## Algebraic Foundations and 4D

In this lecture we set out the algebraic rules that define a geometric algebra in spaces of arbitrary dimensions. The inner and outer products are extended to their most general definition, and we look at rotors operate on arbitrary multivectors. A theme is that many proofs are simplified using the geometric product in intermediate steps, even if the final results only refer to separate inner or outer products.

As an extended example we consider the application of GA to projective geometry. An essential point here is that there is more than one way to picture the algebraic relations in a geometric algebra, and in the projective viewpoint grade-1 objects represent points. The outer product of points returns lines, planes etc., and the inner product tells us about intersection. The projective representation of 3D space requires a 4D geometric algebra, which provides our first example of objects that are homogeneous (pure grade), but cannot be written is a blade.