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]].

''Notation''. We will letLet ''F'' denote an arbitrary [[Field (mathematics)|field]] such as the [[real number]]s '''R''' or the [[complex number]]s '''C'''.

==Trivial or zero vector space==

The simplest example of a vector space is the trivial one: {0}, which contains only the zero vector (see the third axiom in the [[Vector space]] article). Both vector addition and scalar multiplication are trivial. A [[Basis (linear algebra)|basis]] for this vector space is the [[empty set]], so that {0} is the 0-[[Dimension (vector space)|dimensional]] vector space over ''F''. Every vector space over ''F'' contains a [[Linear subspace|subspace]] [[isomorphic]] to this one.

The zero vector space is conceptually different from the [[null space]] of a linear operator ''L'', which is the [[Kernel (linear algebra)|kernel]] of ''L''. (Incidentally, the null space of ''L'' is a zero space if and only if ''L'' is [[injective]].)

==Field==

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==Coordinate space==

[[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, which the axiomatic definition generalizes, 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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===Generalized coordinate space===

Let ''X'' be an arbitrary set. Consider the space of all functions from ''X'' to ''F'' which vanish on all but a finite number of points in ''X''. This space is a vector subspace of ''F''^''X'', the space of all possible functions from ''X'' to ''F''. To see this, note that the union of two finite sets is finite, so that the sum of two functions in this space will still vanish outside a finite set.

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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-FraenkelZermelo–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}}