Examples of vector spaces: Difference between revisions

Revision as of 12:21, 7 October 2020

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==Coordinate space==

{{Main|Coordinate space}}

The original example of a vector space, which the axiomatic definition generalizes, 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''^''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