  This is the first in a series of papers on rank decompositions of the matrix
multiplication tensor. In this paper we: establish general facts about rank
decompositions of tensors, describe potential ways to search for new matrix
multiplication decompositions, give a geometric proof of the theorem of
Burichenko's theorem establishing the symmetry group of Strassen's algorithm,
and present two particularly nice subfamilies in the Strassen family of
decompositions.
