Quaternionic vector space: Difference between revisions

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

SinceIn [[quaternionnoncommutative algebra]], algebraa isbranch of [[division ringmathematics]], thena '''quaternionic vector space''' is a [[moduleModule (mathematics)|module]] over the [[Quaternion|quaternions]]. Since the quaternion algebra is called[[division ring]], these modules are referred to as "vector spacespaces". BecauseHowever, the quaternion algebra is non-commutative,[[Noncommutative ring|noncommutative]] so we must distinguish left and right vector spaces. In left vector spacespaces, linear compositioncompositions of vectors <math> v</math> and <math> w</math> hashave the form <math> av+bw</math> where <math> a</math>, <math> b\in H</math>. In right vector spacespaces, linear compositioncompositions of vectors <math> v</math> and <math> w</math> hashave the form <math> va+wb</math>.

IfSimilar 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 suchthis case we can use a standard basis which has the form

:<math>e_1=(1,0,\ldots,0)</math>

:<math>\ldots</math>

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

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

:<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 vectorvectors over scalarscalars

:<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>