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First of all, my apologies if (well, I know I am but I don't know where to put it) I am posting this in the wrong place. So please feel free to move it to someplace else or to tag it differently if that is possible.

Anyways, I've been reading Lang's Linear Algebra and Lang's Undergraduate Algebra and I feel that they might not be the books I was looking for. They don't seem to be thorough enough and even though I have really been enjoying Lang's style I've started to look for other texts to use instead. I've narrowed it down to Hoffman & Kunze vs. Friedberg, Insel & Spence for Linear Algebra and for Abstract Algebra to Artin vs. Dummit & Foote, but I'm having troubles deciding which ones to go with. I've looked at reviews for all of them and they all seem to be great books and exactly what I'm looking for, but I don't know which ones to take as the main ones for reading. Can anyone who is familiar with these help me decide? What are the differences between each book in each set? These are both for self-study and abstraction isn't a real issue.

Thank-you.

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    It would help if you described what sort of qualities in a book you were looking for. – vadim123 May 03 '13 at 12:49
  • Essentially a book which is comprehensive, thorough, with well chosen examples and with a good set of (theoretical) problems. –  May 03 '13 at 12:53
  • I might add, I'm currently reading Rudin's Principles of Mathematical Analysis and really enjoying it. So I also like a book which can be challenging. –  May 03 '13 at 12:54
  • Since you seem to have some level of experience I would suggest taking both books and going through them to see which one fits better. Personally for abstract algebra I prefer Dummit & Foote especially if you want lots of examples. – Jernej May 03 '13 at 13:00
  • Thanks for the advice. I've decided that Dummit & Foote will suit me better. I don't really like how Artin's book is set up. I'll look through it still when I can, but Dummit & Foote will be the one I'll concentrate on. –  May 03 '13 at 19:29

2 Answers2

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Every good book will contain stuff another good book omits, and the choice is often made on one's personal preferences.

Artin is essentially an undergraduate text combining linear algebra and abstract algebra, with some graduate topics like representation theory and a great emphasis on matrices and geometry. If you want a not so conventional text that covers interesting topics like symmetry groups, go for Artin.

Dummit treats only abstract algebra, but covers all basic topics from undergraduate to beginning graduate level, while Hoffman and Friedberg treat only linear algebra.

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Even though you've decided, perhaps I can add my two cents (or Bitcoins). For algebra, I think Artin is superior in learning the principles of group theory. To that end here is a link to great video lectures by Benedict Gross (Harvard) that follows Artin:

http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra

For rings and fields Dummit & Foote is the best choice. (Probably because of two authors.)

For linear algebra, I prefer "Linear Algebra Done Right" (if you can get past the title). It is a rigorous approach and will greatly benefit your endeavors in algebra if you make some headway in it first. When you get to certain topics in algebra, you will be happy you have familiarity with concepts such as vector space, basis, null set (aka the kernel), etc.

When you get rolling, you will get a great deal out of these short pieces by Keith Conrad (Connecticut) on topics in these areas as you encounter them:

http://www.math.uconn.edu/~kconrad/blurbs/

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    Axler's LADL is as bad a preparation for advanced algebra as one can get, except for probably some matrix-only treatments catering to engineers. It's not like it's written badly (I don't think it is), but it is written from an ideologically anti-algebraic viewpoint. (E. g., the author's decision to work only over $\mathbb{R}$ and $\mathbb{C}$ because "polynomials can then be thought of as genuine functions instead of the more formal objects needed for polynomials with coefficients in finite fields" and similar reasons. Yes, polynomials are defined as certain functions in the text.) – darij grinberg May 04 '13 at 18:37
  • @darijgrinberg Following the link to you website, it is clear that you are substantially more well-versed in math than I. But for all it's flaws - especially eschewing the determinant - I do think one gets a very good, rigorous exposure to the concepts I mentioned. After all it is used at Harvard (including Math 55 in the past), MIT, and Berkeley. Where, e.g., MIT has a more engineering/practical offering by Strange (lectures and text). –  May 04 '13 at 18:47
  • @Andrew Thanks for the video lectures! I considered Axler, but I read it only worked over $\mathbb{R}$ and $\mathbb{C}$ and I wanted a text that worked over arbitrary fields. I think if I had a professor to offer these generalizations I'd be more willing to read it but I'm on my own. I did however come across the paper, on which his book seems to be based, titled "Down with Determinants!" which I plan to read. –  May 04 '13 at 19:35