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My take is that linear algebra should be taught before calculus and that einstein notation should be taught at the same time as matrix multiplication.
by MiroF 6y ago
My 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.