Sheldon Axler
Springer; 2nd edition | 1997 | ISBN-10: 0387982590 | 251 pages | PDF | 201 Mb
This text for a second course in linear algebra is aimed at math majors and graduate students. The approach is novel, banishing determinants to the end of the book and focusing 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. Though this text is intended for a second course in linear algebra - following one that focuses on matrices and computation - there are no prerequisites other than appropriate mathematical maturity. Thus 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. FROM THE REVIEWS: 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."
