Quaternions form Vector Space over Reals

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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 $\R$ is a division ring a fortiori.

Thus we only need to show that $\R$-module $\H$ is a unitary module, by demonstrating the unitary (left) module axioms:

\((\text {UM} 1)\)   $:$     \(\ds x, y \in \H: \forall \lambda \in \R:\)    \(\ds \lambda \paren {x + y} \)   \(\ds = \)   \(\ds \paren {\lambda x} + \paren {\lambda y} \)      
\((\text {UM} 2)\)   $:$     \(\ds x \in \H: \forall \lambda, \mu \in \R:\)    \(\ds \paren {\lambda + \mu} x \)   \(\ds = \)   \(\ds \paren {\lambda x} + \paren {\mu x} \)      
\((\text {UM} 3)\)   $:$     \(\ds x \in \H: \forall \lambda, \mu \in \R:\)    \(\ds \paren {\lambda \mu} x \)   \(\ds = \)   \(\ds \lambda \paren {\mu x} \)      
\((\text {UM} 4)\)   $:$     \(\ds \forall x \in \H:\)    \(\ds 1 x \)   \(\ds = \)   \(\ds x \)      


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