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> 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 won

> 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_ 2y ago
I think one of the coolest examples is probably classical mechanics. See the SIBGRAPI 2021 videos on https://bivector.net/doc.html https://bivector.net/doc.html
- howling 2y ago
  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.
  - hamish_todd 2y ago
    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?
  - srean 2y ago
    > 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

aap_ 2y ago

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

howling 2y ago
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.
- hamish_todd 2y ago
  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?
- srean 2y ago
  > 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