Linear Algebra Done Right

The fourth edition of this textbook is aimed at undergraduate math majors and graduate students, and it offers an expanded treatment of the singular value decomposition and its consequences. It also includes a new chapter on multilinear algebra and increases the use of the minimal polynomial to provide cleaner proofs of multiple results. The text focuses on the central goal of linear algebra: understanding the structure of linear operators on finite-dimensional vector spaces. It has no prerequisites other than suitable mathematical maturity, and it starts by discussing vector spaces, linear independence, span, basis, and dimension.

Format: Unspecified
Length: 390 pages
Publication date: 10 November 2023
Publisher: Springer International Publishing AG

This highly acclaimed textbook, now available in Open Access, serves as a comprehensive guide for a second course in linear algebra, catering to undergraduate math majors and graduate students alike. The fourth edition has been extensively revised and expanded, offering a thorough exploration of the singular value decomposition (SVD) and its profound implications. It introduces a new chapter dedicated to multilinear algebra, covering topics such as bilinear forms, quadratic forms, tensor products, and an approach to determinants through alternating multilinear forms. Furthermore, the book significantly increases the utilization of the minimal polynomial to provide clearer and more concise proofs of various results. To enhance the learning experience, over 250 new exercises have been added.

In contrast to traditional approaches, this textbook takes a novel approach by placing determinants at the end of the book. Instead, it focuses on the central objective of linear algebra: gaining a deep understanding of the structure of linear operators on finite-dimensional vector spaces. The author has gone to great lengths to motivate concepts and streamline proofs, ensuring a clear and accessible journey for students. Each chapter includes a diverse range of engaging exercises that facilitate students' comprehension and manipulation of linear algebra's objects. The meticulous formatting of the pages creates a visually appealing and student-friendly experience, whether accessed in print or electronic formats.

No prior prerequisites are assumed, except for a basic requirement of suitable mathematical maturity. The text begins by delving into vector spaces, linear independence, span, basis, and dimension. It then proceeds to discuss linear maps, eigenvalues, and eigenvectors, laying the foundation for further developments. Inner-product spaces are introduced, leading to the finite-dimensional spectral theorem and its consequences. Generalized eigenvectors are employed to provide insights into the structure of linear operators, adding depth and complexity to the subject matter.

Throughout the book, reviews from previous editions have praised its excellence. One reviewer described it as a "didactic masterpiece," highlighting the author's ability to convey complex concepts in a clear and concise manner. Another reviewer praised the determinant-free proofs offered, deeming them elegant and intuitive.

In summary, this fourth edition of the bestselling textbook in linear algebra is an indispensable resource for students seeking a comprehensive and engaging introduction to the field. With its expanded coverage of the SVD, multilinear algebra, and new exercises, it equips students with the necessary tools to excel in linear algebra and its applications. Whether you are a math major or a graduate student, this textbook will undoubtedly enhance your understanding and appreciation of the subject.

Dimension: 235 x 155 (mm)
ISBN-13: 9783031410253
Edition number: 4th ed. 2024