We have seen how the outer product and the contraction characterize rather different properties of subspaces: qualitative spanning and quantitative measurements. Together, they have given us an enriched view of the linear algebra of subspaces. This much has been known for some time, and is part of the branch of applied mathematics that is called Grassmann-Cayley algebra.

In this chapter we will start afresh and introduce the basics of Clifford algebra to develop a powerful geometric algebra. This geometric algebra will incorporate operators on subspaces into our framework, and permit us to displace the constructions of the subspace algebra in a structure-preserving manner. The crucial construction ...

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