First let me say that I had a somewhat similar experience at one point. I had always done great in math, but at one point I was learning linear programming, in which there's a fundamental algorithm called the simplex method. And I didn't get it. Not even close. I had no idea what I was supposed to be doing nor how the calculation gave me a solution to a problem. Fortunately a good friend was also learning the subject and spent a lot of patient time taking me through it multiple times until I slowly but surely got there.

The point I want to make is: I know what it's like to suddenly experience, really for the first time, absolutely not getting something mathematical, after a lifetime-thus-far of always getting mathematics quickly. It freaked me out, it was unsettling and frightening. But today I am a professional mathematician, so clearly I managed to get through it! I couldn't perform the simplex method today, but I remember the gist of it and what it accomplishes, and I remember that it really is a mechanical procedure, so I'm confident that I could learn it again in a day or a week if I needed to.

So I hope you will have faith that in the future, you too will look back at how you feel now and say "Wow, that was really unsettling and scary, but I got through it and actually understood a reasonable amount of it, and the new math I'm encountering is making sense, so things are okay in the end."

As for how to maybe get from here to there: I would absolutely recommend coming at the subject seeking *understanding* rather than *memorization*. Of course those things can't be 100% separated and reinforce each other, but to me what you're describing is a desire for greater understanding more than a desire for better memorization. Yes, linear algebra and abstract algebra feel, and are, much different sorts of subject than calculus and analysis, and there absolutely is a precision to definitions of individual words that we have to come to grips with. But to me, the best way to come to grips with all these details is to see the overall picture.

For example, I'm not sure what you mean when you say that each theorem has 10 rules and exceptions. A theorem, by definition, doesn't have exceptions; it might have conditions under which a conclusion is valid ("if the matrix has nonzero determinant", for example), but the implication itself that is the content of a theorem always holds true. It is, however, true that the assumptions of linear algebra theorems do tend to be more exacting than the assumptions of calculus theorems: instead of things like "as long as the function is differentiable" (which makes sense for statements about derivatives and is true for most functions anyway, to the point where we forget it's a necessary assumption), we get things like "if a square matrix is nonsingular/is invertible/has nonzero determinant" (yep, equivalent ways of saying the same thing—and that fact is itself a theorem!), or "if the set of vectors is linearly independent" or "is a basis" or "spans the vector space", or "if the rank of the matrix is $r$"—and all of these assumptions are necessary and don't hold as generically as "differentiable".

So, is linear algebra fundamentally different? I would say, "yes, there are enough differences to notice a different feel", but also "no, linear algebra still consists of mechanical procedures, and ways to use them to solve certain kinds of problems, and statements that situation A always implies conclusion B, and leveraging old facts and relationships among mathematical objects to prove new facts and relationships".

As for concrete advice: I would suggest going back to the beginning (I know that can be hard when things are time-sensitive) and getting a more comfortable mastery with smaller pieces of the course, maybe one chapter or lecture-week at a time. Go over the material again in those smaller chunks, and ask yourself dispassionately which parts of it you get pretty well, which parts of it you feel sort of close to getting, and which parts you feel really unsettled by; then concentrate on the more confusing parts and really find a more solid understanding before trying to move on to the next chunk. When so much of a statement's meaning is tied to the exact meaning of the individual mathematical words, we don't want to build higher on a shaky foundation ... so okay, let's be willing to go back to the foundation itself. It's worth the investment of time and energy and should hopefully lead to a more positive experience in the later material.

After a quick search, I can suggest Chapter 1 of this book as a reasonable attempt to organize the main concepts of linear algebra, together with a lot of its terminology. It might be that you find this source (or some other resource) makes more sense to you.

But finally let me re-emphasize my first point: this might well be the first time you've ever felt defeated by mathematics. It's okay to feel that way!—not pleasant, sure, but it doesn't mean that you're incapable of getting through it. Feel the unpleasant feeling, take a deep breath, shrug and say oh well, I'll just be methodical about this particular subject. And just like everything else in mathematics (and life), the more we practice, the better we get; your challenge at this moment is to find ways of practicing linear algebra that work for you.