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I am currently taking a linear algebra course and I have an issue I have never ever experienced before. I understand nothing. Nada. I have never been a genius, but I learn fast. I have never struggled with mathematics before and I just scored near perfect on the mathematical analysis test last week. However next week we have a linear algebra test and my professor has posted the old linear algebra tests, along with solutions, for us students to test ourselves. I got a clean 9 points out of 70 points total.

There are 800+ pages in our linear algebra book and we've been given 10 weeks to study, along with the professor doing classes on Zoom. My issue is with the fact that there are over 150+ theorems and each theorem have an average of 10 rules + exceptions. This means that in 10 weeks not only do we have to memorize about 150*(10 rules + exceptions) which is approximately 1500+ unique things but we also have to understand them to actually get a grade I hope one day achieve.

So my three questions are:

(1) Why does everything have multiple names?

(2) What might be a good way to not confuse these names together?

(3) What makes linear algebra so fundamentally different from other mathematical courses and where do I go to learn these new attributes?

In the book they talk about unit vectors, but the question is about standard basis vectors and the professor is talking about the natural basis (and they mix around unit, basis and natural all the time). I always have to have a table with synonyms or words that at times is something that I already know but is presented in such a way that I get thrown off. This just makes things more tedious because then I learn something only to later learn that one word only means "basis vector" if such and such criteria are met, otherwise they refer to this and that.

I can solve most question in the book easily, but I don't understand why I calculate, I only go after the book and my professors examples. I have gone through things over and over and I just keep getting worse at linear algebra.

J. W. Tanner
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linker
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    The classic book on linear algebra is Halmos' "Finite dimensional vector spaces". It has $<200$ pages. Linear algebra is a mandatory mathematical field, but in terms of difficulty not different from other elementary mathematical fields. – Christian Blatter Dec 27 '20 at 20:02
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    Sounds like the course is way too fast paced (like my undergrad lol). I would focus on trying to find a text you like. There are a lot of questions on this site about good linear algebra texts, so I'd pick a good one and follow it. – Rushabh Mehta Dec 27 '20 at 20:02
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    Well, yes, 'standard basis' is synonym for 'natural basis' but 'unit vector' means slightly different. – Berci Dec 27 '20 at 20:04
  • It's not really different from other fields of mathematics, just more boring. You should concentrate on concepts (ideas), not names, name magic is Neolithic/Bronze Age (or still older), though there is some revival since the invention of search engines. –  Dec 27 '20 at 20:07
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    Nobody memorizes $1500$ things in order to do well in a math class. You memorize a small number of extremely important things and you find a way to understand how the other $1480$ things follow from the extremely important things. – Gerry Myerson Dec 28 '20 at 04:27

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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.

Greg Martin
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  • Extremely helpful answer! I am glad that there is someone who can relate, so that I know I am not the only one who has struggled with this... even though that seems rather implausible when writing it out like this. I do not find linear algebra intuitive but I will take your advice on going back to the beginning and working it through thoroughly so that I hopefully can understand it all. What I meant by "theorems having 10 rules" are things such as "You can multiply matrices together!" (obviously more formally) then there are rules as to where the parenthesis, dimensions and zero matrix fit – linker Dec 28 '20 at 13:16
  • Sure—for that example, I would say that the big picture is "You can multiply matrices of compatible dimensions together!" (and "compatible dimensions" makes total sense once we master the actual mechanics of matrix multiplication) and then the obvious big picture question is "How much does that multiplication act like real-number multiplication?", which has answers like "It is associative" (meaning $(AB)C = A(BC)$), "It's not commutative" (meaning $AB\ne BA$ in general), and "Many but not all matrices have inverses" (the generalization of reciprocals). – Greg Martin Dec 28 '20 at 23:35
  • (continued) Things like "if one of the matrices is a zero matrix, then so is the product", I would consider them examples of matrix multiplications (useful enough to remember, but you could work them out every time if needed), not "extra rules". And characterizing exactly which matrices have inverses leads to other collections of facts involving square matrices and their determinants, which is a great example of mastering one level of topics and then having them to attach the next topics to. – Greg Martin Dec 28 '20 at 23:38