Quaternions form Vector Space over Reals
Theorem
Let $\R$ be the set of real numbers.
Let $\H$ be the set of quaternions.
Then the $\R$-module $\H$ is a vector space.
Proof
Recall that Real Numbers form Field.
Thus by definition, $\R$ is also a division ring.
Thus we only need to show that $\R$-module $\H$ is a unitary module, by demonstrating the module properties:
$\forall x, y, \in \H, \forall \lambda, \mu \in \R$:
- $(1): \quad \lambda \paren {x + y} = \paren {\lambda x} + \paren {\lambda y}$
- $(2): \quad \paren {\lambda + \mu} x = \paren {\lambda x} + \paren {\mu x}$
- $(3): \quad \paren {\lambda \mu} x = \lambda \paren {\mu x}$
- $(4): \quad 1 x = x$
As $\lambda, \mu \in \R$ it follows that $\lambda, \mu \in \H$.
Thus from Quaternion Multiplication Distributes over Addition, $(1)$ and $(2)$ immediately follow.
$(3)$ follows from Quaternion Multiplication is Associative.
$(4)$ follows from Multiplicative Identity for Quaternions, as $\mathbf 1 + 0 \mathbf i + 0 \mathbf j + 0 \mathbf k$ is the unity of $\H$.
$\blacksquare$
Also see
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 $2$