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
and
has form
where
,
. In right vector space, linear composition of vectors
and
has form
.
If quaternionic vector space has finite dimension
, then it is isomorphic to direct sum
of
copies of quaternion algebra
. In such case we can use basis which has form



In left quaternionic vector space
we use componentwise sum of vectors and product of vector over scalar


In right quaternionic vector space
we use componentwise sum of vectors and product of vector over scalar


- Harvey, F. Reese (1990). Spinors and Calibrations. San Diego: Academic Press. ISBN 0-12-329650-1.