Quaternionic vector space: Difference between revisions

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{{Short description|Module over the algebra of quaternions.}}

In [[mathematics]], a '''left''' (or '''right''') '''quaternionic vector space''' is a left (or right) '''H'''-[[module (mathematics)|module]] where '''H''' is the (non-commutative) [[division ring]] of [[quaternion]]s.

In [[noncommutative algebra]], a branch of [[mathematics]], a '''quaternionic vector space''' is a [[Module (mathematics)|module]] over the [[Quaternion|quaternions]]. Since the quaternion algebra is [[division ring]], these modules are referred to as "vector spaces". However, the quaternion algebra is [[Noncommutative ring|noncommutative]] so we must distinguish left and right vector spaces. In left vector spaces, linear compositions of vectors <math> v</math> and <math> w</math> have the form <math> av+bw</math> where <math> a</math>, <math> b\in H</math>. In right vector spaces, linear compositions of vectors <math> v</math> and <math> w</math> have the form <math> va+wb</math>.

Similar to [[Vector space|vector spaces over a field]], if a quaternionic vector space has finite dimension <math> n</math>, then it is isomorphic to the direct sum <math> H^n</math> of <math> n</math> copies of the quaternion algebra <math> H</math>. In this case we can use a standard basis which has the form

The space '''H'''^''n'' of ''n''-tuples of quaternions is both a left and right '''H'''-module using the componentwise left and right multiplication:

:<math> q e_1=(q_11,q_20,\ldots q_n) = (q q_1,q q_2,\ldots q q_n0) </math>

:<math> (q_1,q_2,\ldots q_n) q = (q_1 q, q_2 q,\ldots q_n q)</math>

:<math>e_n=(0,\ldots,0,1)</math>

for quaternions ''q'' and ''q''₁, ''q''₂, ... ''q''_''n''.

In a left quaternionic vector space <math> H^n</math> we use componentwise sum of vectors and product of vectors over scalars

:<math> (p_1, \ldots, p_n)+(r_1, \ldots, r_n) = (p_1+ r_1, \ldots, p_n+ r_n)</math>

:<math> q (r_1, \ldots, r_n) = (q r_1, \ldots, q r_n)</math>

In a right quaternionic vector space <math> H^n</math> we also use componentwise sum of vectors and product of vectors over scalars

:<math> (p_1, \ldots, p_n)+(r_1, \ldots, r_n) = (p_1+ r_1, \ldots, p_n+ r_n)</math>

:<math> (r_1, \ldots, r_n)q = ( r_1q, \ldots, r_nq)</math>

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'''^''n'' for some ''n''.

==See also==

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