# 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:

$q (q_1,q_2,\ldots q_n) = (q q_1,q q_2,\ldots q q_n)$
$(q_1,q_2,\ldots q_n) q = (q_1 q, q_2 q,\ldots q_n q)$

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

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.