After many attempts at formalizing space and spatial relationships, the concept of a vector space emerged as the useful framework for geometrical computations. We use it as our point of departure, and use some of the standard linear algebra governing its mappings. Yet already we will have much to add to its usual structure. By the end of this chapter you will realize that a vector space is much more than merely a space of vectors, and that it is straightforward and useful to extend it computationally.

The crucial idea is to make the subspaces of a vector space explicit elements of computation. To build our algebra of subspaces, we revisit the familiar lines and planes through the origin. We investigate their geometrical ...

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