TITLE: Is advanced college math (eg analysis, abstract/linear algebra, topology) supposed to be as intuitive as elementary math?
QUESTION [15 upvotes]: So I don't know if I'm not smart enough for math, but lately, it seems to me as if some advanced topics are just too unintuitive in my opinion. 
For example, I have no idea what eigenvalues, jacobians or manifolds really are, and it's a similar thing with most of abstract algebra (or at least I've never been told how mathematicians came up with these kinds of concepts -I don't even know what motivated the formulation of a matrix-).
And it's not just algebra. Even though I did pretty well (intuition-wise) in one-variable calculus, I have absolutely no idea why the solutions to differential equations make sense, or the notion behind differential operators.
So, I think one of these things is going on here:
a) I'm not smart enough to gain intuition of these concepts on my own;
b) It is complicated, but possible, to understand intuitively these concepts. Problem is this educational system doesn't put too much emphasis on deep comprehension;
c) Nobody knows very well these areas of math, we just apply rules and we're deeply boggled by how far we can go with them.
Which one of these is the case? I could really use some guidance.

REPLY [3 votes]: All of these topics can be understood intuitively, they just don't tend to be taught intuitively. I remember when starting linear algebra, I already knew what a vector is so it made sense to me, but I pity someone who was trying to understand what a vector is from the axioms of a vector space, which were what was being written onto the blackboard in front of us.
Certainly the topics you mention can be intuitively understood. Intuitively, a manifold is an object of lower dimensionality than the space it's in, so a piece of paper for instance is a 2d manifold in 3d space, or a string is a 1d manifold. Intuitively, if you picture a matrix as representing a transformation of space, and apply it to a unit circle/sphere, then you get an ellipse/ellipsoid. The eigenvectors are the principle axes of the ellipse/ellipsoid and the eigenvalues are the lengths of those axes. The Jacobian I know less about but basically it's the derivative, except of something involving lots of variables, so it has lots of values.
There's some cynical reasons why it's not explained intuitively including that unlike teachers in school, lecturers aren't usually there primarily to teach, and that it's a higher level course so expectations and demands are higher. But a nicer reason is that the power of maths comes from its ability to apply to many things beyond where it was first discovered. Any intuitive understanding of something is limited to a particular domain or application, and if all you're taught is that domain then you don't appreciate or understand the generality of it. Of course in practice, teaching something in a practical context first then generalising would probably be more effective.