Linear Algebra Done Right (Undergraduate Texts in Mathematics) [Paperback] Author: Amazon Prime | Language: English | ISBN:
0387982582 | Format: PDF, EPUB
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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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- Series: Undergraduate Texts in Mathematics
- Paperback: 252 pages
- Publisher: Springer; 2nd edition (April 6, 2004)
- Language: English
- ISBN-10: 0387982582
- ISBN-13: 978-0387982588
- Product Dimensions: 9.2 x 7.7 x 0.6 inches
- Shipping Weight: 1.2 pounds (View shipping rates and policies)
I have used this text for a beginning graduate course in linear algebra, mostly because I prefer its treatment of eigenvalues and eigenvectors over Hoffman and Kunze, and it sticks to the basics: complex scalars. It also has a good treatment of inner product spaces. The basic concepts and theorems are indeed presented cleanly and elegantly. Its use of linearly independent sequences (rather than sets) is a little nonstandard (what if the set of vectors is infinite?) but the adjustment is minor. Two things though I found treated in a less than desirable fashion: He pretends that we don't know about matrices, doesn't want to develop the machinery, and the treatment of coordinate vectors and matrix representations suffers. Students also get no sense of how to compute the solution of concrete vector space problems, which is easily done once the theory is established, and which is an essential skill to have after a second course in linear algebra. I have to give them supplementary notes. Second, the treatment of determinants suffers, apparently for ideological/political reasons. I think students deserve a straightforward development of determinants simply because that theory is widely used in applications, in engineering, and in discrete mathematics, and it has its own beauty. It is not hard to do, and I do it myself from notes, adapted from the treatment of Hoffman and Kunze. Now that undergraduate linear algebra courses have in many places dropped any substantial theorem-proving component, students need a serious course in linear algebra which can take them, e.g. all the way into Jordan form. There are not many good books for this, and this text does a good job with the basics without overkill on the abstraction, so I use it despite the drawbacks mentioned above.
By Bob
I was very much the typical person in the target audience of this. I was a computer science major and I had a semester of linear algebra where all I learned how to solve Ax = b. Then, I happened to pick up Axler one winter evening because the title looked intriguiging. That day changed my life.
Now, I'm a pure math major and Axler is the reason. The exposition is clean and very elegant. By minimizing the use of matrices in his proofs, he presents the subject of linear algebra as an elegant piece of mathematics rather than a subject "spoilt" by applications. He starts with a study of vector spaces and then moves onto transformations, eigenvalues, inner product spaces, etc. all the way upto the jordan form. All along, the use of matrices in minimal. In fact, he introduces them quite late in the book just as a convenient notation and nothing else. This is an admirable aspect because it simplies a lot of the proofs. The proof that every linear operator over a finite-dimensional vector space has an eigenvalue is breathtakingly short and simple. He uses determinants in the last chapter of the book and there too, does an excellent job. (although the point of writing this book was NOT to use determinants, his exposition about determinants is itself one of the best ones I've seen).
Get this book if you wish to understand the theory. It's a typical higher level math text - definition, theorem, proof, exercises (most of which are theorems). If you like math, you won't regret this.
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