I'm posting this as an "answer" since it seems rather long for a comment. Anyway, the commet SW didn't like it.

James, OP of this fine question:

I think, if you want to get "better"answers--by which I mean answers more precisely tailored to your individual level of mathematical development, I think it would help if you edited your question (since you can't make comments until you have 50 reputation points) so as to specify exactly what you mean by "basic". It sounds to me like you have been exposed to single-variable calculus and linear algebra through maybe determinants. To offer a few hints as to what I'm fishing for here, perhaps you could tell us if you have studied: a.) infinite series; b.) partial derivatives and multiple integrals; c.) eigevalues and eigenvectors; d.) characteristic polynomials of matrices; e.) the Hamilton-Cayley theorem; f.) vector calculus--gradient, divergence and curl; g.)linear ordinary differential equations. If you do that, I'll try to answer your question. (You can find my email address on my user profile in case I forget to check back.) Meanwhile, Qiaochu Yuan's answer looks fascinating to me, as does the problem fedja pitched.

@James, OP of this fine question:

I've edited this answer in light of your response. Thanks for getting back to us with the details of your mathematical education to this point. As you can see from one of my comments, I was a little concerned that you might have forgotten us! In any event, my follow up is presented in the paragraph after this next one, which I'm leaving in as part of my original answer to this question.

My original response was:

I think, if you want to get "better"answers--by which I mean answers more precisely tailored to your individual level of mathematical development, I think it would help if you edited your question (since you can't make comments until you have 50 reputation points) so as to specify exactly what you mean by "basic". It sounds to me like you have been exposed to single-variable calculus and linear algebra through maybe determinants. To offer a few hints as to what I'm fishing for here, perhaps you could tell us if you have studied: a.) infinite series; b.) partial derivatives and multiple integrals; c.) eigevalues and eigenvectors; d.) characteristic polynomials of matrices; e.) the Hamilton-Cayley theorem; f.) vector calculus--gradient, divergence and curl; g.)linear ordinary differential equations. If you do that, I'll try to answer your question. (You can find my email address on my user profile in case I forget to check back.) Meanwhile, Qiaochu Yuan's answer looks fascinating to me, as does the problem fedja pitched.

And my addenda are:

First of all, it sounds to me like you have encountered, or are about to encounter, almost everything I mentioned in your course work. Let's see, you've had a full year of calculus, if I understand you, and you are in the first half of your second year. So if your courses are anything like mine were, you have probably seen items (a.) and (b.) on my list--you are probably just getting into partial derivatives etc. right about now. I would guess you've scratched the surface of item (f.), and probably have been exposed to eigenvalues and eigenvectors (item (c.)), and perhaps the characteristic polynomial (item (d.)). I'd bet that items (e.) and (g.) are just up the road in your course work. That being said, I think there are a few really good books you could probably tackle without too much difficulty. First of all, you might check out the book Differetial Equations, Dynamical Systems and and Introduction to Chaos by Hirsh, Smale and Devaney. This is an introductory text on differential equations which includes some very nice explanations of some fairly advanced topics; it should be pretty accessible to a person with your background. If you are interested in abstract algebra, you might have a look at Emil Artin's little book called Galois Theory; it covers some central material on groups and fields, right from the ground up. Incidentally, Smale, Hirsh and Devaney explains most of the linear algebra needed as you go along, so anything you haven't seen will be covered. If you like topology, and are ready for a challenge, you might look into John Milnor's Topology from the Differentiale Viewpoint. Finally, Barrett O'Neill's Elementary Differential Geometry covers the basics of this field, and as I recall only requires knowledge of calculus at your level, plus some linear algebra. All these books are good introductions to topics of great interest to many mathematicians at the present time.

Don't forget to try the problems--math is like music; you've got to practice.

Good luck with it! Let us know how it goes!