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I am a college sophomore in the US with a major in mathematics and am an aspiring mathematician in the fields of computational complexity theory and cryptography. I would like to seek your advice and recommendations on comprehensive, insightful, and challenging books for abstract algebra and abstract linear algebra. I had looked through some books and pulled out some possible candidates:

  • Linear Algebra: Friedberg/Insel/Spence, Hoffman/Kunze, and Axler
  • Abstract Algebra: Herstein (Topics in Algebra), Artin, Dummit/Foote, and MacLane/Birkhoff (Algebra)

This will be my first introduction to abstract algebra and linear algebra. I would like to start with challenging, detailed books as I think my mathematical maturity and knowledge will grow much better than with easy, toned-down books.

As for my background, I studied single-variable calculus using Serge Lang (A First Course in Calculus) and acquired proof-writing skill using the proof-teaching book Chartrand's Mathematical Proofs.

I apologize for any grammatical errors, and look forward to your advice and recommendations!

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    $\begingroup$ I think it is good to challenge oneself. Instead of ignoring the easier, toned down books, you might consider spending a small amount of time with them. For example, take a book you think you have or can master, open to an appropriate page at random, and do the proof or exercise there in your head. If after an hour you find nothing there that challenges you, set aside that book and try the next easy book. There are more systematic ways to assess your knowledge level, but few that are as quick as this way. Gerhard "Reconsidering Randomly Reassesses Reader's Recall" Paseman, 2015.07.15 $\endgroup$ – Gerhard Paseman Jul 15 '15 at 21:57
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    $\begingroup$ Oh, and I would try the exercise with Schaum's Outline Series, both in Linear and Abstract Algebra. You might consider going chapter by chapter, as opposed to skipping some chapters. It's OK to spend a few hours with easy books if it prepares you for days of reading harder ones. Gerhard "Also Try Writing Your Own" Paseman, 2015.07.15 $\endgroup$ – Gerhard Paseman Jul 15 '15 at 22:00
  • $\begingroup$ Dear Mr. Paseman: Thank you very much for the advice! I selected Hoffman/Kunze and Michael Artin as a first introduction to the abstract algebra and abstract linear algebra. Do you mean that the easy books can be used as a supplement to help the understanding of challenging materials? Also how should I use Schaum's Outlines? Should I use them as supplements to my chosen books or should I read them before reading the relevant chapters of H/K and Artin? I thought that the mathematical maturity grows with struggling through the hard books. $\endgroup$ – MathWanderer Jul 15 '15 at 22:35
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    $\begingroup$ You are in a better position to answer your questions than I am. There are a variety of ways to use the easy books. Psychologically, I like to confirm what I know before I tackle a challenge. You may be different. My main point is that you can use the easy books to establish a route into the harder books: don't just ignore the easy books. Also, mathematical maturity grows through experience and effort. I bet Terry Tao was mathematically mature at a very young age, but even he had to read and think and plow through stuff. Gerhard "And Likely He Still Does" Paseman, 2015.07.15 $\endgroup$ – Gerhard Paseman Jul 15 '15 at 22:50
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    $\begingroup$ I would be cautious of tackling such endeavors without knowledge of proof strategies. I am unfamiliar with Lang's Calculus, but the majority calculus one course are almost entirely without proof. Knowledge of induction, Zorn's Lemma, and some logic are needed for rigorous courses in these fields. $\endgroup$ – Chris C Jul 16 '15 at 0:14
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I'm only familiar with the abstract algebra books you listed, not so much the ones on linear algebra. All of those abstract algebra books are awesome, but they are quite advanced and they might not be suitable for someone just setting foot into algebra for the first time. You might try one that's a little more aimed at the average undergraduate first, then keep those other ones around when you are ready for meatier fare.

I'd like to recommend Abstract Algebra: An Inquiry-Based Approach which was written by my colleagues Jon Hodge, Steve Schlicker, and Ted Sundstrom at Grand Valley State University. It's really written so that you are supposed to interact with the book through activities and exercises, and I think you'll get a good initial grounding in the subject this way. Also there are some sections on cryptography in it, so that will satisfy some of your end goals. I used this book to teach both first- and second-semester abstract algebra here at GVSU and both the students and I really liked it.

When you are ready for the deeper stuff don't forget Algebra by Thomas Hungerford. This was my graduate-school text in algebra and I still use it! It has a pretty decent treatment of abstract linear algebra, and I think most graduate level textbooks will as well.

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You might want to take a look at Sergei Treil's "Linear algebra done wrong" for linear algebra. It is sort of an answer to Sheldon Axler's "Linear Algebra done Right" (Springer, 3rd edition 2015).

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