Linear Algebra Done Right
OverviewSan Francisco, USA
Now available in Open Access, this best-selling textbook for a second course in linear algebra is aimed at undergraduate math majors and graduate students. The fourth edition gives an expanded treatment of the singular value decomposition and its consequences. It includes a new chapter on multilinear algebra, treating bilinear forms, quadratic forms, tensor products, and an approach to determinants via alternating multilinear forms. This new edition also increases the use of the minimal polynomial to provide cleaner proofs of multiple results. Also, over 250 new exercises have been added.
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 simplify proofs. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra. Beautiful formatting creates pages with an unusually student-friendly appearance in both print and electronic versions.
No prerequisites are assumed other than the usual demand for suitable mathematical maturity. 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 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.
From the reviews of previous editions:
Altogether, the text is a didactic masterpiece. — zbMATH
The determinant-free proofs are elegant and intuitive. — American Mathematical Monthly
The most original linear algebra book to appear in years, it certainly belongs in every undergraduate library — CHOICE
Pages i-xvii
Pages 383-390
“Each section is followed by a set of original exercises with varying levels of difficulty, designed to thoroughly reinforce the concepts presented within that section. This book can be a valuable resource for linear algebra courses where a determinant free approach to the initial development of the subject is deemed pedagogically appropriate. … the textbook offers a rigorous development of linear algebra over R and C with a unique treatment of determinants.” (Peteris Daugulis, Mathematical Reviews, June, 2025)
Authors and AffiliationsSheldon Axler
Sheldon Axler, Professor Emeritus of the Mathematics Department at San Francisco State University, has authored many well-received books including
Axler has served as Editor-in-Chief of the Mathematical Intelligencer and Associate Editor of the American Mathematical Monthly. He has been a member of the Council of the American Mathematical Society and a member of the Board of Trustees of the Mathematical Sciences Research Institute. He is a Fellow of the American Mathematical Society and has been a recipient of numerous grants from the National Science Foundation.
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