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In [[mathematics]], a '''left''' (or '''right''') '''quaternionic vector space''' is a left (or right) '''<math>\mathbb H'''</math>-[[module (mathematics)|module]] where '''<math>\mathbb H'''</math> is the (non-commutative) [[division ring]] of [[quaternion]]s. One must distinguish between left and right quaternionic vector spaces since <math>\mathbb H</math> is non-commutative. Further, <math>\mathbb H</math> is not a field, so quaternionic vector spaces are not [[vector space]]s, but merely modules.
The space '''H'''<supmath>''\mathbb H^n''</supmath> of ''n''-tuples of quaternions is both a left and right '''H'''-modulequaternionic usingvector thespace using componentwise leftmultiplication. Namely, for <math>q \in \mathbb H</math> and right<math>(r_1, multiplication:\ldots, r_n) \in \mathbb H^n</math>,
:<math> q (q_1,q_2r_1, \ldots, q_nr_n) = (q q_1r_1,q q_2,\ldots, q q_nr_n) ,</math>
:<math> (q_1,q_2r_1, \ldots, q_nr_n) q = (q_1r_1 q, q_2 q,\ldots, q_nr_n q).</math>
for quaternions ''q'' and ''q''<sub>1</sub>, ''q''<sub>2</sub>, ... ''q''<sub>''n''</sub>.
Since '''<math>\mathbb H'''</math> is a [[division algebra]], every [[Finitely generated module|finitely generated]] (left or right) '''<math>\mathbb H'''</math>-module has a [[basis (linear algebra)|basis]], and hence is [[isomorphic]] to '''H'''<supmath>''\mathbb H^n''</supmath> for some ''<math>n''</math>.
==See also==
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