Examples of vector spaces: Difference between revisions

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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'''_''q'', the unique finite field (up to [[isomorphism]], of course) with ''q'' elements. Here ''q'' must be a power of a [[prime number|prime]] (''q'' = ''p''^''m'' with ''p'' prime). Then any ''n''-dimensional vector space ''V'' over '''F'''_''q'' will have ''q''^''n'' elements. Note that the number of elements in ''V'' is also the power of a prime. The primary example of such a space is the coordinate space ('''F'''_''q'')^''n''.

{{DEFAULTSORT:Examples Of Vector Spaces}}