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| {{expert|mathematics|date=October 2011}}
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| {{Unreferenced|date=November 2009}}
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| In [[mathematics]], specifically in [[linear algebra]], the '''coordinate space''', '''F'''<sup>''n''</sup>, is the prototypical example of an ''n''-dimensional [[vector space]] over a [[field (mathematics)|field]] '''F'''. It can be defined as the [[product space]] of F over a finite [[index set]].
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| ==Definition==
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| Let '''F''' denote an arbitrary [[field (mathematics)|field]] (such as the [[real number]]s '''R''' or the [[complex number]]s '''C'''). 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''' called '''coordinate space''' and denoted '''F'''<sup>''n''</sup>.
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| An element of '''F'''<sup>''n''</sup> is written
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| :<math>\mathbf x = (x_1, x_2, \cdots, x_n)</math> | |
| 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>\mathbf x + \mathbf y = (x_1 + y_1, x_2 + y_2, \cdots, x_n + y_n)</math>
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| :<math>\alpha \mathbf x = (\alpha x_1, \alpha x_2, \cdots, \alpha x_n).</math>
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| The zero vector is given by
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| :<math>\mathbf 0 = (0, 0, \cdots, 0)</math>
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| and the additive inverse of the vector '''x''' is given by | |
| :<math>-\mathbf x = (-x_1, -x_2, \cdots, -x_n).</math> | |
| | |
| ===Matrix notation===
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| In standard [[matrix (mathematics)|matrix]] notation, each element of '''F'''<sup>''n''</sup> is typically written as a [[column vector]]
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| :<math>\mathbf x = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}</math>
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| and sometimes as a [[row vector]]:
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| :<math>\mathbf x = \begin{bmatrix} x_1 & x_2 & \dots & x_n \end{bmatrix}.</math>
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| The coordinate space '''F'''<sup>''n''</sup> may then be interpreted as the space of all ''n''×1 column vectors, or all 1×''n'' row vectors with the ordinary matrix operations of addition and scalar multiplication.
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| [[Linear transformation]]s from '''F'''<sup>''n''</sup> to '''F'''<sup>''m''</sup> may then be written as ''m''×''n'' matrices which act on the elements of '''F'''<sup>''n''</sup> via left multiplication (when the elements of '''F'''<sup>''n''</sup> are column vectors) or right multiplication (when they are row vectors).
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| ==Standard basis==
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| The coordinate space '''F'''<sup>''n''</sup> comes with a [[standard basis]]:
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| :<math>\mathbf e_1 = (1, 0, \ldots, 0)</math> | |
| :<math>\mathbf e_2 = (0, 1, \ldots, 0)</math>
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| :<math>\vdots</math>
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| :<math>\mathbf e_n = (0, 0, \ldots, 1)</math>
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| where 1 denotes the multiplicative identity in '''F'''. To see that this is a basis, note that an arbitrary vector in '''F'''<sup>''n''</sup> can be written uniquely in the form
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| :<math>\mathbf x = \sum_{i=1}^n x_i \mathbf{e}_i.</math>
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| ==Discussion==
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| It is a standard fact of linear algebra that every ''n''-dimensional vector space ''V'' over '''F''' is [[isomorphic]] to '''F'''<sup>''n''</sup>. It is a crucial point, however, that this isomorphism is not [[canonical form|canonical]]. If it were, mathematicians would work only with '''F'''<sup>''n''</sup> rather than with abstract vector spaces.
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| A choice of isomorphism is equivalent to a choice of [[ordered basis]] for ''V''. To see this, let
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| :''A'' : '''F'''<sup>''n''</sup> → ''V''
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| be a linear isomorphism. Define an ordered basis {'''a'''<sub>''i''</sub>} for ''V'' by
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| : '''a'''<sub>''i''</sub> = ''A''('''e'''<sub>''i''</sub>) for 1 ≤ ''i'' ≤ ''n''.
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| Conversely, given any ordered basis {'''a'''<sub>''i''</sub>} for ''V'' define a linear map ''A'' : '''F'''<sup>''n''</sup> → ''V'' by
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| :<math>A(\mathbf x) = \sum_{i=1}^n x_i \mathbf a_i.</math>
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| It is not hard to check that ''A'' is an isomorphism. Thus ordered bases for ''V'' are in 1-1 correspondence with linear isomorphisms '''F'''<sup>''n''</sup> → ''V''.
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| The reason for working with abstract vector spaces instead of '''F'''<sup>''n''</sup> is that it is often preferable to work in a ''coordinate-free'' manner, i.e. without choosing a preferred basis. Indeed, many vector spaces that naturally show up in mathematics do not come with a preferred choice of basis.
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| It is possible and sometimes desirable to view a coordinate space [[duality (mathematics)|dually]] as the set of F-valued functions on a finite set; that is, each "point" of '''F'''<sup>''n''</sup> is viewed as a function whose domain is the finite set {1,2....n} and codomain '''F'''. The function sends an element i of {1,2....n} to the value of the i'th coordinate of the "point", so '''F'''<sup>''n''</sup> is, dually, a set of functions.
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| ==See also==
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| *[[real coordinate space]], '''R'''<sup>''n''</sup>
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| *[[complex coordinate space]], '''C'''<sup>''n''</sup>
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| *[[examples of vector spaces]]
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| {{DEFAULTSORT:Coordinate Space}}
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| [[Category:Linear algebra]]
| |
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