How does linear algebra over the octonions and other division algebras work?

As Billy points out, the rule $(rs)\cdot x=r\cdot(s\cdot x)$ implies that $(rs)t$ and $r(st)$ will act identically on your module.

According to Wikipedia's multiplication table we have $$ e_1(e_3e_5) = -e_1e_6 = e_7 \\ (e_1e_3)e_5 = -e_2e_5 = -e_7 $$ so we must have $$ e_7\cdot x = -e_7\cdot x $$ whence (by distributivity) $$ 2e_7\cdot x = 0 $$ and by dividing by $2e_7$, $$ 1\cdot x = 0 $$ Since the module axioms also require $1\cdot x$ to be $x$, this means that the only module that satisfies all of your rules is $\{0\}$ -- not very exciting. Thus:

  • 1. Yes, the basis number is invariant. Every basis contains exactly $0$ vectors.

  • 2. Yes, if you allow $0\times 0$ matrices.

Tags:

Linear Algebra

Modules

Abstract Algebra

Octonions

Division Algebras

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