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| In [[mathematics]], the '''dimension''' of a [[vector space]] ''V'' is the [[cardinal number|cardinality]] (i.e. the number of vectors) of a [[basis (linear algebra)|basis]] of ''V''.<ref>{{cite book|author=Itzkov, 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=http://books.google.com/books?id=8FVk_KRY7zwC&pg=PA4}}</ref><ref>It is sometimes called '''Hamel dimension''' or '''algebraic dimension''' to distinguish it from other types of [[dimension]].</ref>
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| For every vector space there exists a basis (if one assumes the [[axiom of choice]]), and all bases of a vector space have equal cardinality (see [[dimension theorem for vector spaces]]); as a result the dimension of a vector space is uniquely defined. We say ''V'' is '''finite-dimensional''' if the dimension of ''V'' is [[wiktionary:finite|finite]].
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| The dimension of the vector space ''V'' over the [[field (mathematics)|field]] ''F'' can be written as dim<sub>''F''</sub>(''V'') or as [V : F], read "dimension of ''V'' over ''F''". When ''F'' can be inferred from context, often just dim(''V'') is written.
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| == Examples ==
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| The vector space '''R'''<sup>3</sup> has
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| :<math>\left \{ \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} , \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} , \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \right \}</math>
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| as a [[Basis (linear algebra)|basis]], and therefore we have dim<sub>'''R'''</sub>('''R'''<sup>3</sup>) = 3. More generally, dim<sub>'''R'''</sub>('''R'''<sup>''n''</sup>) = ''n'', and even more generally, dim<sub>''F''</sub>(''F''<sup>''n''</sup>) = ''n'' for any [[field (mathematics)|field]] ''F''.
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| The [[complex number]]s '''C''' are both a real and complex vector space; we have dim<sub>'''R'''</sub>('''C''') = 2 and dim<sub>'''C'''</sub>('''C''') = 1. So the dimension depends on the base field.
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| The only vector space with dimension 0 is {0}, the vector space consisting only of its zero element.
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| == Facts ==
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| If ''W'' is a [[linear subspace]] of ''V'', then dim(''W'') ≤ dim(''V'').
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| To show that two finite-dimensional vector spaces are equal, one often uses the following criterion: if ''V'' is a finite-dimensional vector space and ''W'' is a linear subspace of ''V'' with dim(''W'') = dim(''V''), then ''W'' = ''V''.
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| '''R'''<sup>''n''</sup> has the standard basis {'''e'''<sub>1</sub>, ..., '''e'''<sub>''n''</sub>}, where '''e'''<sub>''i''</sub> is the ''i''-th column of the corresponding [[identity matrix]]. Therefore '''R'''<sup>''n''</sup>
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| has dimension ''n''.
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| Any two vector spaces over ''F'' having 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 ''B'' is some set, a vector space with dimension |''B''| over ''F'' can be constructed as follows: take the set ''F''<sup>(''B'')</sup> of all functions ''f'' : ''B'' → ''F'' such that ''f''(''b'') = 0 for all but finitely many ''b'' in ''B''. These functions can be added and multiplied with elements of ''F'', and we obtain the desired ''F''-vector space.
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| An important result about dimensions is given by the [[rank–nullity theorem]] for [[linear map]]s.
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| If ''F''/''K'' is a [[field extension]], then ''F'' is in particular a vector space over ''K''. Furthermore, every ''F''-vector space ''V'' is also a ''K''-vector space. The dimensions are related by the formula
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| :dim<sub>''K''</sub>(''V'') = dim<sub>''K''</sub>(''F'') dim<sub>''F''</sub>(''V''). | |
| In particular, every complex vector space of dimension ''n'' is a real vector space of dimension 2''n''.
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| Some simple formulae relate the dimension of a vector space with the [[cardinality]] of the base field and the cardinality of the space itself.
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| If ''V'' is a vector space over a field ''F'' then, denoting the dimension of ''V'' by dim''V'', we have:
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| :If dim ''V'' is finite, then |''V''| = |''F''|<sup>dim''V''</sup>.
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| :If dim ''V'' is infinite, then |''V''| = max(|''F''|, dim''V''). | |
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| == Generalizations ==
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| One can see a vector space 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.
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| The [[Krull dimension]] of a commutative [[ring (algebra)|ring]], named after [[Wolfgang Krull]] (1899–1971), is defined to be the maximal number of strict inclusions in an increasing chain of [[prime ideal]]s in the ring.
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| === Trace ===
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| {{see also|Trace (linear algebra)}}
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| 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}_{\mathbf{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.
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| Firstly, it allows one to define a notion of dimension when one has a trace but no natural sense of basis. For example, one may have an algebra ''A'' with maps <math>\eta\colon K \to A</math> (the inclusion of scalars, called the ''unit'') and a map <math>\epsilon \colon A \to K</math> (corresponding to trace, called the ''[[counit]]''). The composition <math>\epsilon\circ \eta \colon 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 one requires that this map 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.
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| Alternatively, one may be able 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]].
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| 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\colon 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> One can view the other values <math>\chi(g)</math> of the character 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|''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.<ref>{{Harv|Gannon|2006}}</ref>
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| == See also ==
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| *[[Basis (linear algebra)]]
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| *[[Topological dimension]], also called Lebesgue covering dimension
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| *[[Fractal dimension]]
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| *[[Krull dimension]]
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| *[[Matroid rank]]
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| *[[Rank (linear algebra)]]
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| == References ==
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| {{reflist}}
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| {{refbegin}}
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| *{{Citation | first = Terry | last = Gannon | title = Moonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics | year = 2006 | isbn = 0-521-83531-3}}
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| {{refend}}
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| ==External links==
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| * [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
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| {{DEFAULTSORT:Dimension (Vector Space)}}
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| [[Category:Linear algebra]]
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| [[Category:Dimension]]
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| [[Category:Vectors]]
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