[Sheldon Axler] Linear Algebra Done Right (3rd edition)(pdf){Zzzzz}

This best-selling textbook for a second course in linear algebra is aimed at undergraduate math majors and graduate students. The novel approach taken here banishes determinants to the end of the book. The text focuses on the central goal of linear algebra: understanding the structure of linear operators on finite-dimensional vector spaces. The author has taken unusual care to motivate concepts and to simplify proofs. 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. The book then deals with linear maps, eigenvalues, and eigenvectors. Inner-product spaces are then introduced, leading to the finite-dimensional spectral theorem and its consequences. Generalized eigenvectors are then used to provide insight into the structure of a linear operator.

Reviews

Altogether, the text is a didactic masterpiece.
Zentralblatt MATH

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 force in the service of simplicity and clarity; these are also well served by the general precision of Axler's prose... The most original linear algebra book to appear in years, it certainly belongs in every undergraduate library.
Choice

The determinant-free proofs are elegant and intuitive.
American Mathematical Monthly

Clarity through examples is emphasized... the text is ideal for class exercises... I congratulate the author and the publisher for a well-produced textbook on linear algebra.
Mathematical Reviews

What's New in the Third Edition

This edition contains 561 exercises, including 337 new exercises that were not in the previous edition. Exercises now appear at the end of each section, rather than at the end of each chapter.

Many new examples have been added to illustrate the key ideas of linear algebra.

Beautiful new formatting, including the use of color, creates pages with an unusually pleasant appearance in both print and electronic versions.

Each theorem now has a descriptive name.

New topics covered in the book include product spaces, quotient spaces, and duality.

Chapter 9 (Operators on Real Vector Spaces) has been completely rewritten to take advantage of simplifications via complexification.

Hundreds of improvements have been made throughout the book. For example, the proof of Jordan Form has been simplified.