mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a left (or right) quaternionic vector space is a left (or right) H-

module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modul ...

where H is the (non-commutative) division ring of

quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...

s. The space H^''n'' 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 ''q''₁, ''q''₂, ... ''q''_''n''. Since H is a division algebra, every finitely generated (left or right) H-module has a

basis Basis may refer to: Finance and accounting *Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates *Basis trading, a trading strategy consisting of ...

, and hence is

isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...

to H^''n'' for some ''n''.

References

* Quaternions Linear algebra {{algebra-stub

See also

References