geometric algebra expression for a linear transformation defined on basis vectors

Question

I have a transformation $T(u)$ between 3D vector subspaces of a 4D vector space, $A \rightarrow B$, defined by a mapping of their basis vectors:

$A:\{a_1=e_1\,,\,a_2=e_2\,,\,a_3=e_4\}$

$B:\{b_1=e_2\,,\,b_2=e_3\,,\,b_3=e_1\}$

$T(a_i)=b_i$

I have aligned each space to the standard bases in this example, but I don't want to assume that will always be the case.

My question is: Is there a general process for computing an expression in geometric algebra for this transformation? I know there are ways to define a matrix for this transformation, but is it possible using only the language of geometric algebra?

From reading Linear and Geometric Algebra by Alan Macdonald, I know that there are simple expressions for some classes of linear transformations like symmetric, orthogonal, and skew transformations. For example, symmetric transformations can be expressed as $T(u)=\sum{(u \cdot v_i)\lambda _i v_i}\;$ for normalized eigenvectors $v_i$ and eigenvalues $\lambda _i$ . It does not seem like these were found by a general method. Perhaps the best process is simply to classify the transformation and use the consequences of that classification to construct one (if possible).

According to this question, closed due to its ambiguity, this guess is correct. The accepted answer claims that geometric algebra does not have a standard representation for all linear transformations.

So, if the best process is to classify the transformation and use the consequences of that classification to construct one (if possible), I am still unsure how I should apply this to my example problem. Is there a complete way to classify the set of linear transformations defined from basis maps, and do all classifications have corresponding expressions in geometric algebra? Or, is this a misapplication of ideas, as the linked answer suggests?

Answer 1

It is not possible to represent all linear transformations on R^n within G^n. However the paper Lie groups as spin groups (http://bdml.stanford.edu/twiki/pub/Rise/GeomAlgebra/lie_groups.pdf) claims that it is possible with G^(n,n). For an explanation of G(n,n) see Chapter 10 of my book. I have not read the paper.

Two notes. 1. In your listing of transformations represented in GA you didn't include those from Chapter 7.

2. All of my representations are on a vector space. Your example is between two vector spaces.

Chapter 10, on the conformal model was added in later editions. It is available at the book's web site: http://www.faculty.luther.edu/~macdonal/laga/ .

Yes, you can consider your example to be on a vector space, but you did not phrase it that way.