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General Discussion / Re: Maths thread.
« on: January 20, 2019, 12:28:40 pm »The main reason I don't like the term is that it's confusing to beginners. It's usually the first time that something that is called multiplication doesn't behave like any other type of multiplication that they are used to. I know I at first was wondering why in the world it was defined in the way that it is. "Why not make it like matrix addition and use the component-wise product? That has all of the standard properties of multiplication." Of course we don't use the component-wise product because it's rarely useful, but back then it was radically different than any other kind of multiplication I had seen, and it seemed so arbitrary.
This kind of reminds me of tau vs. pi, or whether we should really call imaginary numbers "imaginary" numbers. It's probably too late to change things now, but it would make more sense, especially for beginners, if the terms were different.
This is all totally valid. Two things I would say:
1. Some terminology, while being confusing to beginners, is very intuitive and natural in a higher level context, or sometimes it's just so well established that you just have to deal with the fact that it's what everyone uses. In either case, though, we preumably want some of these beginners to become non-beginners some day, and we should think about the trade-offs between using counter-intuitive terminology in the introduction of a topic vs. making people re-learn proper terminology later on so that they can communicate with the rest of the community. (As an aside, this reminds me a little bit of learning how to ski. We teach young kids to make wedge-turns and then have to have them un-learn this and make proper turns later on).
2. I like your point about how "we don't use the component-wise product because it's rarely useful, but back then it was radically different than any other kind of multiplication I had seen, and it seemed so arbitrary." I think in a well-taught class there should be some serious time spent on getting students to understand why a component-wise product would be rarely useful, and why the proper definition is in fact not so arbitrary. The problem is that so many classes just introduce matrices and show how to multiply them with no context as to why we might want to do such a thing.
You can define anything you want, but the things worth studying in mathematics have a reason for the definition being what it is and not something else. I can define a binary operation on the set of all functions from the reals to the reals by
(f#g)(x) = f(x)g(x-2)+f(-2x)[g(x)]^2
Perfectly good binary operation. I could even prove theorems about it. But what's it good for? Not much as far as I know. Whereas the binary operation defined by
(f*g)(x) = int_{-infty}^infty f(y)g(x-y)dy (when convergent)
is good for lots of things.
Hmm; I'm not sure if I've ever posted in this thread. Anyway, Linear Algebra is hard to teach.
This is very true. It's also somewhat unique among introductory math courses in that there are two pretty radically different ways to approach the subject, which I generally think of as the "Begin with systems of linear equations" approach and the "Begin with the definition of an abstract vector space" approach. Courses titled "Linear Algebra" might be either one and they have a very different feel to them.
