The Unapologetic Mathematician

Okay, now to lay out the actual objects of our current interest. These are basically collections $latex Phi$ of vectors in some inner product space $latex V$, but with each vector comes a reflection and we want these reflections to play nicely with the vectors themselves. In a way, each point acts as both “program” — an operation to be performed — and “data” — an object to which operations can be applied — and the interplay between these two roles leads to some very interesting structure.

First off, only nonzero vectors give rise to reflections, so we don’t really want zero to be in our collection $latex Phi$. We also may as well assume that $latex Phi$ spans $latex V$, because it certainly spans *some* subspace of $latex V$ and anything that happens off of this subspace is pretty uninteresting as far as $latex Phi$ goes. These are the…

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