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Linear Algebra Done Right

4.33  ·  Rating details ·  884 ratings  ·  53 reviews
This text for a second course in linear algebra is aimed at math majors and graduate students. The novel approach taken here banishes determinants to the end of the book and focuses on the central goal of linear algebra: understanding the structure of linear operators on vector spaces. The author has taken unusual care to motivate concepts and to simplify proofs. For examp ...more
Paperback, 251 pages
Published February 26th 2004 by Springer (first published November 29th 1995)
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Nicolas De Jay

Although by no means official solutions, they are tremendously useful for those us who wish to self-study linear algebra!


Although by no means official solutions, they are tremendously useful for those us who wish to self-study linear algebra!


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Jul 09, 2016 rated it it was amazing  ·  review of another edition
Shelves: mathematics
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 oft
Jul 19, 2020 is currently reading it
Excellent presentation of the material so far. However, there is one glaring omission: no solutions to the exercises are provided.

Fortunately, the bastards are available online here:

Quite accurate thus far, though I’ve only checked through section 2B.
Mar 04, 2012 rated it it was amazing  ·  review of another edition
Shelves: mathematics, algebra
[Second reading]
I read this the second time at the beginning of my physics PhD. It was much more pleasant to read. I am a slow learner, so even if the subject is for beginning undergraduates, a few concepts only make sense to me now. I can also appreciate a little better the whole thing about generalized eigenvectors, nilpotency, Jordan form, and various decomposition (in particular singular value and polar decomposition). I am sure I will forget them soon, considering that I don't use them ofte
Dec 25, 2013 rated it it was amazing
Shelves: math
This book was a real page-turner! Great approach towards linear algebra. Sheldon Axler doesn't introduce determinants until the end, which was a true delight in my opinion because it surely kept me reading until the very last chapter in order to learn about those nasty determinants. The section about Jordan Canonical forms had me hooked for days. The way the Jordan bases were formed was absolutely mind blowing! I did not see it coming at all! Axler truly outdid himself with this one. The book de ...more
Yash Patel
Apr 19, 2020 rated it it was amazing
Oh man, this book is SO GOOD! SO. GOOD. I learned linear algebra from, but this book is SO MUCH better in terms of motivating any number of abstractions. I'm not sure if it's because of my having becoming more familiar with abstraction since having read that book or if it's actually this book, but I thought this was far more approachable, without being any less rigorous. It certainly went through all the same material, so all the material that I didn't un ...more
Mark Moon
I enjoyed teaching out of this book this year, and expect to keep doing so for years to come.
Jan 03, 2021 rated it really liked it  ·  review of another edition
I really liked this book, but I think there is some room for improvement. I will describe my reasoning below. Nevertheless, among the arid and dry landscape of mathematics, this book is an oasis.

The premise of the book is simple and outrageous at first sight. Let us teach linear algebra without the use of determinants. "Anathema!" I hear you all shout. "It is impossible" (I heard myself think). However, Sheldon Axler proved me wrong. And honestly, it makes things much more intuitive and easy.

Dahn Jahn
Mar 12, 2015 rated it it was amazing
Beautiful. One of the books where the reading process doesn't feel like learning more material, but rather like individual components falling into one big picture of the world of (finite-dimensional) linear spaces.

That being said, don't pick this up without also studying additional material. LADR is beautiful, but by no means comprehensive:
1) It lacks a lot of material, either in the topics presented, or additional topics (I would love to see the author's treatment of bilinear forms)
2) Nearly n
Apr 03, 2008 rated it really liked it
This was the textbook used in my undergraduate theory of linear algebra course (taught by Seymour Goldberg, so you know the relevant view). It does have the nice property, basically shared with Lax's book, that it's clearly a book about linear algebra in general (finite-dimensional) linear spaces, and not a mix of some linear algebra, some matrix analysis, and some elementary modeling concepts. Also, Axler is in the "short and sweet" school of mathematical writing, so it doesn't loom large on th ...more
Ronald Lett
May 12, 2011 rated it it was amazing
Shelves: algebra
This is how linear algebra should always be presented. The unfortunate presentation one first encounters that emphasises matrices and determinants without appeal to the algebra and geometry that motivates them is absent from this text. It is a full presentation that will give you an intuitive grasp of linear algebra from both the geometric and algebraic points of view.
Shreeyesh Menon
Jun 05, 2021 rated it it was amazing
Best book on Linear Algebra that I found from a whole bunch that were recommended to me. The author's determination to ignore determinants until the fag end of the textbook and just explaining concepts in more intuitive terms and not the rote, god-awful pedantic tradition of drawing up lines and boxes and putting numbers around them. It's extremely imaginative and makes an effort to get you to truly "see" linear spaces in a new light beyond all the greeks.

If for some reason, you prefer the dete
Coop Williams
Dec 20, 2019 rated it it was amazing  ·  review of another edition
Get ready to write a lot of proofs.
Anlam Kuyusu
May 09, 2020 rated it it was amazing
This is the first math book I couldn't stop reading after the first few chapters. I was hooked after the author's proof of the Rank-Nullity theorem - the elegance of the proof was superb.

The book also is the only book that gives a satisfactory theoretical explanation/justification of determinants. (Hint: It's not pretty that's why it's eschewed.)
Sep 07, 2012 rated it liked it
A much, much better coverage of linear algebra than the book my undergrad class used. As many others have noted, it largely avoids the use of matrices and determinants to focus on the purely algebraic aspects of the material. It is also nice that vector spaces other than R^n or C^n are frequently considered, such as the space of all polynomial functions of a single variable. I was able to read through almost the entire book in one long sitting since it was a lot of review for me was well-present ...more
Apr 05, 2016 rated it it was amazing  ·  review of another edition
Shelves: math
This is a great second book on linear algebra - something you might want to read once you read a foundation text on linear algebra first. It is a full color book which makes the book rather expensive. It's not a large book, but it's packed with just the right explanations and amount of theory to get you to understand the more mathematical aspects of linear algebra - the topics of spaces, linear maps and related theorems and concepts. The book is very accessible, has plenty of exercises (no solut ...more
Dec 31, 2018 rated it it was amazing
Shelves: mathematics
lives up to its name. And as the subhead says, this should not be your first introduction to linear algebra.

All quants should read it eventually. (I define a quant as anyone who uses matrices at work.)
Apr 04, 2008 added it
I really like this text. It's a nice complement to Halmos, which is curious as Halmos was Axler's advisor. ...more
May 21, 2015 added it
Tianyao Chen
Mar 29, 2020 rated it it was amazing
Shelves: math
Extremely lucid explanations for the abstract aspects of Linear Algebra including vector space, etc. Read this for machine learning:D
Jan 10, 2020 rated it it was amazing
Shelves: math
There should be a subtitle "Thinking Like an Algebraist." This is linear algebra from a mathematics perspective, not physics or engineering or anything else. Even in math, it doesn't touch on analysis until the final chapter where he defines determinants.

That said, the book is brilliant for its presentation. The order of topics, examples and results makes so many things just "fall out." For example, the proof Cauchy-Hamilton is pretty much "duh" and that of Jordan Canonical form is dead simple.

Kent Sibilev
Mar 28, 2020 rated it it was amazing
A typical linear algebra textbook would start with the introduction of the matrixes and their various factorizations. So you get a feeling that the whole purpose of the linear algebra is mostly in engineering and applied sciences. This incredible book shows the 'algebra' part of the discipline. I would definitely recommend it as a follow up to the standard approach to the subject. ...more
Thanveer Ahamed
Oct 29, 2017 rated it it was amazing
Excellent book! Author's brilliance is reflected in this book. An in-depth vision for mathematics for linear algebra made me fluent in it! You may need 3 months to finish this book. Since I'm a student at a college, it took me 1 year to finish but worth the time. Must read! ...more
Michael Yu
Sep 15, 2018 rated it really liked it
This is a light and elegant overview of the subject , good for those who want to get the concept quickly.
Aug 09, 2019 rated it really liked it
Good book. UChicago should have required linear algebra for physics majors.
Guilherme Ari
Feb 02, 2020 rated it liked it
Decent Linear Algebra Book.
Jul 09, 2020 rated it it was ok
Hw can I read in tis app
Eric Gao
May 05, 2021 rated it it was amazing
Greatest math textbook of all time
Mar 18, 2017 rated it liked it
Shelves: advanced-math
It's a nice book but I really don't understand why it goes to so much trouble to avoid determinants.

From a purely aesthetic viewpoint there is something to be said for the approach, but in my experience teaching linear algebra 101 students don't have misgivings about using determinants. They just don't know enough yet to realized something might be amiss so why bother?
Xi Gong
Jun 13, 2018 rated it it was amazing
A really great book! It exposes me for the first time to the rigorous side of mathematics that I have long craved for.
Andrew Feist
Apr 20, 2015 rated it it was ok
i read it for a class. it's extremely readable and informative, and for what it is trying to do, it does it well. However, as a far as application goes, this book does not really touch on applying linear algebra, which is an extremely useful feild. This is a great approach for those who wish to go on to higher levels of pure math, eg functional analysis where inner vecotr spaces are not Euclidean, but functions.

this is because the book does not primarily deal with Euclidean vectors and matrices
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