I’m sorry is that in one semester? That’s all over the place, who designed this curriculum? Thats a completely new topic every other week. Horrible pedagogy.

Anyway for exam prep what I do is I collect all the stuff there is to know, write down headings, theorems etc. Exactly like you did there. That’s the battle plan. I’m ticking of topics as I go through them and so I can see what still needs to be done in what time.

Then sort the wheat from the chaff.

1. What are the basics. Basic definitions, like Whats a vector space, whats a linear function.
2. What was the main theorem? Every topic is usually built up like definitions -> examples -> lemmata -> main theorem -> corollaries.
3. What was the main example? Usually there is one function, or set or something that gets refined and used over and over to show the topic at hand.

Write it all out, then try to trim it down. Lots and lots of paper get written on but never read again, it’s the act of writing it, organizing it, rewriting it that helps me make connections. Try to get it all on an A4 page.

What you’re going for is the basics, skip what lemmata seem unimportant (you can always come back later) and go for the main topic whenever you understand the word that get used. Don’t get bogged down in an abstract representation, if you can reproduce the proof of the main theorem on the main example you’re done. That’s more than good enough. For example cyclical group = Z5 or Z6. You know what’s happening there, awesome golden. Yeah you have elements a_1,…,a_n but honestly who goes above n=6 ? Write whatever you got down and then move on. Cycle back on another day.

Fuck corollaries. No one ever asked me about the corollaries.

Also who gives a shit about inner product space, good lord no one ventures outside of euclidean anyway… Basically the inner product guarantees you can talk about angles. If you know what the inner product is on R^n (x_1y_1+x_2y_2+…+x_n*y_n) you can then test on that. Don’t actually learn proofs using positive-definiteness but look at how the proof works on the standard inner product. Once you got that the rest will come almost on its own. Same with abstract algebra really, but I wouldn’t worry about them too much. The basics are way more important, those things might be the difference between an A and a B+. Which kinda stings, but it’s a lot of effort for the last few percentage points, when the easier basic stuff has better pay-off.

Basically for learning there is the formally correct way and the actually usable way. An inner product space is when Rn with that multiplication, summation thingy. <x|x> is positive for all the x except for 0, thats why they call it positive definite (it’s all squares x_1²+x_2²+…) Only wrinkle for complex numbers is that you have that conjugate messing up symmetry. That’s really it tbh.

If you have more questions feel free to ask me. On matrix I’m @marxemathics:matrix.org Kinda hard to explain in a lemmy comment tbh and it’s a bit late.