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.