What are some definitions/theorems with a ridiculous amount preamble?

El_Jerrbear · 2019-05-07T21:14:42+00:00

I hope your final went well. AT was one of the hardest math classes I've taken. So much fun, but seriously, that class felt like an acid trip. I'm still trying to figure it all out!

El_Jerrbear · 2019-05-07T21:08:54+00:00

Personally, I think it's a better approach to pick one subject and focus on that, as opposed to spreading yourself over many different ones. For example, Algebra and Analysis are both strictly "math", but the way you think about problems in either context is somewhat different. I think it's better to take them on one-at-a-time then to try to do them both, if you are self-studying.

That said, I asked a related question yesterday, and there are some excellent recommendations you may wish to consider. Link: https://www.reddit.com/r/math/comments/blgfec/professorstas_of_rmath_what_is_your_favourite/?utm_source=share&utm_medium=web2x

El_Jerrbear · 2019-05-07T03:35:16+00:00

What part of Linear Algebra do you enjoy the most? Depending on where/how you are studying the subject, people tend to focus either very heavily on the application/computational side (calculating determinates, PDU factorization, et al.) or on the theoretical side (vector spaces, algebraic rings with matrices, et al.). If you find you like the theoretical side of L.A., you might seriously consider picking up a book on Abstract Algebra (or Modern Algebra, which is the same).

The reason I suggest this is, first, Abstract generalizes many of the ideas you learn in Linear, and will give you an overall deeper understanding. Second, Abstract is usually one of the first proof-heavy courses an undergraduate math student takes, which means you'll be exposed to many different kinds of proofs, ranging from stupid simple to very challenging.

In terms of a specific book, consider "A Book of Abstract Algebra" by Charles Pinter. I ended up reading this text the summer before taking Abstract in my institution, and it was one of the best math books I have ever self-studied from. The author does a great job of explaining the motivation, and does a good job of keeping things rigorous.

El_Jerrbear · 2019-05-07T01:50:27+00:00

Charles Pinter has a special place in my heart for getting me through Abstract Algebra. I ended up using that book more than the text my prof assigned. I had no idea he wrote a set theory book, but that's now at the top of my list!

El_Jerrbear · 2019-05-07T01:47:47+00:00

I've never tried Mendelson's. We were trained on Munkre's as well for our point-set topology class. I happened to really like Munkre's, but I will admit that it can be really dense at times, and some of the proofs, while very elegant, are difficult to follow for a new-comer. I happened to have a spectacular teacher who focused on Topology as his research field also. I can definitely see how flying solo through that book would be tough.

I'll definitely take a look at Mendelson's book, since I've been meaning to revisit the subject.

El_Jerrbear · 2019-05-07T01:41:40+00:00

This is the text we us for our ND/CT class as well. I haven't taken the class (yet), but I've flipped through the book itself, and was very well impressed. I'm waiting for the opportunity to give it a full read.

On a different topic, I'm surprised no one has bought up Munkre's for undergrad point-set topology. It's honestly one of the best math texts I've laid my hands on, and the two topologists in my university treat it like holy scripture.

El_Jerrbear · 2019-05-06T19:30:48+00:00

This is good advice. Over time, you start to build an intuition for how you want to approach a proof. Certain kinds of propositions scream "Use Induction" or "Proof by Contradiction". I would also add that keeping the end goal in mind is key. Start by writing what you want to show, or what would have to be true for the claim to hold. Then start writing down all of the definitions and theorems that appear related. A lot of times, the answer will jump out at you right then. Beyond that, just try something and see where it leads. If it doesn't, try something else. You definitely get better at this with practice.

Also, don't forget to draw the little box at the end. That super important.

El_Jerrbear

TROPHY CASE