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

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

${\displaystyle q(q_{1},q_{2},\ldots q_{n})=(qq_{1},qq_{2},\ldots qq_{n})}$

${\displaystyle (q_{1},q_{2},\ldots q_{n})q=(q_{1}q,q_{2}q,\ldots q_{n}q)}$

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.