[howling]

> Can you formulate a lot of stuff much more efficiently and concisely, and does it help gain new perspective on some things? yes, absolutely. It provides a wonderful framework for expressing geometric ideas.

Can you elaborate on what stuff does it help to formulate much more efficiently and concisely?

[aap_]

I think one of the coolest examples is probably classical mechanics. See the SIBGRAPI 2021 videos on https://bivector.net/doc.html

[howling]

All of these stuff can be done in normal linear algebra. Some (not all) of the operations can be done more efficiently with GA in low dimensions. It is neither more concise nor more intuitive to understand than normal linear algebra.

[srean]

> All of these stuff can be done in normal linear algebra

On its own that is not a very strong argument. What you can do in linear algebra can be done by scalar add multiply and divide. That additions can be done with logical gates does not mean that programming an accounting application with logical gates as primitives is a good idea.

> It is neither more concise nor more intuitive to understand than normal linear algebra.

The real contention is this one. I have met people who hold opposite views on this

[hamish_todd]

There are ten thousand examples I want to give of why you're wrong. We have to start somewhere so here's a favourite, the "universal projection formula":

(A.B)/B

Projects any A onto any B, in any number of dimensions and with any signature (eg hyperbolic/Euclidean/elliptic). A and B can be lines, planes, points, and with a conformal or anti de Sitter metric a sphere or hyperboloid etc ("blades").

It works because A.B is dimension independently the object "orthogonal to A and containing B or vice versa". And division by B will intersect that orthogonal object with B.

Concise, intuitive, and powerful. What's the linear algebra formula you'd consider to be comparable?