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| {{Unreferenced|date=November 2009}}
| | This can a strategy and often battle activation where that you must manage your own tribe and also protect it. You have so that it will build constructions which will provide protection for your individual soldiers along with the instruction. First concentrate on your protection together with after its recently been taken treatment. You ought to to move forward when it comes to the criminal offense arrange. If you have any kind of questions concerning where and the best ways to make use of [http://prometeu.net hack clash of clans], you can contact us at our own webpage. As well as Military facilities, you also need to keep in mind the way your group is certainly going. For instance, collecting tactics as well as widening your own tribe is the key to good improvements.<br><br> |
| 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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| ''Notation''. We will let '''F''' denote an arbitrary [[field (mathematics)|field]] such as the [[real number]]s '''R''' or the [[complex number]]s '''C'''. See also: [[table of mathematical symbols]].
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| ==Trivial or zero vector space==
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| {{main|Zero vector space}}
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| 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 isomorphic to this one.
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| The zero vector space is different from the [[null space]] of a linear operator F, which is the kernel of F.
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| ==Field==
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| The next simplest example is the field '''F''' itself. Vector addition is just field addition and scalar multiplication is just field multiplication. Any non-zero element of '''F''' serves as a basis so '''F''' is a 1-dimensional vector space over itself.
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| The field is a rather special vector space; in fact it is the simplest example of a '''[[algebra over a field|commutative algebra]]''' over '''F'''. Also, '''F''' has just two [[Linear subspace|subspaces]]: {0} and '''F''' itself.
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| ==Coordinate space==
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| {{Main|Coordinate space}}
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| 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'''<sup>''n''</sup>. An element of '''F'''<sup>''n''</sup> is written
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| :<math>x = (x_1, x_2, \ldots, x_n) \,</math>
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| where each ''x''<sub>''i''</sub> is an element of '''F'''. The operations on '''F'''<sup>''n''</sup> are defined by
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| :<math>x + y = (x_1 + y_1, x_2 + y_2, \ldots, x_n + y_n) \,</math>
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| :<math>\alpha x = (\alpha x_1, \alpha x_2, \ldots, \alpha x_n) \,</math>
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| :<math>0 = (0, 0, \ldots, 0) \,</math>
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| :<math>-x = (-x_1, -x_2, \ldots, -x_n) \,</math>
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| The most common cases are where '''F''' is the field of [[real number]]s giving the [[real coordinate space]] '''R'''<sup>''n''</sup>, or the field of [[complex number]]s giving the [[complex coordinate space]] '''C'''<sup>''n''</sup>.
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| The [[quaternion]]s and the [[octonion]]s are respectively four- and eight- dimensional vector spaces over the reals.
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| The vector space '''F'''<sup>''n''</sup> comes with a [[standard basis]]:
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| :<math>e_1 = (1, 0, \ldots, 0) \,</math>
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| :<math>e_2 = (0, 1, \ldots, 0) \,</math>
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| :<math>\vdots \,</math>
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| :<math>e_n = (0, 0, \ldots, 1) \,</math>
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| where 1 denotes the multiplicative identity in '''F'''.
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| ==Infinite coordinate space==
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| Let '''F'''<sup>∞</sup> 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'''<sup>∞</sup> as
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| :<math>x = (x_1, x_2, x_3, \ldots) \,</math>
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| then only a finite number of the ''x''<sub>''i''</sub> 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'''<sup>∞</sup> is [[countably infinite]]. A standard basis consists of the vectors ''e''<sub>''i''</sub> 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'''.
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| 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'''<sup>'''N'''</sup> - see [[Examples of vector spaces#Function spaces|below]]. '''F'''<sup>'''N'''</sup> is the ''[[product (category theory)|product]]'' of countably many copies of '''F'''.
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| By Zorn's lemma, '''F'''<sup>'''N'''</sup> has a basis (there is no obvious basis). There are [[uncountably infinite]] elements in the basis. Since the dimensions are different, '''F'''<sup>'''N'''</sup> is ''not'' isomorphic to '''F'''<sup>∞</sup>. It is worth noting that '''F'''<sup>'''N'''</sup> is (isomorphic to) the [[dual space]] of '''F'''<sup>∞</sup>, because a linear map ''T'' from '''F'''<sup>∞</sup> to '''F''' is determined uniquely by its values ''T''(''e<sub>i</sub>'') on the basis elements of '''F'''<sup>∞</sup>, and these values can be arbitrary. Thus one sees that a vector space need not be isomorphic to its dual if it is infinite dimensional, in contrast to the finite dimensional case.
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| ==Product of vector spaces==
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| Starting from ''n'' vector spaces, or a countably infinite collection of them, each with the same field, we can define the product space like above.
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| ==Matrices==
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| Let '''F'''<sup>''m''×''n''</sup> denote the set of ''m''×''n'' [[matrix (mathematics)|matrices]] with entries in '''F'''. Then '''F'''<sup>''m''×''n''</sup> 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'''<sup>''m''×''n''</sup> is ''mn''. One possible choice of basis is the matrices with a single entry equal to 1 and all other entries 0.
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| ==Polynomial vector spaces==
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| ===One variable===
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| 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'' + 1.
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| 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'''<sup>∞</sup>.
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| The vector space of polynomials with real coefficients and degree less than or equal to ''n'' is denoted by '''P'''<sub>''n''</sub>.
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| ===Several variables===
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| The set of [[polynomial]]s in several variables with coefficients in '''F''' is vector space over '''F''' denoted '''F'''[''x''<sub>1</sub>, ''x''<sub>2</sub>, …, ''x''<sub>''r''</sub>]. Here ''r'' is the number of variables.
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| :''See also'': [[Polynomial ring]]
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| ==Function spaces==
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| :''See main article at [[Function space]], especially the functional analysis section.''
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| Let ''X'' be an 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''∈'''F'''. We define
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| :<math>(f + g)(x) = f(x) + g(x) \,</math>
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| :<math>(\alpha f)(x) = \alpha f(x) \,</math>
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| where the operations on the right hand side are those in ''V''. The zero vector is given by the constant function sending everything to the zero vector in ''V''. The space of all functions from ''X'' to ''V'' is commonly denoted ''V''<sup>''X''</sup>.
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| If ''X'' is finite and ''V'' is finite-dimensional then ''V''<sup>''X''</sup> has dimension |''X''|(dim ''V''), otherwise the space is infinite-dimensional (uncountably so if ''X'' is infinite).
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| Many of the vector spaces that arise in mathematics are subspaces of some function space. We give some further examples.
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| ===Generalized coordinate space===
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| 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'''<sup>''X''</sup>, 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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| The space described above is commonly denoted ('''F'''<sup>''X''</sup>)<sub>0</sub> 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'''<sup>''n''</sup>. Likewise, if ''X'' is the set of [[natural number]]s, '''N''', then this space is just '''F'''<sup>∞</sup>.
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| A canonical basis for ('''F'''<sup>''X''</sup>)<sub>0</sub> is the set of functions {δ<sub>''x''</sub> | ''x'' ∈ ''X''} defined by
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| :<math>\delta_x(y) = \begin{cases}1 \quad x = y \\ 0 \quad x \neq y\end{cases}</math>
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| The dimension of ('''F'''<sup>''X''</sup>)<sub>0</sub> 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'''<sup>''X''</sup>)<sub>0</sub>.
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| 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''):
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| :<math>(\mathbf F^X)_0 = \bigoplus_{x\in X}\mathbf F.</math>
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| 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'''<sup>''X''</sup>.
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| ===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''<sup>''V''</sup> since it is closed under addition and scalar multiplication.
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| Note that L('''F'''<sup>''n''</sup>,'''F'''<sup>''m''</sup>) can be identified with the space of matrices '''F'''<sup>''m''×''n''</sup> 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'''<sup>''m''×''n''</sup>. This identification normally depends on the choice of basis.
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| ===Continuous functions===
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| If ''X'' is some [[topological space]], such as the [[unit interval]] [0,1], we can consider the space of all [[continuous function]]s from ''X'' to '''R'''. This is a vector subspace of '''R'''<sup>''X''</sup> since the sum of any two continuous functions is continuous and scalar multiplication is continuous.
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| ===Differential equations===
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| 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'''<sup>'''R'''</sup> if the equation is linear. This is because [[derivative|differentiation]] is a linear operation, i.e., <nowiki>(a f + b g)' = a f' + b g', where ' is the differentiation operator.</nowiki>
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| ==Field extensions==
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| Suppose '''K''' is a [[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 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-dimensional vector space over the [[rational number]]s '''Q'''.
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| 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
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| :dim<sub>'''K'''</sub>''V'' = (dim<sub>'''F'''</sub>''V'')(dim<sub>'''K'''</sub>'''F''')
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| For example '''C'''<sup>''n''</sup>, regarded as a vector space over the reals, has dimension 2''n''.
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| ==Finite vector spaces==
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| 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'''<sub>''q''</sub>, the unique finite field (up to [[isomorphism]], of course) with ''q'' elements. Here ''q'' must be a power of a [[prime number|prime]] (''q'' = ''p''<sup>''m''</sup> with ''p'' prime). Then any ''n''-dimensional vector space ''V'' over '''F'''<sub>''q''</sub> will have ''q''<sup>''n''</sup> elements. Note that the number of elements in ''V'' is also the power of a prime. The primary example of such a space is the coordinate space ('''F'''<sub>''q''</sub>)<sup>''n''</sup>.
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| {{DEFAULTSORT:Examples Of Vector Spaces}}
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| [[Category:Mathematical examples|Vector spaces]]
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| [[Category:Vector spaces| ]]
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