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Quaternionic vector space

Module over the algebra of quaternions

Wikipedia contributorsAdapted for Fathomly · View authors
1 min read
Source updated January 21, 2026

In noncommutative algebra, a branch of mathematics, a quaternionic vector space is a module over the quaternions. Since the quaternion algebra is division ring, these modules are referred to as "vector spaces". However, the quaternion algebra is noncommutative so we must distinguish left and right vector spaces. In left vector spaces, linear compositions of vectors v and w have the form av+bw where a, b\in H. In right vector spaces, linear compositions of vectors v and w have the form va+wb.

Similar to vector spaces over a field, if a quaternionic vector space has finite dimension n, then it is isomorphic to the direct sum H^{n} of n copies of the quaternion algebra H. In this case we can use a standard basis which has the form

e_{1}=(1,0,\ldots ,0)
\ldots
e_{n}=(0,\ldots ,0,1)

In a left quaternionic vector space H^{n} we use componentwise sum of vectors and product of vectors over scalars

(p_{1},\ldots ,p_{n})+(r_{1},\ldots ,r_{n})=(p_{1}+r_{1},\ldots ,p_{n}+r_{n})
q(r_{1},\ldots ,r_{n})=(qr_{1},\ldots ,qr_{n})

In a right quaternionic vector space H^{n} we also use componentwise sum of vectors and product of vectors over scalars

(p_{1},\ldots ,p_{n})+(r_{1},\ldots ,r_{n})=(p_{1}+r_{1},\ldots ,p_{n}+r_{n})
(r_{1},\ldots ,r_{n})q=(r_{1}q,\ldots ,r_{n}q)
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This article is adapted from the Wikipedia article “Quaternionic vector space”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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