Lebesgue covering dimension

== Notes ==

{{notelist}}

== References ==

{{reflist}}
{{reflist|group=note}}

== Sources ==

* {{Cite book|last=Axler|first=Sheldon|title=Linear Algebra Done Right|publisher=[[Springer Science+Business Media| Springer]]|year=2015|isbn=978-3-319-11079-0|edition=3rd|series=[[Undergraduate Texts in Mathematics]]|location=|pages=|author-link=Sheldon Axler}}

==External links==

* [http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/lecture-9-independence-basis-and-dimension/ MIT Linear Algebra Lecture on Independence, Basis, and Dimension by Gilbert Strang] at MIT OpenCourseWare

{{Dimension topics}}
{{Linear algebra}}

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[[Category:Dimension]]
[[Category:Linear algebra]]
[[Category:Vector spaces| ]]
[[Category:Vectors (mathematics and physics)]]