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
- 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 $2$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): quaternion
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): quaternion