Quaternionic vector space

In mathematics, a left (or right) quaternionic vector space is a left (or right) H-module where H denotes the noncommutative ring of the quaternions.

The space Hⁿ of n-tuples of quaternions is both a left and right H-module using the componentwise left and right multiplication:

for quaternions q and q₁, q₂, ... q_n.

Since H is a division algebra, every finitely generated (left or right) H-module has a basis, and hence is isomorphic to Hⁿ for some n.