About the Book
Linear algebra is a fundamental tool in many fields, including mathematics and statistics, computer science, economics, and the physical and biological sciences. This undergraduate textbook offers a complete second course in linear algebra, tailored to help students transition from basic theory to advanced topics and applications. Concise chapters promote a focused progression through essential ideas, and contain many examples and illustrative graphics. In addition, each chapter contains a bullet list summarising important concepts, and the book includes over 600 exercises to aid the reader's understanding. Topics are derived and discussed in detail, including the singular value decomposition, the Jordan canonical form, the spectral theorem, the QR factorization, normal matrices, Hermitian matrices (of interest to physics students), and positive definite matrices (of interest to statistics students).
Table of Contents:
Preliminaries; 1. Vector spaces; 2. Bases and similarity; 3. Block matrices; 4. Inner product spaces; 5. Orthonormal vectors; 6. Unitary matrices; 7. Orthogonal complements and orthogonal projections; 8. Eigenvalues, eigenvectors, and geometric multiplicity; 9. The characteristic polynomial and algebraic multiplicity; 10. Unitary triangularization and block diagonalization; 11. Jordan canonical form; 12. Normal matrices and the spectral theorem; 13. Positive semidefinite matrices; 14. The singular value and polar decompositions; 15. Singular values and the spectral norm; 16. Interlacing and inertia; Appendix A. Complex numbers.
About the Author :
Stephan Ramon Garcia is W. M. Keck Distinguished Service Professor and Professor of Mathematics at Pomona College, California. He is the author of two books and over seventy research articles in operator theory, complex analysis, matrix analysis, number theory, discrete geometry, and other fields. He is on the editorial boards of the Proceedings of the American Mathematical Society, Involve, and the American Mathematical Monthly. He received three National Science Foundation (NSF) research grants as principal investigator, five teaching awards from three different institutions, and was twice nominated by Pomona College for the Council for Advancement and Support of Education (CASE) US Professors of the Year award. Roger A. Horn was Research Professor of Mathematics at the University of Utah. His publications include Matrix Analysis, 2nd edition (Cambridge, 2012) and Topics in Matrix Analysis (with Charles A. Johnson, Cambridge, 1991), as well as more than 100 research articles in matrix analysis, statistics, health services research, complex variables, probability, differential geometry, and analytic number theory. He was Editor of The American Mathematical Monthly (1996–2001), the Mathematics Association of America Spectrum book series (1992–5), and the MAA Carus Mathematical Monographs (2002–5). He has also served on the editorial boards of the SIAM Journal of Matrix Analysis, Linear Algebra and its Applications, and the Electronic Journal of Linear Algebra.
Review :
'A Second Course in Linear Algebra by Garcia and Horn is a brilliant piece of work in linear algebra and matrix theory; it is a fascinating text for advanced undergraduate students and graduate students in science, technology, engineering and mathematics (STEM). Covering not only standard material but also many interesting topics, it presents the most fundamental and beautiful ideas in the field with cutting edge applications (such as the singular value decomposition with the Mona Lisa image). The book will become a standard text that is interesting and enjoyable for students as well as researchers.' Fuzhen Zhang, Nova Southeastern University, Florida
'A Second Course in Linear Algebra by Garcia and Horn is an excellent addition to the texts available to instructors of advanced linear algebra courses. The authors have made all the right choices to create a very modern and useful text, for example, the emphasis on block matrices, matrix factorizations, and unitary matrices. The wide-ranging and varied collection of topics provides options to instructors, in addition to covering all the necessities. The book is readable and students will find the lists of important concepts at the end of each chapter useful.' Leslie Hogben, Iowa State University
'Garcia and Horn provide a tasteful mix of linear algebra and matrix theory that covers a wide range of 'must know' material. The book occupies the sparsely populated area between introductory linear algebra texts and research level books such as Horn and Johnson's Matrix Analysis. With its stylish presentation, rich set of exercises, and thorough index, there is no better book from which to learn this body of material.' Nicholas J. Higham, University of Manchester
'This exciting, modern text tells a story that speaks to students of core and applied mathematics as well as many domain sciences. The authors excel at welcoming readers to the subject, with captivating, crystal-clear exposition on every page. The overall composition is uncommonly thoughtful, featuring attractive visuals, energetic and to the point prose, and foundations that pay attention to applications from a traditional, continuous origin as well as the discrete origin of contemporary data analysis. This is a wonderful textbook, and a serious work of reference for experienced researchers.' Ilse Ipsen, North Carolina State University
'… if you are shopping around for a text for a second course in linear algebra, and your idea of a syllabus aligns with those of the authors (matrix oriented, avoidance of reliance on abstract algebra, not much in the way of applications) then this text should definitely be on your definite short list. Even if you're not teaching such a course, this book is worth a look as a general reference for matrix theory. It's a very valuable addition to the literature, and is highly recommended.' Mark Hunacek, MAA Reviews
'This is an impressive book for a second course in linear algebra, and there is much to learn from the excellent exposition. It is also a good reference text on matrix theory; incidentally, the second author is also co-author of the well-received monographs on matrix analysis.' Peter Shiu, The Mathematical Gazette
'The exposition is lively, consistently and uniformly clear, skillfully motivated, well researched, and beautifully illustrated. Nontrivial applications and examples are plentiful, concise, appropriately detailed, and timely (involving, e.g., image resolution and other 'best approximations' of large data sets, linkages between algebra and geometry, and so forth), all of which make the book both a well-honed learning tool and a useful reference.' Russell Merris, The American Mathematical Monthly
'… I highly recommend this book. It is an excellent textbook for a second course in linear algebra. It is also useful for self-study. And it is particularly valuable for anyone beginning research in any area that heavily uses linear algebra, not only because it covers all of the important results and techniques, but also because it teaches you to think like a matrix theorist.' IMAGE: Bulletin of the International Linear Algebra Society
