Quaternions form Vector Space over Themselves

Theorem

The set of quaternions $\H$, with the operations of addition and multiplication, forms a vector space.

Proof

Let the set of quaternions be denoted $\struct {\H, +, \times}$.

From Quaternions form Skew Field, the algebraic structure $\struct {\H, +, \times}$ is a skew field.

By definition, a skew field is a division ring.

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Also see

• Properties of Quaternions

Sources

• 1974: Robert Gilmore: Lie Groups, Lie Algebras and Some of their Applications ... (previous) ... (next): Chapter $1$: Introductory Concepts: $1$. Basic Building Blocks: $4$. LINEAR VECTOR SPACE: Example $2$