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Dimension (vector space)

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(Redirected fromDimension of a vector space)
Number of vectors in any basis of the vector space
A diagram of dimensions 1, 2, 3, and 4

Inmathematics, thedimension of avector spaceV is thecardinality (i.e., the number of vectors) of abasis ofV over its basefield.[1][2] It is sometimes calledHamel dimension (afterGeorg Hamel) oralgebraic dimension to distinguish it from other types ofdimension.

For every vector space there exists a basis,[a] and all bases of a vector space have equal cardinality;[b] as a result, the dimension of a vector space is uniquely defined. We sayV{\displaystyle V} isfinite-dimensional if the dimension ofV{\displaystyle V} isfinite, andinfinite-dimensional if its dimension isinfinite.

The dimension of the vector spaceV{\displaystyle V} over the fieldF{\displaystyle F} can be written asdimF⁡(V){\displaystyle \dim _{F}(V)} or as[V:F],{\displaystyle [V:F],} read "dimension ofV{\displaystyle V} overF{\displaystyle F}". WhenF{\displaystyle F} can be inferred from context,dim⁡(V){\displaystyle \dim(V)} is typically written.

Examples

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The vector spaceR3{\displaystyle \mathbb {R} ^{3}} has{(100),(010),(001)}{\displaystyle \left\{{\begin{pmatrix}1\\0\\0\end{pmatrix}},{\begin{pmatrix}0\\1\\0\end{pmatrix}},{\begin{pmatrix}0\\0\\1\end{pmatrix}}\right\}}as astandard basis, and thereforedimR⁡(R3)=3.{\displaystyle \dim _{\mathbb {R} }(\mathbb {R} ^{3})=3.} More generally,dimR⁡(Rn)=n,{\displaystyle \dim _{\mathbb {R} }(\mathbb {R} ^{n})=n,} and even more generally,dimF⁡(Fn)=n{\displaystyle \dim _{F}(F^{n})=n} for anyfieldF.{\displaystyle F.}

Thecomplex numbersC{\displaystyle \mathbb {C} } are both a real and complex vector space; we havedimR⁡(C)=2{\displaystyle \dim _{\mathbb {R} }(\mathbb {C} )=2} anddimC⁡(C)=1.{\displaystyle \dim _{\mathbb {C} }(\mathbb {C} )=1.} So the dimension depends on the base field.

The only vector space with dimension0{\displaystyle 0} is{0},{\displaystyle \{0\},} the vector space consisting only of its zero element.

Properties

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IfW{\displaystyle W} is alinear subspace ofV{\displaystyle V} thendim⁡(W)≤dim⁡(V).{\displaystyle \dim(W)\leq \dim(V).}

To show that two finite-dimensional vector spaces are equal, the following criterion can be used: ifV{\displaystyle V} is a finite-dimensional vector space andW{\displaystyle W} is a linear subspace ofV{\displaystyle V} withdim⁡(W)=dim⁡(V),{\displaystyle \dim(W)=\dim(V),} thenW=V.{\displaystyle W=V.}

The spaceRn{\displaystyle \mathbb {R} ^{n}} has the standard basis{e1,…,en},{\displaystyle \left\{e_{1},\ldots ,e_{n}\right\},} whereei{\displaystyle e_{i}} is thei{\displaystyle i}-th column of the correspondingidentity matrix. Therefore,Rn{\displaystyle \mathbb {R} ^{n}} has dimensionn.{\displaystyle n.}

Any two finite dimensional vector spaces overF{\displaystyle F} with the same dimension areisomorphic. Anybijective map between their bases can be uniquely extended to a bijective linear map between the vector spaces. IfB{\displaystyle B} is some set, a vector space with dimension|B|{\displaystyle |B|} overF{\displaystyle F} can be constructed as follows: take the setF(B){\displaystyle F(B)} of all functionsf:B→F{\displaystyle f:B\to F} such thatf(b)=0{\displaystyle f(b)=0} for all but finitely manyb{\displaystyle b} inB.{\displaystyle B.} These functions can be added and multiplied with elements ofF{\displaystyle F} to obtain the desiredF{\displaystyle F}-vector space.

An important result about dimensions is given by therank–nullity theorem forlinear maps.

IfF/K{\displaystyle F/K} is afield extension, thenF{\displaystyle F} is in particular a vector space overK.{\displaystyle K.} Furthermore, everyF{\displaystyle F}-vector spaceV{\displaystyle V} is also aK{\displaystyle K}-vector space. The dimensions are related by the formuladimK⁡(V)=dimK⁡(F)dimF⁡(V).{\displaystyle \dim _{K}(V)=\dim _{K}(F)\dim _{F}(V).}In particular, every complex vector space of dimensionn{\displaystyle n} is a real vector space of dimension2n.{\displaystyle 2n.}

Some formulae relate the dimension of a vector space with thecardinality of the base field and the cardinality of the space itself.IfV{\displaystyle V} is a vector space over a fieldF{\displaystyle F} and if the dimension ofV{\displaystyle V} is denoted bydim⁡V,{\displaystyle \dim V,} then:

If dimV{\displaystyle V} is finite then|V|=|F|dim⁡V.{\displaystyle |V|=|F|^{\dim V}.}
If dimV{\displaystyle V} is infinite then|V|=max(|F|,dim⁡V).{\displaystyle |V|=\max(|F|,\dim V).}

Generalizations

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A vector space can be seen as a particular case of amatroid, and in the latter there is a well-defined notion of dimension. Thelength of a module and therank of an abelian group both have several properties similar to the dimension of vector spaces.

TheKrull dimension of a commutativering, named afterWolfgang Krull (1899–1971), is defined to be the maximal number of strict inclusions in an increasing chain ofprime ideals in the ring.

Trace

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See also:Trace (linear algebra)

The dimension of a vector space may alternatively be characterized as thetrace of theidentity operator. For instance,tr⁡ idR2=tr⁡(1001)=1+1=2.{\displaystyle \operatorname {tr} \ \operatorname {id} _{\mathbb {R} ^{2}}=\operatorname {tr} \left({\begin{smallmatrix}1&0\\0&1\end{smallmatrix}}\right)=1+1=2.} This appears to be acircular 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 analgebraA{\displaystyle A} with mapsη:K→A{\displaystyle \eta :K\to A} (the inclusion of scalars, called theunit) and a mapϵ:A→K{\displaystyle \epsilon :A\to K} (corresponding to trace, called thecounit). The compositionϵ∘η:K→K{\displaystyle \epsilon \circ \eta :K\to K} 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, inbialgebras, this map is required to be the identity, which can be obtained by normalizing the counit by dividing by dimension (ϵ:=1ntr{\displaystyle \epsilon :=\textstyle {\frac {1}{n}}\operatorname {tr} }), 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 aHilbert space, or more generallynuclear operators on aBanach space.

A subtler generalization is to consider the trace of afamily of operators as a kind of "twisted" dimension. This occurs significantly inrepresentation theory, where thecharacter of a representation is the trace of the representation, hence a scalar-valued function on agroupχ:G→K,{\displaystyle \chi :G\to K,} whose value on the identity1∈G{\displaystyle 1\in G} is the dimension of the representation, as a representation sends the identity in the group to the identity matrix:χ(1G)=tr⁡ IV=dim⁡V.{\displaystyle \chi (1_{G})=\operatorname {tr} \ I_{V}=\dim V.} The other valuesχ(g){\displaystyle \chi (g)} 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 ofmonstrous moonshine: thej{\displaystyle j}-invariant is thegraded dimension of an infinite-dimensional graded representation of themonster group, and replacing the dimension with the character gives theMcKay–Thompson series for each element of the Monster group.[3]

See also

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  • Fractal dimension – Ratio providing a statistical index of complexity variation with scale
  • Krull dimension – In mathematics, dimension of a ring
  • Matroid rank – Maximum size of an independent set of the matroid
  • Rank (linear algebra) – Dimension of the column space of a matrix
  • Topological dimension – Topologically invariant definition of the dimension of a spacePages displaying short descriptions of redirect targets, also called Lebesgue covering dimension

Notes

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  1. ^if one assumes theaxiom of choice
  2. ^seedimension theorem for vector spaces

References

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  1. ^Itzkov, Mikhail (2009).Tensor Algebra and Tensor Analysis for Engineers: With Applications to Continuum Mechanics. Springer. p. 4.ISBN 978-3-540-93906-1.
  2. ^Axler (2015) p. 44, §2.36
  3. ^Gannon, Terry (2006),Moonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics, Cambridge University Press,ISBN 0-521-83531-3

Sources

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External links

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Dimensional spaces
Animated tesseract
Other dimensions
Polytopes andshapes
Number systems
Dimensions by number
See also
Basic concepts
Three dimensional Euclidean space
Matrices
Bilinear
Multilinear algebra
Vector space constructions
Numerical
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