Linear Algebra Done Right

Sheldon Axler

User Reviews from Amazon

5-star user review on Amazon titled “A Fantastic Second Pass at Linear Algebra”: I used this book in my second course in linear algebra, and I absolutely loved it.

One of my biggest points of irritation in my first course in linear algebra was the fact that we first defined determinant as some unintuitive, computational nonsense and then defined eigenvalues from there. Judging off of the foreword, Axler essentially wrote this book so that students like myself could gain some peace of mind.

The material is developed wonderfully, in my opinion -- it was much more intuitive than my first pass at linear algebra, and the problems within the book (at least those that my professor assigned) were commonly quite good. On top of that, it was one of the first math textbooks I've owned that I could legitimately sit down and read rather than, say, fall asleep or come away more confused than when I first started reading. The proofs are *mostly* quite readable, and Axler definitely has an enjoyable writing style. Whether it's the little surprises like getting to page 22 and seeing it labeled "~7pi" (there are other easter eggs of that variety, by the way) or writing an anecdote from the supreme court where one of the justices is caught up on the word "orthogonal" inside the chapter on inner products and norms (p. 174), he certainly does his best to make reading the book enjoyable.

Apart from the material itself, though, as others have mentioned, the book is also just plain looks nice. I purchased a hardcover edition (which I believe may be the only style printed for the 3rd edition), and whether it's the outer cover or the typeset they use inside, everything just has a nice, colorful pop to it. Coming in at ~330 pages and ~6.5W x 9.5H x 0.8D inches, the book is also compact, an aspect that made carrying it to and from campus each day a trivial task.

As a whole, I have essentially nothing but praise for this text. Some may have wanted a linear algebra text that's more computationally intensive, but I'd rather gain the conceptual understanding that Axler has to offer than manually row reduce or invert some arbitrary matrix any day. I absolutely don't want to sell my copy -- this text is one that I'm going to be happy to leave on my bookshelf and use as a reference whenever I find myself needing to freshen up on my linear algebra later in life.

5-star user review on Amazon titled “BEST LINEAR ALGEBRA TEXT YOU CAN GET”: Yes it's for a second course, and yes it is entirely "proof based" and aimed at math majors, but that's what you want. Even if you're a engineer, computer scientist, programmer, data scientist, etc., you will come away with a thorough, intuitive understanding of linear algebra that will allow you wrap the entire subject into a single, monolithic idea - a "true" understanding (the ineffable "you KNOW it when you know it" type of understanding). Plus it's a beautiful subject in its own right, and the mathematical maturity you will develop makes you a better thinker.

5-star user review on Amazon titled “Wonderful book”: I am a graduate student who studies an area steeped in linear algebra. I have found this book invaluable, particularly the author's hard work on making proofs elegant and readable. I had bought a copy of the third edition previously, but I bought the fourth edition because I greatly appreciate the Axlerization of the tensor product (which I live and breathe) in the last chapter.

5-star user review on Amazon titled “Nice, crisp”: Nice, crisp, short, and to the point textbook with plenty of good exercises. Take your time with it. You will be surprised how much more you know about Linear Algebra after reading it since the author includes motivations and insights everywhere you look. (I'm a second-year undergrad using this textbook for my Linear Algebra class).

5-star user review on Amazon titled “Great for mental imagery and examples”: As a graduate student in engineering, this book has single handedly lifted the veil of my misunderstandings about linear algebra. Concise descriptions of proofs and a steady build up of theoretical ideas make understanding linear algebra so much more enjoyable. This book helped me gain a better mental picture about the essence of linear algebra rather than rote memorization.

5-star user review on Amazon titled “I love this book so much”: I love this book so much. It was my first exposure to "real" math and I continue going back to it when I need my fill of linear algebra. The writing is lucid, the layout very easy on the eyes, and the approach very natural. It isn't for everyone (particularly those looking for a more applied study) and the weak coverage of the determinant is unfortunate (read the determinant chapter in Treil's linear algebra book to see what you missed), but it's a book that I'd recommend to any math major or lover of math wanting a clean presentation of the major results of linear algebra.

5-star user review on Amazon titled “The title is correct”: I got this to help with quantum mechanics, on the recommendation of the prof at MIT. It really is a fine book and goes carefully over linear vector spaces and maps. Highly recommend it.

5-star user review on Amazon titled “Want to actually understand LA”: What can I say. Just fantastic. I studied LA as an undergraduate from a very computational perspective, and left the course scratching my head as to what it all meant. While matrix computations are obviously quite useful in practice, they obscure the underlying structure. Furthermore, once you truly understand LA, the "matrix tricks" all make a lot more sense. A user on stack exchange described this phenomenon as "think in terms of linear maps, compute in terms of matrices", or something to that effect. I couldn't agree more, and Axler teaches you how to think in terms of linear maps.

5-star user review on Amazon titled “The book is clearly explained with boxes and colored boxes to highlight important results/points and some nice examples to illus”: This book deserves a place in every math library. The approach of teaching linear algebra using linear mappings makes a lot more logical sense but since most people don't have the proof writing background at that time, it will fortify their base the second time through. The book is clearly explained with boxes and colored boxes to highlight important results/points and some nice examples to illustrate ideas. The problems are at an appropriate level of difficulty for someone teaching themselves. The book itself is well made with special paper that will last longer.

5-star user review on Amazon titled “This a great linear algebra book even if you think you don't need the abstraction”: Wanted to review linear algebra for a machine learning project and stumbled upon this. Since my project involves multiplying matrices rather than proving theorems, I was initially hesitant to read this book rather than a more applied book like Strang's Linear Algebra with Applications. While it's very abstract compared to Strang, it's a great linear algebra review even if you are primarily interested in matrix computations rather than algebra per se. Compared to the material in a more applied book, Axler's approach to linear algebra plays well with more abstract topics like Functional Analysis which I found to be a very desirable trait. Despite all the praise for this book, I expected not to like it due to the abstraction, but found it to be a very well written and very useful math book. Keep in mind I used this to review linear algebra. Can't say how well Axler would work as an introduction or second course. This is just my opinion and your experience may differ.

5-star user review on Amazon titled “Short and easy to follow”: Not only is this book physically beautiful and light, but the writing style is extremely clear and I've had no problem following the arguments. I'm currently working through each problem with two other people as a fun self-study thing, and everyone loves the book so far.

5-star user review on Amazon titled “A matrix is a Linear map made clear”: You're doing to need Linear Algebra for pretty much any scientific computing. Intro classes usually introduce matrices as sort of containers for equations and this analogy makes it challenging to figure out why certain properties hold for matrices. If you think of a matrix as a linear map between spaces, it becomes far easier to think about them.

5-star user review on Amazon titled “Really good”: This is an amazing book that approaches linear algebra in a really good way. This is a great book for people taking a first course in linear algebra and who are looking for a decent amount of rigour. The excercises are absolutely amazing and are a ton of fun.

5-star user review on Amazon titled “Excellent book”: This is a very good book IMO. It's written in a way that is easy to follow, but at the same time is written in a way that requires you to fill in blanks on your own. I'm not an algebraist, but this book helped develop a liking for Algebra.

5-star user review on Amazon titled “Amazing Abstract Linear Algebra Book”: Amazing book. The colored pages are nice and very refreshing for a math text. I think this would an amazing first time into abstract linear algebra and/or an introductory proof course. When I teach a proof based linear algebra course and can choose the text, I will choose this one.

5-star user review on Amazon titled “best for intended audience”: It is an excellent book, but people should know who the intended audience are:

1. It is for the 2ND CLASS in Linear Algebra. (author says it in preface and back cover)

2. It is a theorem-proof book, about how to think like mathematicians

3. It does NOT cover matrix computations, so that engineers will not get much benefit from it

For the intended audience, it would be one of the best linear algebra books. It avoids matrix expressions and uses abstract symbols only, making the proofs short and elegant.

5-star user review on Amazon titled “Let their education begin”: Excellent textbook for a second look at linear algebra from a strictly theoretical standpoint. It size is small enough so that one may comfortably carry it around and promptly, effortlessly smack around fools that utter: "linear algebra! that's just y =mx + b!!! LOLZ".

Their education is the responsibility of us all, and how often we forget the old ways...

The proofs are clear but do require the reader to fill in some gaps. This is intended. Open to any page and witness the clarity that so often escape the best efforts of a certain class of instructor; Axler will not suffer any unmotivated concepts. Everything builds from previous definitions until there's just enough structure to flesh out the chapter objectives, thus there's little fat to distract the reader.

Moreover, Axler is so badass that he does away with determinants until the last chapter of the book, he's so pimp he just didn't need any stinking determinants in his proofs. That's right, the last chapter introduces trace and determinants and proceeds to bring everything together into a magnificent mic drop.

I now finally understand why determinants are inextricably tied to notions of volume, and why we must multiply by the Jacobian when performing change of variables in multi-variable integrals, and so on.

A newcomer to linear algebra will get very little of use here, save for the clearest definitions I've ever seen regarding the structure of vector spaces, subspaces and linear operators. For a more applied/introductory approach to linear algebra, one can do much worse than Strang.

I now feel much more comfortable moving onto a graduate-level Linear algebra course after visiting Axler's book, as such, it will be an invaluable reference moving forward.

5-star user review on Amazon titled “One of the best books in all of mathematics”: Truth is we in India think we are taught linear algebra from our schools, but are never taught linear algebra. What we are taught is to manipulate a certain set of numbers put in a matrix form that gives us answers for unknown reasons.

Linear Algebra is beautiful. Much of it comes from geometry. But who is to know that based on what we are taught? Enter Axler. He painstakingly builds up your base in various topics like set theory, fields, bases, dimensions, projections to enable you to get to a point where you re-understand a part of maths that is lost to you, because of the way it has been taught.

LADR is not complete - it doesn't cover everything. But what it does cover is built. Not just in the book, but in the reader. It won't be an easy read. But when you are done with about 130 pages of this book, you will get a fresh perspective on all of LA. and LA is pervasive in all of maths. Further, LA is pervasive in much of CS, and in all of machine learning. Getting into those fields without a complete understanding of LA is like entering a flower garden without a sense of smell.

Why Linear Algebra? Matrix Algebra tells you how to get the numbers that you want. Linear Algebra tells you why you get them.

Bonus: I believe Linear Algebra Done Right is Springer's best seller across all the mathematics textbooks that it sells (saw this in an old excel sheet shared by springer with a publisher).


1. For self study: Do Gilbert Strang if you want to get through your exams. But Strang is not rigorous. If you really like mathematics, there is no better book to start your journey in Linear Algebra than Axler.

2. For college courses: Should be primary text for undergraduate LA students.

5-star user review on Amazon titled “Great Intro”: It's the type of book that if you go through without skipping and while doing the problems, you will leave with a great understanding of linear algebra. It leads directly into the same author's more advanced book "Measure, Integration & Real Analysis".