Messing about with actual matrices never gave me the slightest grasp of linear algebra, and the fourier series formulae seemed completly pulled out of thin air, but as soon as I saw the expression of those concepts using abstract linear operators on general vector spaces, all the results and methods seemed obvious. I still feel really pleased when something that's true in my geometrical picture actually works when you stick numbers into matlab.

On the other hand, I first ran into group theory abstractly presented, and it meant nothing to me. I needed to play with lots of examples before I even cared about it, and before I came upon the cycle representation it was all just completely opaque.

They two seemed to be similar in content, introductory first-year maths, similarly presented, and both lecturers were clear and gave beautiful notes, and yet they spoke to me in very different ways. I'm still very happy with linear algebra and rather mystified by groups.

I think in my case the difference is that linear algebra is intrinsically geometrical, and I'm much better at visualizing pictures than at manipulating symbols, but given that one use of groups is to talk about physical symmetry, whereas linear algebra is all about vast tables of numbers, maybe that should be the other way round.

Anyone get the reverse feeling?