<p>i see a potential difference between where the top faculty is in terms of publications compared to where the top grad students go. looking at the faculty at berkeley, there are a ton of harvard and mit phds, which seems to say that those profs were among some of the top grad students (since berkeley is itself a top place). anyway just interesting to hear the divergent pov. i am going to ask around at berkeley what the top schools are but i guess at this point it may be premature of me to think about</p>
<hr>
<p>basic linear algebra is pretty simple (adding matrices, etc.). But linear algebra is where math starts to become more abstract, using vector spaces, etc. I’ve never tried to teach it to myself except in the sense that we all have to teach ourselves through homework, reading examples in the book when taking a class. I’ve tried to read math books but it is hard to make myself do problems without having it assigned. but that’s probably what you’d have to do, try problems and have a book that has the answer in the back. some people like to test themselves so don’t want answers in the back, but I think that’s unwise as you wouldn’t know if you are doing them wrong!</p>
<p>The multi var calc that I took was very computation focused. As soon as you get the hang of it, they are pretty simple (double and triple integrals, etc.). Also review polar coordinates since you can use that sometimes to solve integrals. That was something in precalc that I had trouble with and had to relearn because I thought it was stupid and would never come up again.</p>
<p>You could also try sitting in on some classes at some point. I’ve done that before and it can be very beneficial, albeit with a high time cost. </p>
<p>higher levels of math are mostly about proving stuff instead of computations (real analysis is pretty much all about writing proofs of theorems and other statements or finding counter examples that disprove them). You can probably do ok in real analysis without knowing the computations as much. I think it is two different skills. trying to reproduce the proofs in the book for propositions is a good way to start I think (keeping in mind that even if your proof is different it still may be correct). </p>
<p>The proofs in books tend to try to prove the statement in as few words as possible. but often it is clear to me that the proof has been reverse engineered. so sometimes that can be troublesome in trying to figure out how they came up with that.</p>