Reddit mentions of Mathematical Physics (Chicago Lectures in Physics)

You may be interested in Mathematical Physics by Paul Geroch.

Wow, do you go to some school where mathochism is cool? This is not a junior-level course in my academic worldview. It was not too too long ago that linear algebra was almost exclusively a graduate course. It was pushed down to the undergraduate level because of its extreme usefulness in ODEs and DSP, among other things. Undergraduates did not get that much smarter, instead the curriculum for linear algebra just got that much more streamlined. Your prof is either ignorant of or doesn't care about that evolution. If this is supposed to be a "regular" class, then you might voice a complaint to the chairperson of the department. Junior level courses usually are the introduction to mathematical rigor, not the launchpad for the study of Lie Algebras or other specialized areas. However if you are in an honors class or a hardcore mathematics school, you'll just have to strap in and enjoy the ride.

So here's some rope. All my references are old because I am old(ish). However, you can probably do better with keyword searches in Wikipedia and WolframAlpha based on your lecture notes. Do something like a mind map of the connections. The only thing you are missing online will be problems.

Go to your library and get Linear Algebra and Its Applications. I learned from an earlier edition of the book, but I can't imagine it getting worse. The people who hate the book are the ones who didn't do the exercises. If you stick with it, it is very cool and things start to build and just make sense. Strang is an excellent, excellent expositor, but you have to be a big picture person. He also tells you exactly what the core of the book is, The Fundamental Theorem of Linear Algebra. Grok that and linear algebra is your oyster, e.g., Gram-Schmidt will seem like an obvious thing. (And wouldn't you know, a reference on that Wikipedia page is to a paper by Strang on just that(pdf).)

If you can put up with older notation, you will find a lot in the famous book by Halmos, Finite Dimensional Vector Spaces.

A lot of this carries through to graduate algebra and functional analysis, so find whatever texts your graduate courses require and check their indices. From the above it sounds like your prof is trying to hit all the connections to other areas.

This next book will probably not help you, but it is just crazy enough to make me think you may find some of your professor's thoughts hidden there, Mathematical Physics by Geroch. You don't have time to learn category theory, but his exposition ends up at the spectral theorem, I seem to recall. Seeing another presentation of those powerful theorems might be illuminating. (It's a beautiful book, but I've never heard of it being used in a class.)

If you don't have MATLAB, get a (free?/cheap?) student edition and play with it for "real" examples of what you are doing. Going through the Theorem-Proof process never worked for me with things like linear algebra: Seeing how you can pull things apart and put them back together is what makes the power of linear algebra come alive and gives you some motivation.

The last piece of advice is not a guarantee, but has always worked for me when in a draconian course: Drill yourself on your old tests and quizzes and homework. When everyone is failing and the final comes around, chances are good (for various reasons, including pity and laziness) that the earlier exams are almost exactly recapitulated. Use your prof's office hours to go over the subtleties of the exam problems. If you are engaged with the material, the chances are good that he will extend the scope of discussion and pull in examples from the current lectures. That's a very handy insight to have.

If the notes of your class do make it online, please think of linking it back here. I'm curious as to how deep this course is since it is pretty wide.