Editing Examples of vector spaces (section)

==Coordinate space==
[[File:Line equation qtl4.svg|thumb|Planar [[analytic geometry]] uses the coordinate space '''R'''<sup>2</sup>. ''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}}
A basic 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''<sup>''n''</sup>.<ref>{{Harvard citations|last = Lang|year = 1987|loc = ch. I.1|nb = yes}}</ref>  An element of ''F''<sup>''n''</sup> is written
:<math>x = (x_1, x_2, \ldots, x_n) </math>
where each ''x''<sub>''i''</sub> is an element of ''F''. The operations on ''F''<sup>''n''</sup> 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>
Commonly, ''F'' is the field of [[real number]]s, in which case we obtain [[real coordinate space]] '''R'''<sup>''n''</sup>. The field of [[complex number]]s gives [[complex coordinate space]] '''C'''<sup>''n''</sup>. 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'''<sup>''n''</sup> is a ''2n''-dimensional real vector space.

The vector space ''F''<sup>''n''</sup> has 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 ''F''.