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This post is not constructive, so maybe this rather should be posted on CW, but since there is a 'soft-question' tag, i'm posting it here.

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I believe the best way to memorize theorems is to draw. It was not hard to illustrate theorems by pictures when it is about analysis & topology.

However, i have no idea how to memorize theorems in algebra. Specifically, i'm studying linear algebra right now and it is hard to visualize theorems. For example, "If $V$ is a finite-dimensional vector space, $T:V\rightarrow V$ is linear, $W$ is a $T$-invariant subspace such that $V=\text{rng}(T)\oplus W$, then $W=\text{ker}(T)$" is a theorem in linear algebra. Well, it is easy to prove this, but it is not that easy to memorize to use this theorem whenever i need this. I cannot visualize this by drawing a big circle named $V$ and two small circles in this big circle, namely $\text{rng}(T)$ and $W$. (Because this diagram tells nothing about their direct sum is $V$. Plus, since i cannot draw a picture illustrating this, i don't understand why it should be.

What would be a good idea to memorize theorems related to algebra?

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Try to understand why the theorem works, try to understand why you proved it, what applications it has. For this example, you can decompose every linear map into a part where $T(U)$ stays in $U$ and a part where $T(W) = 0$. So now you can restict yourself to calculate the values of $T(b_i)$ where $B\{b_i\}_{i \in I}$, because everything else will map to 0. Often times if you know what the applications for a certain theorem are, it's easier to remember what it was good for. –  Stefan Feb 22 '13 at 3:20
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Why would you ever need to memorize that theorem? –  Qiaochu Yuan Feb 22 '13 at 3:22
    
@Qiachu Yuan That is just an example. If i don't deeply understand one subject as a learner, then how would i know one is relatively more important and less important –  Jj- Feb 22 '13 at 3:28
    
If you need to memorize anything, you are doing it wrong –  Alex Feb 22 '13 at 3:59
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1 Answer

up vote 11 down vote accepted

My two cents: during all my university studies and after that, the best method I've ever had to study and memorize is to teach an imaginary class what I'm trying to grasp...over and over until I could actually teach that piece of stuff to a real class.

This imaginary class poses tough questions, asks for examples and counter-examples, analyzes each tiny aspect of what's been taught, and you as the teacher must be able to address and give satisfactory answers and insights.

Big secret: keep by your side several books on the subject (tablets now are incredibly handy for this), which you'll consult constantly until you're able to give an AAA lecture to your imaginary class on the subject being taught.

Last one big secret: be tough on yourself and demand from yourself excellency while doing this "teaching".

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Second best way is to get a bunch of classmates to teach ;-) –  vonbrand Feb 22 '13 at 3:27
    
I'm afraid you'll have to pay them and a lot...at least that's what I would have done to suffer a student friend trying to study by giving me class...! –  DonAntonio Feb 22 '13 at 3:40
    
@DonAntonio: this is a very good method, but the point of mathematics is that you do not need to memorize anything at all. IF you do, you are doing it wrong. –  Alex Feb 22 '13 at 3:59
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@Alex, no need to memorize "anything at all"?? I think that in any learning we need to memorize, and quite a bit indeed. How can you do integrals if you didn't memorize "anything" and you can't remember what is the derivative of something? How can you do equation if you didn't memorize the multiplication tables...? You need memory, and a whole lot. What you do NOT need is to have repetitive memory (like , say, recite all the english kings, when they were born and etc.). –  DonAntonio Feb 22 '13 at 4:03
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Even then, @Alex: you do need memory, and a lot, to apply the things you've studied, no matter how good you understood them. You need to remember stuff you learned in high school to apply it in university (algebra, geometry, trigonometry, etc.), then what you learned in undergrad. to apply it in grad. and etc. –  DonAntonio Feb 22 '13 at 4:32
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