Examples of vector spaces: Difference between revisions

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This page lists some '''examples of vector spaces'''. See [[vector space]] for the definitions of terms used on this page. See also: [[dimension (vector space)|dimension]], [[basis (linear algebra)|basis]].

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[[File:Line equation qtl4.svg|thumb|Planar [[analytic geometry]] uses the coordinate space '''R'''². ''Depicted:'' description of a [[line (geometry)|line]] as the [[equation solving#Solution sets|solution set]] in <math>\vec x</math> of the vector equation <math>\vec x \cdot \vec n = d</math>.]]

{{Main|Coordinate space}}

TheA originalbasic example of a vector space is the following. For any [[Positive number|positive]] [[integer]] ''n'', the [[Set (mathematics)|set]] of all ''n''-tuples of elements of ''F'' forms an ''n''-dimensional vector space over ''F'' sometimes called ''[[coordinate space]]'' and denoted ''F''^''n''.<ref>{{Harvard citations|last = Lang|year = 1987|loc = ch. I.1|nb = yes}}</ref> An element of ''F''^''n'' is written

:<math>x = (x_1, x_2, \ldots, x_n) </math>

where each ''x''_''i'' is an element of ''F''. The operations on ''F''^''n'' are defined by

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===Several variables===

The set of [[polynomial]]s in several variables with coefficients in ''F'' is vector space over ''F'' denoted ''F''[''x''₁, ''x''₂, …..., ''x''_''r'']. Here ''r'' is the number of variables.

:''{{See also'': [[|Polynomial ring]]}}

==Function spaces==

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==Field extensions==

Suppose ''K'' is a [[Field extension|subfield]] of ''F'' (cf. [[field extension]]). Then ''F'' can be regarded as a vector space over ''K'' by restricting scalar multiplication to elements in ''K'' (vector addition is defined as normal). The dimension of this vector space, if it exists,{{efn|Note that the resulting vector space may not have a basis in the absence the [[axiom of choice]].}} is called the ''degree'' of the extension. For example, the [[complex number]]s '''C''' form a two-dimensional vector space over the real numbers '''R'''. Likewise, the [[real numbers]] '''R''' form a vector space over the [[rational number]]s '''Q''' which has (uncountably) infinite dimension, if a Hamel basis exists.{{efn|There are models of [[Zermelo–Fraenkel set theory|ZF]] without [[Axiom of choice|AC]] in which this is not the case.}}

If ''V'' is a vector space over ''F'' it may also be regarded as vector space over ''K''. The dimensions are related by the formula

:dim_''K''''V'' = (dim_''F''''V'')(dim_''K''''F'')

For example, '''C'''^''n'', regarded as a vector space over the reals, has dimension 2''n''.

==Finite vector spaces==

Apart from the trivial case of a [[zero-dimensional space]] over any field, a vector space over a field ''F'' has a finite number of elements if and only if ''F'' is a [[finite field]] and the vector space has a finite dimension. Thus we have ''F''_''q'', the unique finite field (up to [[isomorphism]]) with ''q'' elements. Here ''q'' must be a power of a [[prime number|prime]] (''q'' = ''p''^''m'' with ''p'' prime). Then any ''n''-dimensional vector space ''V'' over ''F''_''q'' will have ''q''^''n'' elements. Note that the number of elements in ''V'' is also the power of a prime (because a power of a prime power is again a prime power). The primary example of such a space is the coordinate space (''F''_''q'')^''n''.

These vector spaces are of critical importance in the [[representation theory]] of [[finite group]]s, [[number theory]], and [[cryptography]].

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==References==

{{Reflist}}

* {{cite book | last1=Lang | first1=Serge | author1-link=Serge Lang | title=Linear Algebra | publisher=[[Springer-Verlag]] | ___location=Berlin, New York | isbn=978-0-387-96412-6 | year=1987}}

{{DEFAULTSORT:Examples Of Vector Spaces}}