Linear Algebra and Real Analysis Review

[Kindayr]

I'm going into my 3rd year as a pure math major at UWO. I have completed both second year Real Analysis and Linear Algebra with decent marks. However, I really feel that I didn't take too much from both other than the general concepts, especially second semester of Linear Algebra (due to laziness mostly).

We used:

Stephen R Lay's "Analyis: with an introduction to proof"

Friedberg, Spence, and Insel "Linear Algebra"

I was wondering which textbooks I could pick up from the library or amazon that would do a better effort, whether it be more rigour or just more information, at teaching me the more specific concepts.

I'll be taking a full course load of math this coming year, including analysis, topology, and algebra oriented courses. So just hoping to expand my grasp on the prerequisite topics.

I'm already interested in picking up Spivak's "Calculus" and Artin's "Algebra". Are these the best, or are there better options I could look into?

 

[micromass]

Firstly, you won't need THAT much linear algebra in the topics that are about to come, even not in abstract algebra. Real analysis is more serious though.

First, linear algebra. Friedberg is an excellent book, in my opinion. If you know Friedberg, then you basically know enough linear algebra to continue. A more rigorous book is of course Hoffman & Kunze, which is my favorite book on linear algebra. It does the results in complete generality. For the fine details, this book is excellent!

For real analysis, you might want to pick up Rudins "Principles of mathematical analysis". I don't like this book for a first course. But if you want more details and a decent review of the topic, then this book is excellent.
You also might want to pick up Yeh's real analysis book. It's also quite nice.

Artin's "abstract algebra" is a very good book, certainly take it!

I don't know why you want Spivak's calculus. It'll only teach you calculus, and you (most likely) know this topic already. It's great for reviewing calculus, but it won't teach you real analysis...

 

[Kindayr]

Thanks for the advice!

I've heard Hoffman & Kunze is a good book as well! I'm glad you mentioned this because it is precisely the textbook that a prof of mine recommended to me last summer to buy and use concurrently in his Linear Algebra course. I had forgotten the authors until now, heh. (The person running the Linear Algebra curriculum prefers Friedberg, that's why it was the course material as opposed to Hoffman & Kunze, to which the professor teaching me preferred.)

Another professor mentioned Spivak during a summer lecture series that introduced us casually to Galois theory. Not that Spivak had anything with the course material, was just brought up in conversation. I have also heard of it all over these forums. That's basically my reasoning to pick it up and check it out. I really don't like Stewart's "Calculus" which is what my school uses, as I feel it lacks a lot of rigour.

I will definitely look into the other texts you have mentioned! A few are in the campus library, so when I go to pay my tuition cheque I'll certainly pick some of them up.

Thank you!