Linear Algebra Done Right / Edition 2

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Overview

This text for a second course in linear algebra, aimed at math majors and graduates, adopts a novel approach by banishing determinants to the end of the book and focusing on 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 has an eigenvalue. The book starts by discussing vector spaces, linear independence, span, basics, 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. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra. This second edition features new chapters on diagonal matrices, on linear functionals and adjoints, and on the spectral theorem; some sections, such as those on self-adjoint and normal operators, have been entirely rewritten; and hundreds of minor improvements have been made throughout the text.

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Editorial Reviews

From the Publisher

From the reviews:

ZENTRALBLATT MATH

"This second edition of an almost determinant-free, none the less remarkably far-reaching and didactically masterly undergraduate text on linear algebra has undergone some substantial improvements. First of all, the sections on selfadjoint operators, normal operators, and the spectral theorem have been rewritten, methodically rearranged, and thus evidently simplified. Secondly, the section on orthogonal projections on inner-product spaces has been extended by taking up the application to minimization problems in geometry and analysis. Furthermore, several proofs have been simplified, and incidentally made more general and elegant (e.g., the proof of the trigonalizability of operators on finite-dimensional complex vector spaces, or the proof of the existence of a Jordan normal form for a nilpotent operator). Finally, apart from many other minor improvements and corrections throughout the entire text, several new examples and new exercises have been worked in. However, no mitigation has been granted to determinants. Altogether, with the present second edition of his text, the author has succeeded to make this an even better book."

AMERICAN MATHEMATICAL MONTHLY
"The determinant-free proofs are elegant and intuitive."

CHOICE
"Every discipline of higher mathematics evinces the profound importance of linear algebra in some way, either for the power derived from its techniques or the inspiration offered by its concepts. Axler demotes determinants (usually quite a central technique in the finite dimensional setting, though marginal in infinite dimensions) to a minor role. To so consistently do without determinants constitutes a tour de forces in the service of simplicity and clarity; these are also well served by the general precision of Axler’s prose. Students with a view towards applied mathematics, analysis, or operator theory will be well served. The most original linear algebra book to appear in years, it certainly belongs in every undergraduate library."

ZENTRALBLATT MATH
"Altogether, the text is a didactic masterpiece."

From the reviews of the second edition:

S. Axler

Linear Algebra Done Right

"The most original linear algebra book to appear in years, it certainly belongs in every undergraduate library."—CHOICE

"A didactic masterpiece."—ZENTRALBLATT MATH

“This book can be thought of as a very pure-math version of linear algebra … . it focuses on linear operators, primarily in finite-dimensional spaces … . Axler has come up with some very slick proofs of things that … makes the book interesting for mathematicians. The book is also very clearly written and fairly leisurely. … Axler concentrates on the properties of linear operators, and doesn’t introduce other concepts unless they’re really necessary.” (Allen Stenger, The Mathematical Association of America, December, 2010)

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Product Details

  • ISBN-13: 9780387982588
  • Publisher: Springer New York
  • Publication date: 7/18/1997
  • Series: Undergraduate Texts in Mathematics Series
  • Edition description: 2nd ed. 1997. Corr. 7th printing 2004
  • Edition number: 2
  • Pages: 251
  • Sales rank: 236,419
  • Product dimensions: 0.58 (w) x 7.50 (h) x 9.25 (d)

Table of Contents

Preface to the Instruction
Preface to the Student
Acknowledgments
Ch. 1 Vector Spaces 1
Ch. 2 Finite-Dimensional Vector Spaces 21
Ch. 3 Linear Maps 37
Ch. 4 Polynomials 63
Ch. 5 Eigenvalues and Eigenvectors 75
Ch. 6 Inner-Product Spaces 97
Ch. 7 Operators on Inner-Product Spaces 127
Ch. 8 Operators on Complex Vector Spaces 163
Ch. 9 Operators on Real Vectors Spaces 193
Ch. 10 Trace and Determinant 213
Symbol Index 247
Index 249
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Customer Reviews

Average Rating 3.5
( 8 )
Rating Distribution

5 Star

(4)

4 Star

(1)

3 Star

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2 Star

(2)

1 Star

(1)

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Sort by: Showing all of 8 Customer Reviews
  • Anonymous

    Posted April 23, 2000

    A Terrific Book for a Second Taste of Linear Algebra

    We used this as a textbook for the first quarter of a year long abstract algebra course. I can't recommend it enough. The proofs are clean, clear and beautiful. The text is uncluttered by unnecessary examples. Although many engineers interested primarily in computation will not likely enjoy this elegant work, anyone with good taste in mathematics will. Particularly beautiful and revealing is Axler's proof that every operator on a finite-dimensional, nonzero complex vector space has an eigenvalue--a brief proof completed without the use of determinants! This and so many other proofs throughout the book make linear algebra seem as simple and beautiful as it really is.

    3 out of 3 people found this review helpful.

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  • Anonymous

    Posted April 30, 2003

    Linear Algebra made Exciting

    This is a spellbinding book on linear algebra. Unlike the mechanistic thinking encouraged by the matrix-based approach, I found the direct view of vector spaces delightfully refreshing. Visualizing operators purely as mappings without the matrix-vector multipication 'crutch', provided an insightful understanding, that kept me enthralled. I read the entire book in a B&N store, often gasping alound at the simplicity and beauty of the proofs of several nontrivial results. By the way I bought the book to keep! Im an engineer and have used linear algebra matrix theory, but found this exposition deepened my understanding, while being a terrific read. Why not 5 stars? I did not want it to end!!Professor, plese write another book like this with more advanced topics.

    2 out of 2 people found this review helpful.

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  • Anonymous

    Posted December 27, 2002

    An elegant introduction to linear algebra from the *right* perspective

    As suggested by the wide spectrum of opinions on it, this is a book written from one perspective and one perspective only -- what many mathematicians consider the *right* one -- and therefore won't satisfy some engineering students who can't read through a math book that doesn't have concrete, carefully worked out examples, and nice little diagrams. But it seems awfully unfair to call it horrible just because you're used to linear algebra texts that all but spoon-feed you how to do a bunch of problems on matrices algebra. Most mathematics students, however, will agree that this indeed is the more elegant and intuitively-satisfying approach to writing a linear algebra textbook than the way the conventional lower-division linear algebra texts were written. The theory is built from a very intuitive and physically relevant concept of vector spaces, not some artificially concocted mathematical tools like matrices, which indeed arose from the need to describe linear maps on these vector spaces. If you're new to the subject or do not yet possess the mathematical maturity that most junior-level undergraduate math students have, you won't have nearly as much fun reading this book as the rest of us. (I recommend Howard Anton's Elementary Linear Algebra, which has all those nice examples and cute diagrams, if you're indeed not equipped well enough to appreciate the beauty of this textbook.) But if you care about understanding and appreciating linear algebra as a theory in itself, and not just as a tool, then by all means purchase this fine work by Professor Axler. Trust me, you will have more fun than you are allowed to have as an engineering student taking a math class.

    1 out of 2 people found this review helpful.

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  • Anonymous

    Posted January 7, 2002

    one of the best

    This a nice and straightforward book on linear algebra. If I was the author, I would not call it Linear Algebra Done Right (it does sound bad). Some people are obviously not happy with this book, but they need to realize that linear algebra is not a study of matrices; matrices only serve as a tool in some problems of linear algebra. Overall, highly recommended to math majors; not recommended to engineering majors!

    1 out of 2 people found this review helpful.

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  • Anonymous

    Posted April 14, 2000

    Loved It

    I think the one-star reviewer here is missing the point of the book. Matrices are indeed an important part of linear algebra, but what is much more important is a fundamental understanding of what a linear space is all about. This book communicates that essence in a clear and original way. I found it much easier to learn from than the books I had in college (that did things via determinants, matrices, etc). I'm a 3D game programmer so linear algebra is important to me. And this is a great book.

    1 out of 1 people found this review helpful.

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  • Anonymous

    Posted December 11, 2001

    The title gives a false hype to the text

    After the semester was over I realized how poorly the text was put together. I ended up having to read out of two other books from the library and aided with Schaum's Outline for Linear Algebra, which I strongly recommend if you are stuck having to use this text for class. There were not enough problems worked out to gain a conceptual idea of the theory that is involved. Even those problems that are given have no answers so there is no way to judge your own answer to the questions. There is no beginning explanation of matrices and when they are introduced it is in a format that proves their validity but still gives no concrete examples of their use. This would definitely not be recommended if no class on matrix theory has been taken or possibly some previous work with the theory of linear algebra. The author understands the subject but he doesn't present it so that others can share his understanding.

    0 out of 2 people found this review helpful.

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  • Anonymous

    Posted March 29, 2000

    It Should be Called 'Linear Algebra Done Wrong'

    In a word: horrible. The author takes a totally wrong approach to linear algebra. He introduces matrices towards the end of the book, and spends about a page on them. The book is written in an essay format, with very few examples and worked-out problems. A better choice would be Bernard Kolman's Elementary Linear Algebra.

    0 out of 5 people found this review helpful.

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  • Anonymous

    Posted January 22, 2011

    No text was provided for this review.

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