Reddit reviews Linear Algebra: An Introduction to Abstract Mathematics (Undergraduate Texts in Mathematics)

Intro Calculus, in American sense, could as well be renamed "Physics 101" or some such since it's not a very mathematical course. Since Intro Calculus won't teach you how to think you're gonna need a book like How to Solve Word Problems in Calculus by Eugene Don and Benay Don pretty soon.

Aside from that, try these:

Excursions In Calculus by Robert Young.

Calculus:A Liberal Art by William McGowen Priestley.

Calculus for the Ambitious by T. W. KORNER.

Calculus: Concepts and Methods by Ken Binmore and Joan Davies

You can also start with "Calculus proper" = Analysis. The Bible of not-quite-analysis is:

[Calculus by Michael Spivak] (http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918/ref=sr_1_1?s=books&ie=UTF8&qid=1413311074&sr=1-1&keywords=spivak+calculus).

Also, Analysis is all about inequalities as opposed to Algebra(identities), so you want to be familiar with them:

Introduction to Inequalities by Edwin F. Beckenbach, R. Bellman.

Analytic Inequalities by Nicholas D. Kazarinoff.

As for Linear Algebra, this subject is all over the place. There is about a million books of all levels written every year on this subject, many of which is trash.

My plan would go like this:

1. Learn the geometry of LA and how to prove things in LA:

Linear Algebra Through Geometry by Thomas Banchoff and John Wermer.

Linear Algebra, Third Edition: Algorithms, Applications, and Techniques
by Richard Bronson and Gabriel B. Costa.

2. Getting a bit more sophisticated:

Linear Algebra Done Right by Sheldon Axler.

Linear Algebra: An Introduction to Abstract Mathematics by Robert J. Valenza.

Linear Algebra Done Wrong by Sergei Treil.

3. Turn into the LinAl's 1% :)

Advanced Linear Algebra by Steven Roman.

Good Luck.

A First Course in Graph Theory by Chartrand and Zhang

Combinatorics: A Guided Tour by Mazur

Discrete Math by Epp

For Linear Algebra I like these below:

Lecture Notes by Tao

Linear Algebra: An Introduction to Abstract Mathematics by Robert Valenza

Linear Algebra Done Right by Axler

Linear Algebra by Friedberg, Insel and Spence

If you are still an undergrad and your school offers a "how to prove stuff and how to think about abstract maths" course take it anyway. No matter how far along you have come.

An example text for such a course is this one:

https://www.amazon.com/Introduction-Mathematical-Reasoning-Numbers-Functions/dp/0521597188

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As for Linear Algebra (the most useful part of all higher mathematics for sure (R/math: if you disagree, fight me on this one...i'll win) ) I will tell you i learned a LOT from these two texts:

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https://www.amazon.com/Linear-Algebra-Introduction-Mathematics-Undergraduate/dp/0387940995

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https://www.amazon.com/Linear-Algebra-Right-Undergraduate-Mathematics/dp/3319110799/ref=pd_lpo_sbs_14_img_0?_encoding=UTF8&psc=1&refRID=APH3PQE76V9YXKWWGCR9

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There is one more suggestion I can offer, which is at the level of learning habits and psychology: https://www.amazon.com/Mind-Numbers-Science-Flunked-Algebra/dp/039916524X

It's written for a much more popular audience than the earlier suggestions, but I still found it helpful.

The instinct for a lot of people is that when they get stuck, they think that the way forward is to isolate oneself to that problem and batter themselves at it until they solve it. The author does a good job explaining why this is almost always the wrong approach, and offers some psych-ish suggestions on better approaches. For example, she describes the difference between "diffuse" vs "focused" thinking, and how important it is to learn to switch between the two modes, so you don't get stuck performing focused thinking in the wrong area. Or how memory needs to be allowed to "chunk" so that it can form larger mental maps.

Good luck!

Edit to say, as for where homomorphisms are used, one cool application is that linear transformations (vector space homomorphisms) are a close analog of group homomorphisms. Having taken your group theory class, you may find something like this interesting? https://www.amazon.com/Linear-Algebra-Introduction-Mathematics-Undergraduate/dp/0387940995

I'm recently going through "Linear Algebra" by Robert Valenza.

It is very compact and elegant. I find it excellent for an abstract introduction to Linear Algebra.

I'd love to hear your opinion on it, if you're familiar with it.