Is "Linear Algebra Done Right 3rd edition" good for a beginner?
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.
My Opinion
After reading the whole 3rd edition of Axler's book, I might want to say the answer to your question is:
This book definitely worth reading but is not absolutely good for a beginner!
Reasons
$1$. 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 (As I did)! However, I might say that there are really elegant proofs in the book too!
$2$. 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.
$3$. 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 common, no Gaussian elimination, no LU and related decompositions! Although the Gram-Schmidt procedure is mentioned but it's 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 so emphasis on 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.
Conclusion
Finally, here are my suggestions. Depending on your needs, if you want to study theory more than applications, read Hoffman/Kunze once and for all to close the case of linear algebra. I think this is the best choice for a beginner. This book teach you the subject by an algebraist thought. If you want to get some analyst thoughts, the last part of Halmos's masterpiece is a nice complement for Hoffman/Kunze and will prepare you to study functional analysis later. If you want a good textbook which emphasizes the application and works a lot with matrices with digging into the theory, read Strang. The Strang's book is a perfect companion for any theoretical book. After reading one of these books, it is fun to go over Axler 's Linear Algebra Done Right to see how the theory can be represented in a different way. Axler's book has the potential to be the best linear algebra book ever; however, it still needs lots of polishing and editions!
A Brief Review of Linear Algebra Done Right, 3rd edition
My background: In the second half of my fresh year, we used the second edition of this book in our first (and only) course in linear algebra. (I’m an econ major in China, and everything is scheduled, you know…) Previously I didn’t have any knowledge about linear algebra, nor did I know anything about rigorous mathematical proofs. So at the beginning I was a little bit confused, but at the same time greatly impressed by the the new fancy world that opened to me: the rigor, the proofs, the abstractions. Later, as the course moved on and as I continued reading through the book, I was able to get better and better understanding, and enjoy the book even more. After the course, I finished the remaining chapters not touched in class. When the $3$rd edition came in the winter of 2014-2015, I bought it from Amazon, and flipped through it again and again. To me, this masterpiece is the most deeply respected and beloved, for there began my whole journey to the wondrous world of pure mathematics.
What are there in this book?
Linear Algebra Done Right is a theory book. It focuses on building the theory of linear algebra using rigorous proofs and on understanding the structure of vector spaces and linear maps. As a textbook targeted toward undergraduates, it also aims to increase one’s mathematical maturity and to let one appreciate the beauty of the subject. The main objects of discussion are linear maps and linear operators (which are coordinate-independent), not the specific details and techniques about matrices. See the contents of this book on Amazon.
Why should one choose a theory book for a first encounter to linear algebra?
Linear algebra is foundational in pure mathematics, applied mathematics, and almost all scientific and engineering fields. For pure math it is a building block in functional analysis, PDE, differential geometry, algebraic geometry and representation theory, just to name a few; if you are in data science, then matrices will be your basic tool; if you work in economics or engineeriing, then a close friend of yours is optimization, which uses linear algebra intensively. In all cases, giving the importance of the subject, giving that you will live with it, see it and use it for a long time, you want to have an early in depth understanding of the subject. After you have mastered the theory, it is a lot easier to understand the subsequent applied matter if you need it. No matter what you do in the future, a theoretical training can be of tremendous help: if you are going to do pure math, then you probably don’t need too much details (like various criteria for a matrix to be negative semidefinite), but without a true understanding of the subject, the study of the ever more abstract theories in pure math would be shaky. On the other hand, if you are going to use linear algebra as a tool for modeling and calculations, then you will be dealing with (very) specific aspects of matrices. Theory helps you navigate through those details, not letting you lost in them. (By the way, even you would choose a more traditional textbook as a first learning, you are probably not going to see all the quite sophisticated results you need for research, nor do you want to dive into the details too early; you also have to wait for a second learning)
That being said, it is indeed true that one should not shy away from determinant; it is often unmotivated, but that does not mean we should abandon it. So the best combination would be: first go through Linear Algebra Done Right to have a firm theoretical background, and then after that read a more traditional textbook for the missing parts. For those math-minded I recommend Michael Artin’s Algebra. For the more applied other posters’ recommendations are fine. I also humbly suggest my own notes on bilinear forms. Since you have already gone through the hard theory, reading those books and notes should be fast and easy.
What’s the feature of this book?
Perfectionism, Elegance, and Extreme Beauty. Anyone reading this book will discover how the author has put great efforts in perfecting every detail: every proof has been scrutinized and polished again and again, to make them as elegant as possible; every example has been greatly considered and carefully selected; the arrangement of materials is neat and compact, without any waste of words; the $3$rd edition features luxuriously beautiful formatting, like usage of colors and boxes, which is rare among theory books. I would also like to mention that every theorem in this book has a descriptive name. For example, see this theorem on page $281$ of the book:
$9.16$ Nonreal eigenvalues of $T_{\mathbb{C}}$ come in pairs
Suppose $V$ is a real vector space, $T\in\mathcal{L}(V)$, and $\lambda\in\mathbb{C}$.Then $\lambda$ is an eigenvalue of $T_{\mathbb{C}}$ if and only if $\bar{\lambda}$ is an eigenvalue of $T_{\mathbb{C}}$.
Here $T_{\mathbb{C}}$ is the complexification of $T$. In this way, key facts are a lot easier to remember. And most importantly, readers get to know what each theorem is about. Compare this style to some notoriously abstract and terse books, where reader may sometimes have no idea what the author is talking about.
Criticism
To be more neutral, let me include here some voice from the other side. Recently I saw a book review for Linear Algebra Done Right on The American Mathematical Monthly, which is largely negative:
Leslie Hogben, Reviewed Work: Linear Algebra Done Right, Third Edition. By Sheldon Axler. The American Mathematical Monthly Vol. 123, No. 6 (June-July 2016), pp. 621-624
The article said that the book contains insufficient materials for a second course in linear algebra, and concluded that it is more suitable as a textbook for a challenging first course. Well, this I agree indeed. I agree that the book is best used in a challenging and demanding first course on linear algebra, as I have talked earlier.
Conclusion
I have never seen a textbook author who is so enthusiastic and considerate for the audience. (I have seen many authors that are reluctant to take time and effort to consider pedagogical issues, and their “hand-waving” often leaves the pain to the reader) The book is brilliant, outstanding, and well-known, as it has been widely adopted in many universities around the world. Yet it is demanding at the same time, as it constantly pushes you forward to high mathematical maturity. Even if you do not plan to take an unorthodox and challenging path, this rare intellectual treasure is nonetheless still worth a look, for its unforgettable impression and enjoyment.