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 $\H$.

Also see

 * Properties of Quaternions