Amazon book reviews say it takes unorthodox approach and is for a second exposure to linear algebra. I didn't have a first exposure to linear algebra.
Is this book going to be bad for me, then? Or, should I read another linear algebra book after reading it? I want to avoid reading two linear algebra books because reading such a textbook consumes a lot of time.
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After reading the whole 3rd edition of Axler's book and a glance on 4th edition, I might want to say the answer to your question is:
This book definitely worth reading but may not be a good fit for a beginner!
$0$. The main feature of Axler's book is its determinant-free development of the theory which leads to cleaner proofs. I am sure that you will enjoy this approach. However, determinant has its own theoretical importance and neglecting it is not possible and not even beneficial!
$1$. Although Axler tries to give geometric pictures for some definitions and concepts, I would argue that he could do a better job at this. So, you might not get enough intuition from this book.
$2$. Axler prefers writing proofs with words instead of equations! I mean that he likes using words and the mind of reader instead of writing it down. As an example see this post. This may be a little annoying for some beginners or those who prefer detailed equations instead of words. Also, this may cause you feeling lost in some places when this tradition combines with the typos in the proof! However, I might say that there are really elegant proofs in the book too!
$3$. Axler's book is different in most of the aspects from the all books on linear algebra so it may cause you confusion when you want to take a look at other resources for reviewing or learning some topics. However, in most of the cases he mentions the differences. One of the differences not mentioned in the text (but mentioned in the preface for instructor) is the definition of polynomials.
$4$. The material is a little insufficient to me. No topic about multi-linear forms and tensor products is included! No examples or discussions are made for vector spaces over finite fields! No emphasis is made in the book on algebraic structures like fields, modules, rings, groups and algebras that one should know in a theoretical book. Also, some important concepts like double dual space are not in the text and just some exercises are included for them. Also, there is nothing about the inverse matrix of an operator in the book! Worse than that is you do not get used to work with matrices and linear algebraic equations in this book. I mean come on, no Gaussian elimination, no LU and related decompositions! Although the Gram-Schmidt procedure is mentioned, its relevant decomposition, the QR decomposition, is not addressed. In general, the book does not give you matrix pictures so much! I understand that Axler is trying to emphasize the abstraction of the concept of the vector space; however, these pictures really help you to keep the ideas in mind and have some examples for yourself!
$4$. Also there is no solution manual of the book yet! So you are on your own when dealing with exercises. But I would say that there are nice exercises in the book so be sure to look at them while reading the book.
This edition has major improvements and I am really satisfied with it. He has tried to give more geometric intuitions. As an example, see how the norm of an operator is related to deformation of a ball to an ellipsoid on Chapter 7. QR decomposition is included. SVD is treated with more details and in a more general form for operators from a linear space to a possibly different space and used to prove the Polar decomposition neatly. However, the polar decomposition is still proved for operators but it could be proved for more general linear maps and then specialized to such operators. Determinants are treated by multi-linear forms which is a much better approach than the previous edition. The only thing is that I am not really a fan of Axler anti determinant point of view. Tensor products are also introduced. I can just say "Well Done, Sheldon"!
If you have encountered a standard abstract linear algebra course, then it is fun to go over Axler's Linear Algebra Done Right to see how the theory can be presented in a different way. Axler's book has the potential to be the best linear algebra book ever and it has improved substantially over time!
If you also want to read a thorough and technical review on 3rd editions of his book, you can take a look at this paper.
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It is a nice book. It is easy to follow and understand if your a beginner. After you read it, I would recommend the book Linear Algebra and Its Applications by Gilbert Strang.
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This is merely me coming from a statistical perspective. As mentioned in the comments, you're talking about computer science and AI, which strongly suggests that phrase "data science" to me.
I read Axler in my undergraduate and started a M.S. statistics program this year, and have read up on data science. I would consider data science and stats to have very, very similar applications for linear algebra.
Do note that I asked a very similar question very recently, and the comments suggest pursuing this link. I have heard good things about Strang, but haven't bought it yet.
As for the actual question, I didn't find Axler at all useful for learning linear algebra for the reasons I needed it (statistics, data science). I recall reading in a review of Axler's book that it is more of an algebraic take of linear algebra: from what I understand of "algebraic," I think this is a valid point. There is little emphasis (from what I recall of that book) of things you would find useful in data science (and should really know), such as $QR$ decompositions, $LU$ decompositions, singular-value decompositions, generalized inverses, pseudoinverses, calculating the rank of a matrix, etc. That isn't to say Axler couldn't help you with this, but Axler, from what I recall, doesn't cover a lot of this material that I'm mentioning here.
If I were to recommend a course of action for you, it would be:
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I used an earlier edition as an undergraduate. Your question presumes self-study, and I think Linear Algebra Done Right is ideal for self-study as a beginner for the following reasons:
As the other answers point out, this is a theoretical book. If you don't think you will ever need to write a proof, and are only interested in computational linear algebra and its applications, then this book may not be for you.
However, if you do appreciate proofs and theoretical understanding, then I recommend learning the theory of linear algebra before worrying about the applied stuff, and this is a good book for that. Axler's book won't teach you everything you might ever want to know about linear algebra, but it will provide the foundations necessary to make it relatively easy to learn anything from linear algebra not in his book. For instance, much of the applied stuff you can learn on demand as you need it, from Wikipedia and other sources, so long as you have a good theoretical foundation.
I should point out that Axler's book was unfortunately not my first exposure to linear algebra. My first exposure was with Friedberg, Insel and Spence (FIS). My experience with FIS was miserable, and I still have PTSD from it. I think that FIS might be tolerable if you have a really good instructor to guide you through the jungle. My instructor was not great and I would never even consider FIS for self-study as a beginner.
However, the one benefit from my first bad experience with linear algebra is that it led me to appreciate Axler's avoidance of determinants. Determinants are truly messy creatures with non-trivial combinatorial properties, and they tend to lead to technical proofs. For a beginner, they are definitely a distraction and hinderance to learning the core abstract concepts and proof techniques in linear algebra.
This is not to say that there is nothing interesting about determinants. Quite the contrary. Determinants are interesting enough that they are worth studying by themselves as a separate subject, and you can find books that focus on them. Even Lewis Carroll of Alice in Wonderland fame wrote a book on determinants.
But trying to wrap your head around determinants while learning the basic theoretical concepts of linear algebra is fighting two wars at the same time. Axler showed that this does not need to be done. So kudos to him for distilling linear algebra down to a simple form that is easy to digest.
How to get the most out of Linear Algebra Done Right as a beginner:
First of all, you need to have enough mathematical maturity to have some ability to read and follow proofs. If you have never done that before, or taken a math course that required you to do that, then self-study of theoretical linear algebra may not be a great idea. You need to at least understand why proofs are important.
Second, my recommended regime for going through this book is a bit tedious, so you need to have some patience. It will involve reading most of the proofs multiple times. The goal is to reach the point where you could re-derive every proof in the book on your own:
As mentioned above, I followed this process with the latter 60% of the book, and it got me an A+ on an exam for a class I mostly slept in. The total cost of that for me was one week. Hypothetically, the above process should work with other linear algebra books as well, but would likely be a lot more painful. I wouldn't even try it with a book like FIS. Axler's exposition is good and he doesn't try to overwhelm you with an "everything but the kitchen sink" approach.
I guess I should point out that I don't find linear algebra to be the most intellectually interesting subject in the world, despite the fact it can be useful.
I think a good goal for yourself is to become proficient with the proof techniques and tricks of the trade used in Axler's book. For me, this was more important that the actual content or intuition behind linear algebra. If you study the proofs the way I suggest, you will start to internalize patterns in the proof strategies that Axler uses. Perhaps, Axler doesn't use every trick one might want to know, but his methods are good enough to prove all the major results in linear algebra, so it's good enough for me.
I suppose it's also worthwhile doing some of the exercises in the book, although I found the exercises to be too simple to leave a mental imprint. My process of drilling the proofs will probably help you more than doing the exercises, although the exercises will sanity check that you understand what's going on and aren't just blindly memorizing proofs.
It's unconventional in the sense that it works mostly with lists, as opposed to sets (a minor adjustment that makes certain proofs, like the complex spectral theorem, easier) and it avoids determinants until the very end. Also, by developing the theory of linear transformations first, then about matrices, it really emphasizes a key thought to keep in mind with linear algebra: Think in terms of linear transformations, compute with matrices. It's a very good book and easy to follow. And even when he skips a few steps, he explicitly says, "I'm skipping steps here, you should do it" so you aren't left feeling lost.