How to take maths notes, and generally to work with them, in order to understand as much as you can as best as you can?

Question

I have hit a bit of a roadblock in my undergraduate studies. Our lectures assume that we should be taking notes, and that is entirely understandable: after all, what else can you be doing during a lecture besides listening and note-taking? However, I have not the foggiest idea as to how to integrate notes into my studying process. I understand that if I take notes during class, I can engage much more with my material, and therefore give myself a head-start on reviewing it later on, (and I am not even talking about the case when you simply don’t know how to take notes matching the tempo of the lecture,) but consider this: my Linear Algebra course, for example, has a rather unorthodox professor teaching it, and the arrangement of the topics is rather different from how it is in the textbook, not to mention that it doesn’t cover a lot of the material that, frankly, would be really nice to know. If I take notes during that class, I will, in the best case scenario, be left with a set of notes that are rather scanty, and hard to elaborate upon if trying to combine processing those notes with processing the material of textbook as well. So what should I do? Should I just trust my lecturer (however hard it may be) with the comprehensiveness and structure of the course, and mostly study the notes, or should I spend god-awful stretches in order to create as god-awful a set of notes that is a blasted Frankenstein of my lecturer’s words and those of the textbook? How do you generally work with maths notes after class?