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| In [[mathematics]], a '''left''' (or '''right''') '''quaternionic vector space''' is a left (or right) '''H'''-[[module (mathematics)|module]] where '''H''' denotes the [[ring (mathematics)|noncommutative ring]] of the [[quaternion]]s.
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| The space '''H'''<sup>''n''</sup> of ''n''-tuples of quaternions is both a left and right '''H'''-module using the componentwise left and right multiplication:
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| :<math> q (q_1,q_2,\ldots q_n) = (q q_1,q q_2,\ldots q q_n) </math>
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| :<math> (q_1,q_2,\ldots q_n) q = (q_1 q, q_2 q,\ldots q_n q)</math>
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| for quaternions ''q'' and ''q''<sub>1</sub>, ''q''<sub>2</sub>, ... ''q''<sub>''n''</sub>.
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| Since '''H''' is a [[division algebra]], every [[Finitely generated module|finitely generated]] (left or right) '''H'''-module has a [[basis (linear algebra)|basis]], and hence is isomorphic to '''H'''<sup>''n''</sup> for some ''n''.
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| ==See also== | |
| * [[Vector space]] <!-- Doesn't mention the quaternions, but should!-->
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| * [[General linear group]] <!-- Doesn't mention the quaternions, but should!-->
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| * [[Special linear group]] <!-- Doesn't mention the quaternions, but should!-->
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| * [[List of simple Lie groups#A2n.E2.88.921_II_.28n_.E2.89.A5_2.29|SL(n,H)]] <!-- No definitions here!-->
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| * [[Symplectic group]] <!-- Hoorah, it mentions Sp(n) and Sp(p,q)-->
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| ==References==
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| *{{cite book
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| | first = F. Reese
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| | last = Harvey
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| | year = 1990
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| | title = Spinors and Calibrations
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| | publisher = Academic Press
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| | location = San Diego
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| | id = ISBN 0-12-329650-1
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| }}
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| {{algebra-stub}}
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| [[Category:Quaternions]]
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| [[Category:Linear algebra]]
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