I was pondering what it means to understand math… consider Linear Algebra, a relatively simple abstract theory with many applications.
I studied linear algebra and got a decent grade. I know how to use most of the (elementary) theory to solve problems. But I don’t feel I understand linear algebra.
For example take determinants. To me this is just a peculiar rule to assign a number to a matrix that has many useful uses. Of course this grants the determinant a special status among other inconsequential rules that could be devised to assign numbers to matrices. But I don’t know what a determinant is or why it is the way it is. A determinant to me is a weird thing. I can’t visualize it. And looking at the definition I haven’t got the faintest clue what permutations and whatnot has to do with anything.
So would a mathematician be, compared to me, just more proficient in memorizing and applying these definitions and theorems or would a determinant be a familiar entity to him? Would he see why a determinant HAS to be zero for a singular matrix? (Well, not just because it follows from the definition of course).
When I look at a real integral… even though I might have trouble calculating a complicated integral analytically, I have no problem visualizing what it represents. I “get” integrals and derivatives. It helps that I can sketch them in a graph. I don’t “get” determinants, eigenvectors, kernels, etc.
They are more abstract concepts, so that would presuppose more effort. So am I just unable to elevate my understanding to a higher level or would a mathematician share my feelings?
I think it’s mainly a matter of familiarity and having some knowledge of related concepts. There’s an old joke that once you get to advanced math, you stop understanding things and just learn what’s true, but I’m not convinced that there’s any truth to it.
For one thing, you can use determinants to find the solutions of a sysytem of equations in 2 unknowns by using Cramer’s Rule: http://www.pballew.net/cramer.html
As for permutations, among other things you will learn about probability. Study about lotteries and casino games and you’ll learn that they are not in business to make you rich.
wolf_meister for starters that is pretty weak. Obviously I know about Cramer’s Rule, having taken a course in linear algebra. You should take the time to read more carefully before jumping at the chance to plug your website. Your leap to lotteries and casinos also comes off as condescending.
I know what a permutation is but I’d be interested to know why it arises in the context of linear systems of equations. In other words, a motivation for the definition.
ultrafilter so would you say you share my feelings? I’d take some satisfaction in that! Is understanding determinants just knowing how to use them in a text book fashion? Do you feel it’s important to know the definition of a determinant? I think the arbitrary quality it has to me leads to a quick rusting of my linear algebra skills.
From what you’ve described of your class, it sounds more like a class in matrix theory than one in linear algebra. For the first, knowing the definition of the determinant is vital. For the second, it’s more important to know the properties of it.
In order to really understand determinants, you need to study ring homomorphisms and linear algebra. At that point, you’ll get why a singular matrix has determinant 0.
wolf_meister I apologize for my tone in the previous post. I probably gave you a wrong impression when I said “but I don’t know what a determinant is or why it is the way it is” and the thing about permutations. I shouldn’t have been so harsh on you. Sorry.
OK, that helps a lot. You’re saying I need to study advanced abstract algebra to get it. Makes sense. My knowledge of linear algebra is from the book “Elementary Linear Algebra” by Howard Anton. It doesn’t go as far as ring homomorphisms.
It means, to understand (patterns of) behaviour. In the real world, we have physical objects, which have a tangible form, and which also behave in a certain way. Things fall to the ground, things normally expand upon supplying heat (energy)…etc. Form is usually interleaved with function. Both evolved due to the other, with no clear forerunner, perhaps function. If you want to study behaviour in an abstract manner, you need to embody these behavioural agents as objects dor study. That’s what a determinant is. It acts in a certain way. But we need to provide it a form so that we can “see” it and recognise it from stuff that is not a determinant. It turns out that a lot of mathematical structures replicate or resemble physical phenomenas very well. So, instead of inductively predicting future behaviour, we do so by deducing, based on operations on mathematical structures.
I was very good at math in both high school and at university took perhaps nineteen math courses during my engineering degree – essentially many of the same courses a math major would have taken. I was a T.A. for university complex calculus and several other courses. I did tolerably well in national and international invitational math competitions.
Pure mathematicians, physicists, engineers – all of these people might use high level mathematics. Often, physicists and engineers are essentially interested in using math as a tool and want to use it to figure out a practicial problem. Knowing the rules is enough, you don’t always need to be able to “picture” abstract concepts. The more you use these concepts, and the more situations you use them in, the more you can “picture” them. Taking advanced courses helps you picture more and more of these concepts in a concrete way. I can (or could) picture many of the things that gave you difficulty. That said, there are a great many things I studied where I knew the rules, and could apply, but did not feel I had an “instinctive” grasp of. I suspect many mathematicians feel this way, but that if you use that specific area of math often enough you feel this way about fewer abstractions.
In sum, learning and using more math gives you a better picture of some abstractions, but some abstract things remain abstract to most mathematicians (although for a given mathematician, different things probably remain abstract).
Now your moving into the realm of applications, or maybe the ilustrious question “why does math work?”, which is not what I intended. Does the determinant resemble any physical phenomena?
Interesting post Dr_Paprika. I agree that sums it up nicely.
I would just argue, following ultrafilter’s contribution, that it’s very difficult to picture some concepts in elementary theory without a “higher level” grasp of the theory. If you look into the book I mentioned (a good book for its level in my opinion), the introduction to determinants lacks any motivation and the definition truly is arbitrary. It’s basically starting with this, you end with that, and this is good because it works for a number of things.
I see what your point is now. The deductive reasoning is the crux of the issue. The deducing starts from “the sum of all signed elementary products of a square matrix”, which comes out of the blue, to my unsatisfaction.
By the way I wasn’t limiting my difficulty in visualizing concepts to determinants so I think I also needed more effort to get a feel for linear algebra. If anyone has more opinions about understanding (meaning visualizing?) higher math as opposed to knowing how to use it I’d be interested in hearing it.
I’m not sure how I could explain eigenvectors and kernels any better than most books and classes already do, but determinants are near-universally badly explained.
The best motivation for determinants is volume. Think in R[sup]n[/sup], and consider the natural notion of n-volume. For n=1 it’s length. For n=2 it’s area, and so on. Now, if you take a S set of volume V(S) and apply a linear transformation T to it, the volume of the result V(T(S)) is det(T)*V(S). The really nice thing about this method is that I’ve defined the determinant without referring to any specific basis. It’s purely geometry.
Now, you go back and pick an orthonormal basis and work out an expression for the determinant of T in terms of the matrix for T by taking the unit cube and calculating from that. Then, this expression is generalized.
“Really”, this is a consequence of the “real” definition of determinants, but I’ve found that definition is best reserved for a second (or later) pass at the subject.
True. It helps a lot but it’s not always possible. But visualize could mean things became obvious or natural. That’s usually my goal (when I’m interested enough). When you “visualize” the logical progression is all there from the basics and you don’t forget things as easily, I think.
But yeah, the higher you go the harder it becomes I suppose until you just have to let that notion go.
Cool Mathochist. Very interesting as always. That’s a nice way to think of determinants. By the geometrical property they express of a linear transformations. Some things are clearer now… why the determinant would be invariant under two similar matrices… and if a transformation is not invertible (that is if the matrix is singular), I think V(T(S)) is always zero (the transformation “flattens” n-volumes).