Linear Algebra Done Right / Edition 2by Sheldon Axler
Pub. Date: 07/18/1997
Publisher: Springer New York
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… See more details below
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.
Table of Contents
|Preface to the Instruction|
|Preface to the Student|
|Ch. 1||Vector Spaces||1|
|Ch. 2||Finite-Dimensional Vector Spaces||21|
|Ch. 3||Linear Maps||37|
|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|
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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.
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.
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.
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.
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!
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.
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.