Generalize to Pseudo Euclidean spaces

[lukeburns]

The construction here only applies to Euclidean spaces. Namely, given two vectors $a$ and $b$, the decomposition $a = a\parallel + a\perp$ such that $a \cdot b = a\parallel b$ and $a \wedge b = a\perp b$ only applies when $b^2 \not= 0$ (i.e. is invertible).

[lukeburns]

The purpose of this paper is to show that GA can be constructed axiomatically without a metric – i.e. that the contractive property of the symmetric product can be derived without imposing it via a quadratic form. The problem as things currently stands is that the postulated decomposition of the inner product into parallel and perpendicular parts does not work with null vectors.

[lukeburns]

When $a^2 = 0$ or $b^2 = 0$ this doesn't work. What about an axiom that says for any two vectors $a, b$, they can be decomposed into $a = a* + a\perp$ and $b = b* + b\perp$ such that $a* b = b_ a*, a\perp b = - b a\perp$, and $a b\perp = - b_\perp a$?

[lukeburns]

A better approach may be to require that the wedge product as defined by $a \wedge A_r = \frac{1}{2}( a A_r + (-1)^r A_r a)$ is associative so that $a \wedge (b \wedge c) = ( a \wedge b ) \wedge c$. This is enough to show that $a \cdot c$ is a scalar.