quaternionic vector space

Since quaternion algebra is division ring, then module over quaternion algebra is called vector space. Because quaternion algebra is non-commutative, we distinguish left and right vector spaces. In left vector space, linear composition of vectors $v$ and $w$ has form $av+bw$ where $a$, $b\backslash in\; H$. In right vector space, linear composition of vectors $v$ and $w$ has form $va+wb$.

If quaternionic vector space has finite dimension $n$, then it is isomorphic to direct sum $H^n$ of $n$ copies of quaternion algebra $H$. In such case we can use basis which has form

:$e\_1=(1,0,\backslash ldots,0)$

:$\backslash ldots$

:$e\_n=(0,\backslash ldots,0,1)$

In left quaternionic vector space $H^n$ we use componentwise sum of vectors and product of vector over scalar

:$(p\_1,\; \backslash ldots,\; p\_n)+(r\_1,\; \backslash ldots,\; r\_n)\; =\; (p\_1+\; r\_1,\; \backslash ldots,\; p\_n+\; r\_n)$

:$q\; (r\_1,\; \backslash ldots,\; r\_n)\; =\; (q\; r\_1,\; \backslash ldots,\; q\; r\_n)$

In right quaternionic vector space $H^n$ we use componentwise sum of vectors and product of vector over scalar

:$(p\_1,\; \backslash ldots,\; p\_n)+(r\_1,\; \backslash ldots,\; r\_n)\; =\; (p\_1+\; r\_1,\; \backslash ldots,\; p\_n+\; r\_n)$

:$(r\_1,\; \backslash ldots,\; r\_n)q\; =\; (\; r\_1q,\; \backslash ldots,\; r\_nq)$