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==Finite vector spaces==
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]]) 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 (because a power of a prime power is again a prime power). The primary example of such a space is the coordinate space ('''F'''<sub>''q''</sub>)<sup>''n''</sup>.
These vector spaces have been exploited as [[projective geometry|projective]] [[finite geometry]] called [[Galois geometry]] PG(''n, q'') which uses ''n'' + 1 coordinates from Galois field GF(''q'').<ref>[[Oswald Veblen]] (1906) [http://www.ams.org/journals/tran/1906-007-02/S0002-9947-1906-1500747-6/S0002-9947-1906-1500747-6.pdf Finite Projective Geometries], [[Transactions of the American Mathematical Society]] 7: 241–59</ref> Linear transformations on the vector spaces represent projectivities of geometry which provide representation of some [[finite group]]s.
==References==
{{Reflist}}
{{DEFAULTSORT:Examples Of Vector Spaces}}
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