Can anyone recommend an advanced linear algebra book?

[Eclair_de_XII]

I have already taken two elementary linear algebra courses, and have taken the upper-division linear algebra course offered at my school. However, I feel that I did not learn as much from the latter as I should have. I can owe this to not applying myself as much as I should have, due to other issues I have been coping with as of late.

Anyway, what I covered in the last linear algebra course I have taken was most of the basics of linear algebra that were made more abstract, polynomials, bilinear forms, adjoints, and possibly more. The book that I used last semester (that I had to return because I was renting it) was Advanced Linear Algebra by Cooperstein. I covered chapters one, two, four, six, and eight. I also learned about those isomorphism theorems, though I did not have to use them that often.

In any case, can anyone recommend me an advanced linear algebra book? Preferably one that is free and can be downloaded in a pdf format? What are your thoughts about this book called "Linear Algebra Done Right"? I've heard of it, so I figure that it must have some reason for being popular.

 

[StoneTemplePython]

"advanced" in what sense? I.e. what are you looking to develop after reading it?

it could mean applications to real world or to other parts of math (or stats or optimization or cs or ...)? If you have a goal of developing (i) multi-linear algebra that would lead you to a rather different place than focusing on developing some interesting results with (ii) spectral theory + blocked matrices + say Perron Frobenius theory.

If you find (ii) to be to your interests, I'd probably suggest doing the second half of Meyer's Matrix Analysis which is not free but has some very good stuff in there... the book comes with a thoughtful solutions manual too. There's also a bunch of 'bonus items' amongst the exercises covering things like commuting matrices, kronecker products and Newton's Identities -- important stuff that you typically would miss. Whether a lot of this stuff formally is linear algebra or matrix theory and analysis I suppose is a linguistic issue.
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On the other hand, Axler's book is fine, though again not free. He says ridiculous things about determinants but the book has some good things in there otherwise. It's popular enough that you can find various websites from universities that use it for a course, and just clone and work through the syllabus/ homework schedule that said courses use. Again it depends on what you are looking to develop. To help balance things out and properly understand determinants, I'd suggest at least working through chapter 3 of Linear Algebra Done Wrong, first. This one is freely available as a PDF here:
https://www.math.brown.edu/~treil/papers/LADW/LADW.html

you may enjoy some other chapters (e.g. 7) as well

I think all 3 of the above books only use fields of characteristic zero, so if you are interested in other algebraic properties, they may not be a good fit. Again, your goals matter here.

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I recommended a free PDF on linear algebra from Kuttler to you a while back. I'm not sure how much progress you made with it.

 

[Eclair_de_XII]

what are you looking to develop after reading it?

Like I said, I was hoping to learn what I should have in the course I took last fall, which would probably warrant repurchasing the book I had used. But now that I think about it, I really want to get into linear programming (optimization, like you said). It sounds like a useful skill to apply to programming, and perhaps something that can be used in the workplace.

I recommended a free PDF on linear algebra from Kuttler to you a while back. I'm not sure how much progress you made with it.

Sorry, I had actually realized I didn't need the book to do my project two years ago, and had forgotten about the book you'd linked me to, until now.

 

[StoneTemplePython]

But now that I think about it, I really want to get into linear programming (optimization, like you said). It sounds like a useful skill to apply to programming, and perhaps something that can be used in the workplace.

If you want linear algebra with an eye toward optimization, my suggestion would be the book by Meyer and definitely not Axler who tries to avoid matrices, doesn't tell you much anything algorithmic, not even gaussian elimination, omits big things like Cramers Rule (which e.g. can be used to prove integer valued solutions for special classes of Linear Programs), and so on.

For a gentle look at Linear Programming itself, the book "Understanding and Using Linear Programming" by Matousek and Gartner is pretty good. (I am part way through their semidefinite programming book and finding it is rather... less gentle.)

 

[The Bill]

Another excellent introductory book on linear programming and other topics such as computational complexity is Combinatorial Optimization: Algorithms and Complexity by Papadimitriou and Steiglitz.

It's a Dover book, so it won't cost much to add to your library. I've found it useful over the past couple of decades, and the presentation still feels fairly modern despite the fact that it was published in the early 80s.

Also, I'm not suggesting it as an alternative to anything suggested earlier in the thread, but as a potential supplementary source.

 

[mpresic3]

I like Strang's book on linear algebra. He does some optimization. Also linear programming is addressed in many books on operations research like Hiller and Lieberman. Luenberger writes a good book on dynamics and linear systems.