3 ms·
Geometric algebra and Clifford algebras are technically the same thing. The difference is that the term geometric algebra is generally used to refer to the work
by yiyus 6y ago
Geometric algebra and Clifford algebras are technically the same thing. The difference is that the term geometric algebra is generally used to refer to the work of David Hestenes, while Clifford algebra is used in more formal settings.
David Hestenes is himself a physicists. He did a lot to bring Clifford algebras into the main public attention (mainly applying it to the study of space-time). He used the term "geometric algebra" because it looks like this was the term used by Clifford in his work.
There is nothing wrong with teaching the cross product, some people (me included) just think that teaching about exterior (and geometric) products would be more productive.
- MiroF 6y agoMy take is that linear algebra should be taught before calculus and that einstein notation should be taught at the same time as matrix multiplication.
- reikonomusha 6y agoDo you also think the nuance of co- and comtravariant tensors should be understood, and the conventions of raising and lowering indexes, which is where a lot of value of Einstein notation comes from?
- MiroF 6y agoYes, much more preferable to 4 years of talking about "row vectors" and "column vectors." Perhaps not so early though. Introducing the concepts in the context of a euclidean metric seems unmotivated?
- ssivark 6y agoWhile not necessarily in a first course, it is an important difference to appreciate which saves a lot of headache later. Eg: Vanilla gradient descent involves naively adding a covariant and a contravariant quantity, which is generically a recipe for disaster, and nicely motivates second order methods that use the Hessian as the "metric" to raise/lower indices. The picture about Hessian eigenvalues determining the step-size is of course complementary and helps gain an intuition for the meaning of a "metric" and its importance.
- adamnemecek 6y agoMy take is that linear algebra dances around the lack of inverse. Math gets more interesting when you assume closure under inverse.
- jessaustin 6y agoUsing the term "geometric algebra" as Hestenes does is in keeping with the "don't name things after mathematicians" essay that was on HN yesterday: https://news.ycombinator.com/item?id=24385389 https://news.ycombinator.com/item?id=24385389