ABSTRACT
This chapter shows how subspaces can be described by introducing an operation called “outer product” on vectors. It becomes the basis of the Clifford algebra to be described in the next chapter. We first specify subspaces of different dimensions (the origin, lines passing through the origin, planes passing through the origin, and the entire 3D space) in terms of the outer product, which makes clear the properties of the outer product operation. Then, we introduce an operation called “contraction” and define the norm and the duality of subspaces. Finally, we show that two methods exist for specifying subspaces: the direct representation and the dual representation.
