Product Description

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.

Product Details

• Amazon Sales Rank: #170961 in Books
• Published on: 2004-04-06
• Original language: English
• Number of items: 1
• Binding: Paperback
• 272 pages

Customer Reviews

``Done right'' is right
A brief book on linear algebra that develops the theory by emphasizing vector spaces and linear maps; this leads to clearer, more elegant proofs than the traditional, matrix-based approach. This approach manages to be both more lucid and more abstract.

Among the many fine features of this book are the author's marginal notes highlighting important points, commenting on strategy, and mentioning other names that a concept may go by (e.g., an injective mapping is also known as one-to-one, this is quite useful for beginning students).

Great for a deeper understanding of linear algebra
I picked this book up, after a friend recommended it to me. I have done linear algebra before, probably the same way anyone else has done it at some point; via linear equations, matrix algebra, and a lot of exposure to the ubiquitous, yet strangely unmotivated, determinant. Contrary to many books on linear algebra, this book takes a more abstract look at linear algebra, and it gets to the heart of the subject very quickly. I consider myself somewhat of a purist, and I could therefore rather quickly find myself at home in this book. Axler starts off discussing vector spaces, and linear maps, trying to keep them as abstract and general as possible. This book adopts the standard theorem, proof, example, exercise way of describing mathematics. Something that may seem like a daunting way to learn mathematics to many people. Although I strongly believe that these people miss the point: Mathematics is not about remembering theorems and their proofs from the outside, it is about using theorems and definitions to construct proofs of other theorems, and to ponder these.

The exercises might therefore seem a bit too difficult, but they are certainly not impossible to do. In fact, almost all the exercises in this book build upon material that was previously described, and unless you haven't understood something in the text, they are straightforward to solve. Axler has done a very good job at getting to the core of linear algebra with this book, and I can wholeheartedly recommend it to anyone who considers him or herself a serious mathematician.

Excellent for a second course in Linear Algebra
The book does a better job of explaining what is happening at the heart of linear spaces and linear transformations than most. This is mostly due to the fact that linear maps and operators are used more often that matrices in the proofs, and that determinants are relegated to the end of the book. Overall a very good bare bones, gives you what you need book.