Quaternionic vector space explained

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\inH

. 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

Hn

of

n

copies of quaternion algebra

H

. In such case we can use basis which has form

e1=(1,0,\ldots,0)

\ldots

en=(0,\ldots,0,1)

In left quaternionic vector space

Hn

we use componentwise sum of vectors and product of vector over scalar

(p1,\ldots,pn)+(r1,\ldots,rn)=(p1+r1,\ldots,pn+rn)

q(r1,\ldots,rn)=(qr1,\ldots,qrn)

In right quaternionic vector space

Hn

we use componentwise sum of vectors and product of vector over scalar

(p1,\ldots,pn)+(r1,\ldots,rn)=(p1+r1,\ldots,pn+rn)

(r1,\ldots,rn)q=(r1q,\ldots,rnq)

References

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Quaternionic vector space".

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