Transformation of a vector via geometric algebra

Question

$\begin{pmatrix} 1 & 3 & 2 \\ 0 & 2 & 5 \\ 2 & 2 & 1 \end{pmatrix} \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} = \begin{pmatrix} x \\ y \\ z \end{pmatrix} $

Let's take this general transformation of a vector. How do I calculate this in the framework of geometric algebra ?

Answer 1

You should really think that of geometric algebra (the study of certain Clifford algebras over the reals) as an extension of linear algebra -- not as a replacement for matrix algebra. Geometric algebra provides new ways to describe things like subspaces, invariant subspaces, certain specific linear transformations like rotations and reflections, etc. But that doesn't mean you can find an equivalent geometric algebra expression to simplify any given linear transformation.

So the simple answer is: you don't.