In chapter 2 we introduced geometric algebra in two and three dimensions. We now turn to a discussion of the full, axiomatic framework for geometric algebra in arbitrary dimensions, with arbitrary signature. This will involve some duplication of material from chapter 2, but we hope that this will help reinforce some of the key concepts. Much of the material in this chapter is of primary relevance to those interested in the full range of applications of geometric algebra. Those interested solely in applications to space and spacetime may want to skip some of the material below, as both of these algebras are treated in a self-contained manner in chapters 2 and 5 respectively. The material on frames and linear algebra is important, however, and a knowledge of this is assumed for applications in gravitation.

The fact that geometric algebra can be applied in spaces of arbitrary dimensions is crucial to the claim that it is a mathematical tool of universal applicability. The framework developed here will enable us to extend geometric algebra to the study of relativistic dynamics, phase space, single and multiparticle quantum theory, Lie groups and manifolds. This chapter also highlights some of the new algebraic techniques we now have at our disposal. Many derivations can be simplified through judicious use of the geometric product at various intermediate steps. This is true even if the initial and final expressions contain only inner and outer products.