Linear Algebra Done Right | Sheldon Axler | Simple introduction to abstract concepts

Linear Algebra Done Right | Sheldon Axler | A classic introduction for those who like to reason abstractly

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Linear Algebra Don...

	Linear Algebra Done Right Sheldon Axler Springer, 2004 - 272 pages average customer review:based on 37 reviews view larger image
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highly recommended

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 example, the book presents--without having defined determinants--a clean proof that every linear operator on a finite-dimensional complex vector space (or an odd-dimensional real vector space) has an eigenvalue. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra. No prerequisites are assumed other than the usual demand for suitable mathematical maturity. Thus, the text starts by discussing vector spaces, linear independence, span, basis, and dimension. Students are introduced to inner-product spaces in the first half of the book and shortly thereafter to the finite-dimensional spectral theorem. This second edition includes a new section on orthogonal projections and minimization problems. The sections on self-adjoint operators, normal operators, and the spectral theorem have been rewritten. New examples and new exercises have been added, several proofs have been simplified, and hundreds of minor improvements have been made throughout the text.

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Simple introduction to abstract concepts

I read this book a few years ago and it's one of the most enlightening books I've read. What makes it unique is it's way of very simply communicating the basic concepts of vector spaces and linear maps between them. It's one of the few books where you usually immediately realize how to prove each lemma, proposition and theorem before even looking at the proof. Everything just comes out "naturally".

Some people here seem to complain about the lack of breadth in the exposition. However, I see this book as a way of teaching the reader to understand the concepts and proof techniques of abstract linear algebra (of course you won't see how to use determinants in proofs) and it prepares a student very well for a course in functional analysis. Probably a natural continuation would be to read a book focusing on modules. The new edition of Roman's Advanced Linear Algebra seems very good (compared to the 1st edition which was bad) and is probably a good choice.

Those who want formulas and applications should probably take a look at a book titled "engineering mathematics" or maybe Strang's Linear Algebra which is quite practically oriented (and has corresponding video lectures at MIT OCW).

It's true that Axler lacks solutions to exercises, but most are very straightforward. When I read the book I typed up solutions to every exercise in the book. These seem to have spread online last spring through some of the people who I gave them to for self-studying, so I've put my solutions out online. They should be found by googling "solutions axler linear algebra done right" or "fagerholm site:tkk.fi".

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A classic introduction for those who like to reason abstractly

When I was a university student a long time ago our first teacher, Bob Bryce, left a lasting impression on me with a very clean, abstract style of lecturing that I came to expect as the model for good exposition of mathematics. I was disappointed in many cases after that, but this book brings me back to that golden period.
This book is a tribute to the power of abstract mathematics, without being overly abstruse. You have to like abstraction to like this book, -not abstraction for its own sake, but because it solves so many problems in one go. In other words it is a good step on the way to becoming a mathematician.
I was always suspicious of the introduction of determinants as being a hack and the author's approach (of leaving it to the end) has confirmed my feelings on this issue.
The proof that every linear transformation has an eigenvector (in the Complex field) is different to the usual demonstration by determinants and in my opinion more beautiful.

If you want to apply linear algebra, for example if you are a statistician you will need to complement this book with another matrix-oriented book, there are plenty of those. Leave those books for a second course.

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Great text

This was a required book for an undergraduate 3rd year Linear Algebra course for Math majors. I am finding it very relevant and interesting. Most books and courses are geared only for matrices. This book explains most of the concepts without matrices and then introduces determinants, eigenvalues, etc. I found it very interesting and helpful.

better than average but not superb.

this book's approach to linear algebra is not anything unusual; its approach is typical development one sees in algebra books. My main complaint is that this book uses many algebraic notions such as quotient structures and polynomials without enough development of the ideas.

i think it is best to study algebra and learn the linear algebra as a part of algebra not as a separate subject.

despite my complaint and prejudice, i think this book quite good for as an introductory undergraduate text.

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Just a gadget, buy it for fun, not for serious study.

I do not think this book should be complemented by any other books; it is an unnecessary complement to any other books. Buy this for fun, not for a serious study. I just regard this book as a gadget. As the author said, he never had other books beside him when writing this book; I do not think this is the right atitude for writing a book.

reviews: page 1, 2, 3, 4, 5, 6, 7, 8

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