I'm studying mathematics, with some statistics also, and I've always chosen applied courses. I'm getting to the point where I'm studying 3rd year undergraduate to graduate level material.
My first problem is that I have the feeling that some of the time I'm just hacking, without really being sure of why I'm doing what I'm doing or whether it's even reasonable. For example with Taylor series expansions or limits I'm not really comfortable about convergence or discarding smaller terms.
Moreover a bunch of techniques seem to crop up over and over in different guises, decomposition of a function into basis functions being a case in point -- I only noticed this was a common theme the second or third time it came up however. I keep hearing terms like subspaces but don't know what they mean.
I'm starting to think that unless I want to just be a hacker there are some fundamentals from pure maths I ought to look at. Limits being one. Vector spaces, subspaces and bases being another. An example of something that wouldn't be relevant is number theory. Problem is I don't know what I don't know so it's hard to make a list of what I should know!
My question is, what areas should an applied mathematician know at the end of a typical undergraduate course in order to have solid foundations (or at least a good intuition) for using the applied maths they have learnt?