I am looking for an introductory book on Linear Algebra. But the posts that I have found related to this question (for example this one) doesn't meet (neither address) my specific requirements. So I thought that it will not be a bad idea to post another question asking for introductory Linear Algebra texts. Some of my specific requirements are,

  1. In the beginning of each chapter (at least most of the chapters), the book should discuss what were the main problems for which the necessity of idea(s) of the chapter was(were) needed.

  2. The book should provide motivations for each (at least most) of the theorems.

  3. Proofs should be very clear, rigorous and precise. In place of "jumps" some indication should be given so that "jumps" are indeed made.

  4. The pace of the book should be slow.

  5. The book's focus should be more (if not exclusively) on conceptual matters.

It may appear that I am claiming too much from the author. If that is so, then let me emphasize that it is not necessarily needed that all the requirements should be satisfied exactly but the more the book satisfies the requirements, the more better it will be for me.

Now let me tell some books that I really admire (though it may be that the books doesn't satisfy all the requirements I have given above) I don't like (at least for the beginners). I have marked the books I like by $(\color{green}{\checkmark})$ and those I don't by $(\color{red}{\times})$


$(\color{green}{\checkmark})$ Analysis by Terence Tao.

$(\color{green}{\checkmark})$ Calculus by Tom M. Apostol.

$(\color{green}{\checkmark})$ Understanding Analysis by Stephen Abbott.

$(\color{green}{\checkmark})$ How We Got From There To Here: A Story of Real Analysis by R. Rogers and E. Boman

$(\color{red}{\times})$ A Course in Pure Mathematics by G. H. Hardy.

$(\color{red}{\times})$ Introduction to Real Analysis by R. G. Bartle and D. R. Sherbert.

Set Theory

$(\color{green}{\checkmark})$ Introduction to Set Theory by T. Jech and K. Hrbáček.

$(\color{green}{\checkmark})$ Elements of Set Theory by H. B. Enderton.

$(\color{green}{\checkmark})$ Abstract Set Theory by A. A. Fraenkel.

$(\color{green}{\checkmark})$ Foundations of Set Theory by A. A. Fraenkel.

$(\color{green}{\checkmark})$ Axiomatic Set Theory by P. Suppes.

$(\color{red}{\times})$ Naive Set Theory by P. R. Halmos.

Number Theory

$(\color{green}{\checkmark})$ Elementary Number Theory by D. M. Burton.

$(\color{green}{\checkmark})$ Higher Arithmetic by H. Davenport.

$(\color{red}{\times})$ Introduction to the Theory of Numbers by G. H. Hardy and E. M. Wright.

The books listed under either $(\color{green}{\checkmark})$ or $(\color{red}{\times})$ doesn't follow any particular order of "liking" or "disliking".

Can you give some suggestions of Linear Algebra text books in accordance with my requirements as elaborated above?

  • 1
    $\begingroup$ My instructor used Hungerford, which had a good pace for me (at the time). I also like what I've read of Goldhaber and Ehrlich's text. $\endgroup$ – Sinister Cutlass Nov 20 '15 at 3:37
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    $\begingroup$ Linear Algebra Done Right by Sheldon Axler. It's great. It's all concept, great pace, and he has a great expository voice. I really enjoyed it. $\endgroup$ – Em. Nov 20 '15 at 3:42
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    $\begingroup$ You are way too worried about which book you read, IMHO. Just pick up any old introductory linear algebra book and start working through it. Trying to find the perfect book is just wasting valuable time that could be spent exploring cool new mathematical concepts. $\endgroup$ – user137731 Nov 20 '15 at 3:42
  • $\begingroup$ @Bye_World: I agree, but I sympathize with the OP. I think it's easy to get caught up or somehow convinced that "if you don't have the right book, then you won't learn everything you want/need/are expected to know". What I (eventually) learned is that all books on a particular topic (at this level) are basically the same: they'll contain the same theorems, mostly the same proofs, and at least some very similar exercises. The most meaningful difference is probably only in the harder exercises, and you can hopefully find time to look those up in other books and mull them over as you go along. $\endgroup$ – Will R Nov 20 '15 at 4:22
  • $\begingroup$ @WillR: I don't think I have any philosophy (or any variant of it) like, "if you don't have the right book, then you won't learn everything you want/need/are expected to know". Actually my reason for being so "worried" about the selection of books is due to my experience with Set Theory books. One of my senior suggested that Halmos's Naive Set Theory is a very good book for an introduction to Set Theory. I read the book completely and came to the conclusion that it is a very bad book for the beginners and it has certainly been a wastage of my valuable time. $\endgroup$ – user 170039 Nov 20 '15 at 4:49

Sheldon Axler's "Linear algebra done right" is a good text, as probablyme said in the comments. Also I think "Linear algebra" by Jim Hefferon is a book that satisfies some of your requirements.

  • 1
    $\begingroup$ I think Axler's book fits the questioner's requirements admirably. However, anyone out there looking to use it as an intro should bear in mind that Axler intends it as a second course, not an intro. Axler focuses entirely on proofs and concepts and doesn't tell you how to calculate anything (almost), which is not the way most people should be introduced to linear algebra. He is assuming you already took a course where you learned how to do computations such as Gaussian elimination. $\endgroup$ – Nate C-K Mar 28 '16 at 16:57

One good way to provide motivation for linear algebra is to see how it's helpful for understanding ODEs. It's possible that much of the historical motivation for developing linear algebra came first from ODEs (but I'm not sure about that, so let me know if I'm wrong). If that sounds interesting, you might be interested in Differential Equations, Dynamical Systems, and Linear Algebra by Hirsch and Smale, and also Differential Equations and Linear Algebra by Gilbert Strang.


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