How can I improve my understanding of algebra?

[0rthodontist]

I am taking modern algebra and I'm not having an easy time. The problem, I think, is the symbolic manipulations. I like discrete/graph theory/algorithms/computing type math because the arguments tend to be high level, conceptual, and mostly in English. However, when a proof depends on a bunch of symbolic twiddling, I can verify every step and finish reading it with zero understanding. It doesn't sink in. Maybe it is a problem of visualizing; I can think, OK, a coset, but I don't have a clear way of thinking about a coset other than a couple of symbols on paper. The only actual mental image I have for it is the same mental image I have for partitions, which leaves out a lot of information. How do you think about proofs that depend on symbolic manipulation? Should I just start memorizing?

I can work my way through most of the homework problems, but it takes me a lot of time and even though they seemed challenging and interesting, I forget a great deal of them later-unlike graph theory homework, roughly equally challenging, where I remember just about everything. In the lectures particularly, I am usually a minute or two behind the professor and absorb little.

 

[ircdan]

I can work my way through most of the homework problems, but it takes me a lot of time and even though they seemed challenging and interesting, I forget a great deal of them later-unlike graph theory homework, roughly equally challenging, where I remember just about everything. In the lectures particularly, I am usually a minute or two behind the professor and absorb little.

I'm learning algebra now so maybe this will help. First let me say I think everyone has a hard time.

This is what works for me when studying for say an exam. I prove every single theorem in the text(for whatever chapters we covered) without looking at the proofs in the book. I also do "most" of the homework, I rarely have time to do all of it.

Anyways after I write a proof, I step back and look at it, carefully, and analyze it step by step. I try to see why I didn't see the solution right away. Why did I make so many wrong turns when the proof was so obvious and so seemingly natural? When you ask these questions, when you think about these things, you can start to get a better feel for what can work in the future. Of course when proving anything, you also need to use everything you know about whatever is being asked. For example if H is cyclic then right away we can say lots of things about H.

If you analyze the proofs carefully and learn how they are constructed you just start to "get it". So remembering how to do the homework problems is not necessary, because you want to get to the point where you can get a problem, and you can just rethink it except this time, you have a better "sense" for what will work, so the rethinking process won't take nearly as long. In fact, with lots of the proofs, it's the exact same ideas that are used over and over again, so the "rethinking" part will sometimes be just a matter of recognizing what will work, and in general, using everything you have learned up to that point to tackle the problem.

I think several of the proofs in algebra are for lack of a better word, beautiful. Just keep at it, keep proving things, look at the proofs carefully, and you will get it. Remember it takes a lot of work, it's not easy for anyone(or for most people heh). Goodluck.

 

[0rthodontist]

Yes, those are generally good ideas for any kind of proof. I have had many proof based courses. However, my memory of the homework problems definitely reflects my understanding of a subject. If you remember something after thinking about it, chances are you understand it. If you don't remember your reasoning even after thinking something through, chances are it didn't sink in. Of course the test problems will require you to prove something you haven't proved before, but memory of problems is still a useful measure. There was an article in Scientific American a while ago about how experts develop a specialized memory for their subject--I believe that's true.

My problem really is specifically with the more symbolic nature of algebra as opposed to the conceptual nature of the other proof-based courses I have had. A question for anyone who understands a fair amount of algebra, how do you think about it? Do you just crunch symbols or do you have a visualization or other internal representation of the ideas?

 

[Hurkyl]

It's not necessarily a bad thing not to attach a concrete meaning to symbols -- groups are an abstract thing, and it is useful to let context fill in the meaning.

(e.g. think about vector spaces: you first learned about things like Rⁿ, but then you learned about abstract vector spaces, and then could suddenly start proving things about polynomials and differential equations)


That said, there are at least two big conceptual things that you should be developing:
(1) You should be building up the mental concept of "arithmetic in a group". Many interesting things can be collected into a group. While each individual example will have its own pecularities, there are general overarching themes common to many or all of them... the abstract setting makes it easier to see and study those themes.

(2) You should be building up the mental concept of "arithmetic with groups". You can do arithmetic on the groups themselves. At the very least, you'll learn how to multiply and divide groups. The arithmetic of groups is not as nice as the arithmetic of, say, the integers. (although the arithmetic of finite abelian groups comes very close)



Quotient groups are like division. It's obvious for product groups

$$(G \times K) / K \cong G$$

but the same idea holds for any normal subgroup of any group: modding out by K "divides out" the K-component of your group.

Another way of saying it is that, G / K is the "best" way to start with G, and then force everything in K to be identity.


It doesn't really matter precisely what G / K "is" -- just that it has the above properties. In fact, when interested in the arithmetic of G / K, it is not uncommon to really do all your arithmetic in G, but remember that $g \equiv h \pmod K$ iff $gh^-1 \in K$.

The cosets do form a partition -- presumably you know how partitions relate to equivalence relations? Does that make it more clear what the point of a coset is?

 

[0rthodontist]

It's not necessarily a bad thing not to attach a concrete meaning to symbols -- groups are an abstract thing, and it is useful to let context fill in the meaning.

(e.g. think about vector spaces: you first learned about things like Rⁿ, but then you learned about abstract vector spaces, and then could suddenly start proving things about polynomials and differential equations)


That said, there are at least two big conceptual things that you should be developing:
(1) You should be building up the mental concept of "arithmetic in a group". Many interesting things can be collected into a group. While each individual example will have its own pecularities, there are general overarching themes common to many or all of them... the abstract setting makes it easier to see and study those themes.

(2) You should be building up the mental concept of "arithmetic with groups". You can do arithmetic on the groups themselves. At the very least, you'll learn how to multiply and divide groups. The arithmetic of groups is not as nice as the arithmetic of, say, the integers. (although the arithmetic of finite abelian groups comes very close)



Quotient groups are like division. It's obvious for product groups

$$(G \times K) / K \cong G$$

but the same idea holds for any normal subgroup of any group: modding out by K "divides out" the K-component of your group.

Another way of saying it is that, G / K is the "best" way to start with G, and then force everything in K to be identity.


It doesn't really matter precisely what G / K "is" -- just that it has the above properties. In fact, when interested in the arithmetic of G / K, it is not uncommon to really do all your arithmetic in G, but remember that $g \equiv h \pmod K$ iff $gh^-1 \in K$.

You're saying it's just about getting used to manipulations of symbols (arithmetic)? You aren't thinking about "something else" when you write the symbols?

The cosets do form a partition -- presumably you know how partitions relate to equivalence relations? Does that make it more clear what the point of a coset is?

I know that, my problem is that thinking of them as _just_ equivalence classes does not include all the information about them. I mean, I can think of a vertex and an edge, and my mental picture pretty much captures all the information there is about a vertex and an edge. Then I can reason with that image and turn it into a proof of something, or use it to picture and understand something I'm reading. I don't have any image like that that captures all the relevant information for most things I'm trying to learn in algebra.

 

[JasonRox]

For me, I just see it as a game with rules. Play the game while following the rules.

I personally can't really explain how I do my stuff. For some reason I can see the consequences of the "rules". Like when I see a theorem, I try to interpret it in the rules that I know. Then ask myself, does it make sense? What else does this theorem imply? Does that make sense? And so on...

 

[symbolipoint]

How? Repeated study and repeated practice. It should finally work like a language. The study and practice must be done every day.

Always be willing to restudy something which you already studied and earned credit. If you use the attitude that since you already "took that course and don't need to learn it again", then you are probably cutting down what more you might learn well.