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First of all, I know that there are many posts on this topic, but my question, as you will see, has important differences.

I am in the last year of my undergraduate studies, and I have been taught introductory courses of Group Theory and Ring Theory. Now, I want to attend Galois Theory, but I see that I have no good knowledge of Linear Algebra. So, I thought that this is a good chance to refresh and complete my knowledge. And I look for a good book with many exercises to solve.

In other words, I'm looking for a good Linear Algebra book, which will have all the necessary chapters with exercises and will be useful for Galois Theory, Algebraic Geometry, Representation Theory, Coding Theory and other more advanced algebra courses.

What are your recommendations?

PS: I didn't like Gilbert Strang's book "Linear Algebra and Its Applications".

PPS: What's your opinion for the books: 1) Linear Algebra, Serge Lang, 2) Introduction to Linear Algebra, Serge Lang and more generally your opinion about Lang's books, 3) Linear Algebra: Step by Step, Kuldeep Singh?

PPS: I apologize for my English!

Thank you in advance!

J W
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Chris
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    Rather than an entire book, consider Chapter 4 Vectors Spaces and Modules (about 35 pages) in Herstein's Topics in Algebra. Chapter 4 is designed to fill in linear algebra gaps before Galois theory, which is in Herstein's Chapter 5. In the same book, Chapter 6 is a fairly thorough treatment of linear functions (representation and decomposition theorems). – Dave L. Renfro Jun 27 '17 at 13:46
  • Thank you for your comment. I will check it! And I like Herstein's book. But, If i want to do coding theory which needs matrixes etc is this enough? – Chris Jun 27 '17 at 13:56
  • Herstein is more abstract, so it's not the place I'd recommend for someone wanting a lot of work with matrices, but since you will be able to look at the book, this is something you can determine. I do know there are several abstract algebra texts published in the last 25 years or so that include a chapter on coding theory. In fact, I have one at home, a unsolicited publisher's sample copy that was sent to me about 12 or 13 years ago (because I had taught an abstract algebra course at a certain university a couple of times in the early 2000s), but unfortunately I don't remember its title. – Dave L. Renfro Jun 27 '17 at 14:11
  • By the way, Sheldon Axler's book that @Aditya Kumar mentioned is also well worth considering. Of course, as I see in the comments there, it has similar concerns with respect to matrices. Personally, if you're concerned about matrices, I'd recommend getting something like Ayres's Schaum's Outline ... Matrices (freely available) or Browne's Introduction to the Theory of Determinants and Matrices for reference purposes. – Dave L. Renfro Jun 27 '17 at 14:39
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    Lang's Introduction to Linear Algebra and Linear Algebra are both good. The first book consists of selected chapters from the second book, plus two introductory chapters on vectors and matrices. If you include information about which languages you are able to read, you may get better advice. There are good books that have been written in other languages. Good books that are in English or have English translations are Lectures on Linear Algebra by Gelfand, Linear Algebra and Geometry by Kostrikin and Manin, and Finite-Dimensional Vector Spaces by Halmos. – user49640 Jun 27 '17 at 20:02
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    See the recommendations here too. https://www.ocf.berkeley.edu/~abhishek/chicmath.htm#i:linear-algebra – user49640 Jun 27 '17 at 20:06
  • @user49640 Thank you for your comment! I will check these books! – Chris Jun 29 '17 at 16:21
  • I am starting to read David Lay's book : "Linear Algebra and It's Applications", although I can just go with Howard Anton "ELementary Linear Algebra". Serge Lang and Sheldon Axlers' are typical for Math majors wanting for rigors with theorems-proofs with all abstract things introduced from the start; that's why they begin with Vector Spaces in the 1st chapter. There is one book (an old book) that is quite rigorous : "Linear Algebra" by Kenneth Hoffman and Ray Kunze. Hoffman is from MIT, so we can expect a good text. It begins with Linear Equations, followed by Vector Spaces. – Lex Soft Jul 08 '22 at 05:53

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I think the best book for you will be Linear Algebra done Right by Sheldon Axler. The reasons being

1.) It takes an abstract but intuitive approach, is heavy on proofs.

2.) Exercises are not too numerous, about 20-25 in each chapter and are not unreasonably hard; and cover most of the techniques used in the chapter.

3.) Since you will be doing a course on galois theory, you will find the discussions in the later chapter on minimal polynomial etc illuminating.

4.) Finally, I was in a similar state 6 months back and had to pick up linear algebra properly before course on galois theory. It took me about 10 days to finish it. It is not dense but is still quit thin. And the writing is superlative when compared to other books on the same topic.

Rest other books that you have mentioned have one or two of the flaws that I mentioned that Axler doesn't have. GO for it.

  • Thank you for your answer! Is this book complete? – Chris Jun 27 '17 at 13:58
  • Complete as in? –  Jun 27 '17 at 14:00
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    If you meant that will you need to read anything else for galois theory, then yes it is. I think the only result you will need is Dedekinds lemma which is not proved there. But that is generally covered in Galois theory courses. –  Jun 27 '17 at 14:01
  • My worry is if it's ok for Coding Theory (which uses Matrixes etc). – Chris Jun 27 '17 at 14:02
  • Tbh, Axler banishes matrices to the end of his work. However in my experience, one I finished Axler, I understood matrices way better than I did after reading strang. Let me try to find the 5-handout that remedies this fault of Axler –  Jun 27 '17 at 14:06
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    https://ocw.mit.edu/courses/mathematics/18-700-linear-algebra-fall-2013/study-materials/ I used these handouts. They contain all matrices you will need and are written to complement Axler. –  Jun 27 '17 at 14:08
  • Nice source! Thanks! – Chris Jun 27 '17 at 14:11
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    In addition to the book, you might also like to look at the set of 50 videos that were recently produced to accompany Linear Algebra Done Right: http://www.linear.axler.net/LADRvideos.html – Sheldon Axler Jun 28 '17 at 04:56
  • @SheldonAxler Oh my; I never expected that you'd comment here. Thank you for writing such an amazing text :) –  Jun 29 '17 at 16:31
  • @SheldonAxler Thank you for your comment and it's my honor that you answered me. Both of the videos and your books are great. I will do my revision from them. – Chris Jun 29 '17 at 17:08
  • It's not the best book for all audiences; it fits more to people who want rigor, like the guys from Math majors to which the books is written for. "Linear Algebra" by Kenneth Hoffman (MIT) and Ray Kunze can be an alternative for those who don't want to use texts by Howard Anton, David Lay, etc. It was actually written for Math major, but the authors considered audiences from other disciplines like Engineering. They avoided the introduction of too many abstract ideas from the very beginning; Appendix is provided for such things. So, like others, the 1st chapter begin with Linear Equations. – Lex Soft Jul 08 '22 at 06:08
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I think it better to study both algebraic and geometric intuition of linear algebra, it helps you to understand how theorems work, in this way you can apply Linear algebra technique in other fields.

For geometric intuition: I recommend "Essence of linear algebra" series which aimed at animating the geometric intuitions.

For Algebraic intuition: you can use Stephen Boyd and Lieven Vandenberghe book "Introduction to Applied Linear Algebra" , It is used as the textbook for the course EE103 (Stanford) and EE133A(UCLA).

indeed, you can use Sheldon Axler book " Linear Algebra Done Right" for more algebraic intuition.

Amir Nikabadi
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If you want a self-study book on linear algebra then go for 'Linear Algebra Step by Step' as it has complete solutions to all the problems online.

matqkks
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