Examples of vector spaces: Difference between revisions

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==Coordinate space==

{{Main|Coordinate space}}

PerhapsThe the most importantoriginal example of a vector space, which the axiomatic definition generalizes, is the following. For any [[Positive number|positive]] [[integer]] ''n'', the spaceset 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

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

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Commonly, '''F''' is the field of [[real number]]s, in which case we obtain [[real coordinate space]] '''R'''^''n''. The field of [[complex number]]s gives [[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 vector space '''RF'''^''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 '''RF'''.

==Infinite coordinate space==