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Linear Algebra Done Right

4.27  ·  Rating details ·  704 ratings  ·  39 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!


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Average rating 4.27  · 
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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...


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
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 the
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
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 solutions thoug ...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.
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.
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!
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.)
Diego Gomez
Jul 18, 2018 rated it it was amazing
Excellent introduction to linear algebra and abstract mathematics in general.
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.
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.
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)
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
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 a
Dec 22, 2015 rated it really liked it
Shelves: math-engineering
The material started with Vector Spaces, which makes the most sense to me for linear algebra. The text focus was more on "what is going on" and less on rigorous mathematics, making it a good reference for use with books that are more proof-oriented.

The problem with linear algebra is its inherit difficulty: It is a subject that is extremely hard to really understand. Despite much study, and a grad class, it still has me well into the level of very limited abilities. Best thing? Press
Apoorv Vikram Singh
Jul 30, 2016 rated it it was amazing
An exceptional book on Linear Algebra. Preferably a good read after some basic linear algebra course, so that you can actually appreciate the linear algebra in the book, and also have a mindfuck moment, when you relate it to your existing knowledge of linear algebra.
Aug 17, 2015 added it
Wonderful!!!! No determinants til the end. Emphasis on concept over formal, so rare in Lin alg. Weak on determinants - Artin or Halmos, Finite Dimensional Vector Spaces, are better. The only reason I understand Graham-Schmidt conceptually and not as an algorithm to be memorized!!!
Hannes Salin
Oct 01, 2013 rated it really liked it
Gave me a different perspective on linear algebra, interesting.
May 31, 2010 rated it liked it
For math majors. But actually pretty useful for learning the notation, since it's built from the ground up.
May 21, 2015 added it
Apr 07, 2016 rated it it was amazing  ·  review of another edition
Shelves: math, favorites
The third edition is an excellent update. Previously the book was good, but the formatting and figures were subpar. The third edition rectifies this, and it makes it much more of a joy to read.
May 13, 2014 rated it really liked it
Assuming you already are educated in linear algebra, this is a great explanation of the intuition behind what you've learned to calculate with determinants.
Jun 16, 2018 rated it it was amazing
a great book for self-learning with reasonable and very beautiful proofs. 5 stars!
Jan 14, 2016 rated it it was amazing  ·  review of another edition
Shelves: mathematics
A refreshing abstract approach to the subject (banishing determinants to the final chapter), yet throughly readable with very slick proofs. I would recommend it as a second book on LA.
Jan 30, 2012 rated it really liked it
Shelves: mathematics
I really wish it worked out examples - having the student do the examples themselves is silly. How will they know how to do it in the first place if there's no example? Other than that, I like it.
Sep 09, 2012 marked it as to-read
I think I need to study this book. Imagine: presenting Linear Algebra algebraically!!!
Peteris Erins
Jan 17, 2012 rated it liked it
Not particularly enlightening. Unassuming yet thorough coverage of finite vector spaces with only a brief mention of the infinite.
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