Is "Linear Algebra Done Right 3rd edition" good for a beginner?
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.
$\endgroup$ 88 Answers
$\begingroup$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.
$\endgroup$ 2 $\begingroup$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 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.
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 teaches 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 application and works a lot with matrices with digging into the theory, read Strang. 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 presented 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!
$\endgroup$ 4 $\begingroup$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:
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.
$\endgroup$ 2 $\begingroup$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:
- Get a standard first-course treatment in Linear Algebra. I personally like Lay's Linear Algebra and its Applications (3rd ed.), but apparently there's a 5th edition out now that I don't have.
- After finishing the first-course treatment, read Linear Algebra Done Wrong or any of the books that are mentioned in the links above.
It is a nice book. It is easy to follow and understand if your a beginner. After you read it, I would recommend Linear Algebra and It's Applications by Gilbert Strang.
$\endgroup$ $\begingroup$MIT OCW has a "Scholar's" (independent study) edition of Strang's Linear Algebra course, including videos of his lectures, grad student recitations, notes, homework, and exams (with solutions for homework and exams). It's free, but non-credit. Can be used with the 4th or 5th edition of Strang's book. Strang uses the traditional, determinants-early approach.
FWIW, Benedict Gross, in his Abstract Algebra lectures expressed his preference for the "done right" approach to the prerequisite Linear Algebra course to facilitate better understanding of Abstract Algebra.
$\endgroup$ 1 $\begingroup$I have the read the entire book and learned a few new things, even as a mature mathematician. However, I would definitely not recommend it for a first course in linear algebra. If you are going to use this text, I recommend learning RREF (Row Reduced Echelon Form) beforehand as it does not cover this most useful tool. Even without a focus on matrices, the book still contains problems on linear combinations, span, basis and other topics that are greatly simplified with RREF. I would also recommend studying fields and the first few chapters of Friedberg or David. C. Lay before using linear algebra done right. This rigorous proofs are much more simple once you have encountered numerous standard practice questions allowing you to learn the computational techniques before proving them. Friedberg is a far superior book for a second course in LA and David C Lays book for a first course (that is, if you learn chapter 4 before 1).
$\endgroup$ 2 $\begingroup$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:
- Axler's proofs are easy to reduce to first principles and follow step-by-step.
- The book is organized and formatted well, which makes it easy to later use for review.
- It is fairly compact and not unwieldy in length, so you can actually get through the book cover-to-cover without missing anything in it. I went through roughly 60% of the book's proofs in a week of intensive studying for an exam where I needed to know all the proofs in the book by heart. Axler is also selective about only covering the most important topics in theoretical linear algebra, which is a large reason the book's length is manageable.
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:
- Follow each proof step-by-step. Proof steps may follow from a previous definition, lemma/theorem or some simple algebraic manipulation. When Axler says that something follows from something earlier, don't just take his word for it. Go back to the earlier definition or theorem, and figure out how the proof step follows from it. Sometimes, Axler will combine steps or not explicitly mention a particular step. When that happens, make sure you can break down what he is doing into the individual proof steps that you can derive from first principles, such as the substitution of a definition or algebraic identity. If you run across a proof step where you don't understand how it follows, do not proceed until you can figure it out. Once you do, you might want to make a note of it.
- At the end of every chapter, write out a statement of every Lemma and Theorem in that chapter but without the proof. Then, without looking at the book, try to re-derive the proofs from scratch on your own.
- For any proofs you were unsuccessfully able to reproduce on your own, go back and restudy them. Also, check to make sure you didn't make any mistakes in your proofs. Then, repeat STEP 2, testing yourself on all the proofs from that chapter again. Don't proceed to the next chapter, until you have 100% success at rederiving all the proofs.
- Every 3-4 chapters, give yourself a longer test, where you try to reproduce all the proofs of every lemma and theorem from those 3-4 chapters.
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.
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