I found what seems to be a good linear algebra book. However, I want a more theoretical as opposed to computational linear algebra book. The book is Linear Algebra with Applications 7th edition by Gareth Williams. How high quality is this? Will it provide me with a good background in linear algebra?

3Not too familiar with the book you listed, but Sheldon Axler's Linear Algebra Done Right seems to offer a pretty thorough theoretical treatment of the subject. – InterestedGuest Dec 01 '11 at 12:47

There has been a long discussion of this on http://mathoverflow.net/questions/16994/linearalgebratexts . – darij grinberg Dec 01 '11 at 16:03

5As for Axler, it is more impractical than theoretical. Pretty much every other text is better in my opinion. For details, see the comments underneath one of the replies in the abovelinked MathOverflow discussion. – darij grinberg Dec 01 '11 at 16:04
9 Answers
I don't really know the book you are talking about, so I can't give you an opinion on that. I suggest that you take a look at:
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For what it covers I, too, think this is a very good book. Unfortunately, it's not very comprehensive and more often than not if I try to use it as a reference what I'm looking for just isn't there... – ItsNotObvious Dec 02 '11 at 23:12

1Axler shines in some way. E.g. I like the section that he compares Taylor polynomial to orthogonal polynomial approximation and motivates the use of orthonormal bases. His emphasis on generalized eigenvectors is also refreshing. But I think his take on determinant is too radical or even problematic. For instance, he avoids using determinant to define eigenvalue, but ends up having different definitions of eigenvalues for different fields ($\mathbb{R}$ and $\mathbb{C}$). – user1551 Dec 03 '11 at 17:35

As the title might suggest, Axler has a weird obsession with *not* teaching the determinant, an important multilinear function with uses in many areas of pure and applied math. Determinants aren't defined until the last section of the last chapter, and proofs avoid it just to make a point. For a more orthodox presentation Hoffman and Kunze or Halmos are suggested. – WMe6 Jun 04 '17 at 05:42
I like the book of Hoffman & Kunze. It gives a very nice and quite rigorous treatment of linear algebra. The selection of problems is excellent.
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Hoffman and Kunze is very good for the theoretical side of LA. But not only is it too difficult for any but the very best students,it completely omits applications and I don't think that's appropriate for a serious linear algebra course. – Mathemagician1234 Dec 01 '11 at 20:45

6Depending on what do you mean by "Linear Algebra". I would say Hoffman and Kunze is an excellent text for linear algebra but not for matrix algebra. – user1551 Dec 03 '11 at 17:25

1Hoffman and Kunze really isn't that hard. Mostly, it's hard to read because (to me) it lacks any human feel... it's very dry. If you can do proofs, you can work through H&K. – Aug 03 '14 at 01:32

@Mathemagician1234 Hello! Sorry to bother you, but I know you are very knowledgeable about textbooks and I was wondering if I could ask you for a recommendation. Would you know a book which gives a good intuition for matrix algebra, or Numerical Linear Algebra? I am currently reading "Numerical Linear Algebra" by Trefethen and Bau, which seems pretty good, but I would highly value any recommendations you have. Thanks a lot! – Ovi Sep 15 '19 at 20:22
I like Gilbert Strang's book, Introduction to Linear Algebra, but it may not be as advanced as some of the other suggestions here.
His video lectures are a useful companion to the book and a joy to watch.
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4Strang's book is the great classic that shows how to write an intelligent book on applied mathematics without making it a mindless plugandchug trick manual. – Mathemagician1234 Dec 07 '11 at 05:35
I have always preferred Linear Algebra Done Wrong, a set of notes by Sergei Treil, over Axler's book, which, while being completely rigorous, is a practical introduction with a view to its applications.
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I strongly recommend Steven Roman's Advanced Linear Algebra, for an abstract treatment of linear algebra.
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An even better treatment can be found in T.Blyth's MODULE THEORY: AN APPROACH TO LINEAR ALGEBRA. – Mathemagician1234 Dec 02 '11 at 22:53

2@Mathemagician1234 Instead of giving your recommendations as comments, why not post it as an answer? – Srivatsan Dec 02 '11 at 23:12

@Srivatsan I'm too lazy.Besides, the comments look better in context in response to another answer.: ) – Mathemagician1234 Dec 03 '11 at 04:03

Hmmm... I didn't much care for the book myself. First edition had a many errors, too, so if you must, make sure you are working off a later edition. – Arturo Magidin Dec 07 '11 at 05:27
I'm not an expert on Linear Algebra, but I would seriously recommend the book Linear Algebra by Jim Hefferson. On top of being free to download, it has a highly motivational approach. Every other textbook on the subject that I read went like this (my reactions in italic):
 Linear Algebra is the study of the solving of systems of linear equations. No problem!
 Matrices are a nice shorthand for expressing linear systems and Gauss' method is an elegant method of solving them. Neato!
 Anyway here are pages and pages of unmotivated definitions which apparantly have nothing to do with lienar systems, equations, or even Gauss's method. Learn them off by heart and please don't pay attention to the fact that we're apparantly just shuffling matrix entries around according to meaningless and artificial rules. Wwhat?
Linear Algebra explained things I thought very nicely. Even the parts that I didn't quite understand (like why "echelon form" matrices are worth study, when at first glance they appears to be a merely typographical rather than mathematical notion) were presented in such a way as to leave me with a sense of mystery rather than frustration ("Wow, why would an apparantly merely typographical notion have such interesting mathematical properties? I should learn more." rather than "What? This is just typography, not math. Screw off with your arbitrary definitions, book." as too often happens when definitions are unmotivated).
Some of the exercices are also excellent. In the early chapters you'll often find an exercice to which your initial reaction is "...what does this have to do with Linear Algebra?". That is, working out how linear algebra can be used to solve the problem is part of the exercice, thus developping your feel for the applications and motivations of the subject.
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The books by Axler and Hoffman & Kunze come to mind, as mentioned by others.
Let me suggest Halmos' FiniteDimensional Vector Spaces (published by Springer). It is very neat, and it works mostly in a coordinatefree way. Another way of seeing it is that it does things in finitedimensional vector spaces in a style that is used for infinitedimensional vector spaces.
A book that is not a linear algebra book, but contains some excellent bits of linear algebra is "Differential Equations, Dynamical Systems and Linear Algebra", by Hirsch and Smale. I'm talking about the 1974 edition, not the recent one with Devaney as a coauthor (I don't know that edition, I heard it is quite changed; maybe someone can chip in here). It is truly a gem.
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I may be a little late responding to this, but I really enjoyed teaching from the book Visual Linear Algebra. It included labs that used Maple that I had students complete in pairs. We then were able to discuss their findings in the context of the theorems and concepts presented in the rest of the text. I think for many of them it helped make abstract concepts like eigenvectors more concrete.
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1) Linear Algebra by Hoffman and Kunge, 2) Linear Algbra by G. Strang, 3) Linear Algebra by Helson, 4) Introduction to linear algebra by V. Krishnamurthy 5) University Algebra by N. S. Gopalkrishanan 6) A First Course in Abstract Algebra by Fraleigh, J. B. If you explain what you exactly want? then I can suggest u better also.
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