Examples of vector spaces: Difference between revisions

Revision as of 01:08, 22 January 2018

Latest revision as of 22:37, 30 November 2023

(23 intermediate revisions by 17 users not shown)

Line 1:

{{Short description|none}} <!-- This short description is INTENTIONALLY "none" - please see WP:SDNONE before you consider changing it! -->

{{Unreferenced|date=November 2009}}

{{More citations needed|date=February 2022}}

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 [[fieldField (mathematics)|field]] such as the [[real number]]s '''R''' or the [[complex number]]s '''C'''. See also: [[table of mathematical symbols]].

==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 3in ofthe [[vectorVector space]]s article). Both vector addition and scalar multiplication are trivial. A [[basisBasis (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]] [[Isomorphism|isomorphic]] to this one.

{{main|Zero vector space}}

The simplest example of a vector space is the trivial one: {'''0'''}, which contains only the zero vector (see axiom 3 of [[vector space]]s). 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-dimensional vector space over '''F'''. Every vector space over '''F''' contains a subspace [[Isomorphism|isomorphic]] to this one.

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

==Field==

The next simplest example is the field '''F''' itself. Vector addition is just field addition, and scalar multiplication is just field multiplication,. and thisThis property can be used to prove that a field is a vector space. Any non-zero element of '''F''' serves as a basis so '''F''' is a 1-dimensional vector space over itself.

The field is a rather special vector space; in fact it is the simplest example of a '''[[algebraAlgebra over a field|commutative algebra]]''' over '''F'''. Also, '''F''' has just two [[Linear subspace|subspaces]]: {0} and '''F''' itself.

==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}}

PerhapsA the most importantbasic example of a vector space is the following. For any [[Positive number|positive]] [[integer]] ''n'', the space[[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

Perhaps the most important example of a vector space is the following. For any [[Positive number|positive]] [[integer]] ''n'', the space of all ''n''-tuples of elements of '''F''' forms an ''n''-dimensional vector space over '''F''' sometimes called ''[[coordinate space]]'' and denoted '''F'''^''n''. 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

:<math>x + y = (x_1 + y_1, x_2 + y_2, \ldots, x_n + y_n) </math>

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

:<math>0 = (0, 0, \ldots, 0) </math>

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

The most common cases are whereCommonly, '''F''' is the field of [[real number]]s, givingin thewhich case we obtain [[real coordinate space]] '''R'''^''n'',. or theThe field of [[complex number]]s giving thegives [[complex coordinate space]] '''C'''^''n''. The ''a + bi'' form of a complex number shows that '''C''' itself is a two-dimensional real vector space with coordinates (''a'',''b''). Similarly, the [[quaternion]]s and the [[octonion]]s are respectively four- and eight-dimensional real vector spaces, and '''C'''^''n'' is a ''2n''-dimensional real vector space.

The [[quaternion]]s and the [[octonion]]s are respectively four- and eight-dimensional vector spaces over the reals.

The vector space '''R'F''^''n'' comes withhas a [[standard basis]]:

:<math>e_1 = (1, 0, \ldots, 0) </math>

:<math>e_2 = (0, 1, \ldots, 0) </math>

:<math>\vdots </math>

:<math>e_n = (0, 0, \ldots, 1) </math>

where 1 denotes the multiplicative identity in '''R'F''.

==Infinite coordinate space==

Let '''F'''^∞ denote the space of [[infinite sequence]]s of elements from '''F''' such that only ''finitely'' many elements are nonzero. That is, if we write an element of '''F'''^∞ as

:<math>x = (x_1, x_2, x_3, \ldots) </math>

then only a finite number of the ''x''_''i'' are nonzero (i.e., the coordinates become all zero after a certain point). Addition and scalar multiplication are given as in finite coordinate space. The dimensionality of '''F'''^∞ is [[countably infinite]]. A standard basis consists of the vectors ''e''_''i'' which contain a 1 in the ''i''-th slot and zeros elsewhere. This vector space is the [[coproduct]] (or [[direct sum of modules|direct sum]]) of countably many copies of the vector space '''F'''.

Note the role of the finiteness condition here. One could consider arbitrary sequences of elements in '''F''', which also constitute a vector space with the same operations, often denoted by '''F'''^'''N''' - see [[Examples of vector spaces#Function spaces|below]]. '''F'''^'''N''' is the ''[[productProduct (category theory)|product]]'' of countably many copies of '''F'''.

By [[Zorn's lemma]], '''F'''^'''N''' has a basis (there is no obvious basis). There are [[uncountably infinite]] elements in the basis. Since the dimensions are different, '''F'''^'''N''' is ''not'' isomorphic to '''F'''^∞. It is worth noting that '''F'''^'''N''' is (isomorphic to) the [[dual space]] of '''F'''^∞, because a [[linear map]] ''T'' from '''F'''^∞ to '''F''' is determined uniquely by its values ''T''(''e_i'') on the basis elements of '''F'''^∞, and these values can be arbitrary. Thus one sees that a vector space need not be isomorphic to its double dual if it is infinite dimensional, in contrast to the finite dimensional case.

==Product of vector spaces==

Line 49 ⟶ 48:

==Matrices==

Let '''F'''^''m''×''n'' denote the set of ''m''×''n'' [[matrixMatrix (mathematics)|matrices]] with entries in '''F'''. Then '''F'''^''m''×''n'' is a vector space over '''F'''. Vector addition is just matrix addition and scalar multiplication is defined in the obvious way (by multiplying each entry by the same scalar). The zero vector is just the [[zero matrix]]. The [[dimension (vector space)|dimension]] of '''F'''^''m''×''n'' is ''mn''. One possible choice of basis is the matrices with a single entry equal to 1 and all other entries 0.

When ''m'' = ''n'' the matrix is [[Square matrix|square]] and [[matrix multiplication]] of two such matrices produces a third. This vector space of dimension ''n''² forms an [[algebra over a field]].

==Polynomial vector spaces==

===One variable===

The set of [[polynomial]]s with coefficients in '''F''' is a vector space over '''F''', denoted '''F'''[''x'']. Vector addition and scalar multiplication are defined in the obvious manner. If the [[Degree of a polynomial|degree of the polynomials]] is unrestricted then the dimension of '''F'''[''x''] is [[countably infinite]]. If instead one restricts to polynomials with degree less than or equal to ''n'', then we have a vector space with dimension ''n''&nbspthinsp;+&nbspthinsp;1.

One possible basis for '''F'''[''x''] is a [[monomial basis]]: the coordinates of a polynomial with respect to this basis are its [[coefficient]]s, and the map sending a polynomial to the sequence of its coefficients is a [[linear isomorphism]] from '''F'''[''x''] to the infinite coordinate space '''F'''^∞.

The vector space of polynomials with real coefficients and degree less than or equal to ''n'' is often denoted by '''P'''_''n''.

===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==

:''See main article at [[Function space]], especially the functional analysis section.''

Let ''X'' be a non-empty arbitrary set and ''V'' an arbitrary vector space over '''F'''. The space of all [[function (mathematics)|function]]s from ''X'' to ''V'' is a vector space over '''F''' under [[pointwise]] addition and multiplication. That is, let ''f'' : ''X'' → ''V'' and ''g'' : ''X'' → ''V'' denote two functions, and let ''α''∈' in ''F'''. We define

:<math>(f + g)(x) = f(x) + g(x) </math>

:<math>(\alpha f)(x) = \alpha f(x) </math>

Line 77 ⟶ 78:

===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.

The space described above is commonly denoted ('''F'''^''X'')₀ and is called ''generalized coordinate space'' for the following reason. If ''X'' is the set of numbers between 1 and ''n'' then this space is easily seen to be equivalent to the coordinate space '''F'''^''n''. Likewise, if ''X'' is the set of [[natural number]]s, '''N''', then this space is just '''F'''^∞.

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.

A canonical basis for ('''F'''^''X'')₀ is the set of functions {δ_''x'' | ''x'' ∈ ''X''} defined by

The space described above is commonly denoted ('''F'''^''X'')₀ and is called ''generalized coordinate space'' for the following reason. If ''X'' is the set of numbers between 1 and ''n'' then this space is easily seen to be equivalent to the coordinate space '''F'''^''n''. Likewise, if ''X'' is the set of [[natural number]]s, '''N''', then this space is just '''F'''^∞.

A canonical basis for ('''F'''^''X'')₀ is the set of functions {δ_''x'' | ''x'' ∈ ''X''} defined by

:<math>\delta_x(y) = \begin{cases}1 \quad x = y \\ 0 \quad x \neq y\end{cases}</math>

The dimension of ('''F'''^''X'')₀ is therefore equal to the [[cardinality]] of ''X''. In this manner we can construct a vector space of any dimension over any field. Furthermore, ''every vector space is isomorphic to one of this form''. Any choice of basis determines an isomorphism by sending the basis onto the canonical one for ('''F'''^''X'')₀.

Generalized coordinate space may also be understood as the [[direct sum of modules|direct sum]] of |''X''| copies of '''F''' (i.e. one for each point in ''X''):

:<math>(\mathbf F^X)_0 = \bigoplus_{x\in X}\mathbf F.</math>

The finiteness condition is built into the definition of the direct sum. Contrast this with the [[direct product]] of |''X''| copies of '''F''' which would give the full function space '''F'''^''X''.

===Linear maps===

An important example arising in the context of [[linear algebra]] itself is the vector space of [[linear map]]s. Let ''L''(''V'',''W'') denote the set of all linear maps from ''V'' to ''W'' (both of which are vector spaces over '''F'''). Then ''L''(''V'',''W'') is a subspace of ''W''^''V'' since it is closed under addition and scalar multiplication.

Note that L('''F'''^''n'','''F'''^''m'') can be identified with the space of matrices '''F'''^''m''×''n'' in a natural way. In fact, by choosing appropriate bases for finite-dimensional spaces V and W, L(V,W) can also be identified with '''F'''^''m''×''n''. This identification normally depends on the choice of basis.

===Continuous functions===

If ''X'' is some [[topological space]], such as the [[unit interval]] [0,1], we can consider the space of all [[Continuous function (topology)|continuous function]]s from ''X'' to '''R'''. This is a vector subspace of '''R'''^''X'' since the sum of any two continuous functions is continuous and scalar multiplication is continuous.

===Differential equations===

The subset of the space of all functions from '''R''' to '''R''' consisting of (sufficiently differentiable) functions that satisfy a certain [[differential equation]] is a subspace of '''R'''^'''R''' if the equation is linear. This is because [[derivative|differentiation]] is a linear operation, i.e., <nowiki>(''a'' ''f'' + ''b'' ''g)'')′ = ''a'' ''f''{{space|hair}}′ + ''b'' ''g''′, where '′ is the differentiation operator.</nowiki>

==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 an (uncountably) infinite-dimensionala 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]].

These vector spaces have been exploited as [[projective geometry|projective]] [[finite geometry]] called [[Galois geometry]] PG(''n, q'') which uses ''n'' + 1 coordinates from Galois field GF(''q'').<ref>[[Oswald Veblen]] (1906) [http://www.ams.org/journals/tran/1906-007-02/S0002-9947-1906-1500747-6/S0002-9947-1906-1500747-6.pdf Finite Projective Geometries], [[Transactions of the American Mathematical Society]] 7: 241–59</ref> Linear transformations on the vector spaces represent projectivities of geometry which provide representation of some [[finite group]]s.

==Notes==

{{notelist}}

==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}}