Linear transformations of a vector space image change its vectors. When this happens, the blades spanned by those vectors change quite naturally to become the spans of the transformed vectors. That defines the extension of a linear transformation to the full sub-space algebra. This embedding gives us more powerful tools to apply linear transformations immediately to subspaces, without needing to first decompose those subspaces into vectors.

We study the resulting structure in this chapter. The algebra dictates how we should do the outer products and contractions of transformed blades, and in that way gives us the ...

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