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 $a$, $b\in H$. 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

$e_1=(1,0,\dots ,0)$

$\dots$

$e_n=(0,\dots ,0,1)$

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

$(p_1,\dots ,p_n)+(r_1,\dots ,r_n)=(p_1+r_1,\dots ,p_n+r_n)$

$q(r_1,\dots ,r_n)=(q{r}_1,\dots ,q{r}_n)$

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

$(p_1,\dots ,p_n)+(r_1,\dots ,r_n)=(p_1+r_1,\dots ,p_n+r_n)$

$(r_1,\dots ,r_n)q=(r_{1}q,\dots ,r_{n}q)$

See also

• Vector space
• General linear group
• Special linear group
• SL(n,H)
• Symplectic group

References

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