This is the 78th book in the American Mathematical Society’s series of Graduate Studies in Mathematics . It is a wonderful book: very accessible and rigorous in the same time, containing basic and not-so-basic facts, discussing many (sometimes unexpected) applications, and with parts suitable to be used in an undergraduate course, or a graduate course, or as research references.
As the author explains in its Preface, “this book is the result of many courses taught over the years…” He tells us his basic motivation: “I wish someone had taught me this material when I was a graduate student.”
The first chapters in the book (Vector spaces; Gaussian elimination; Eigenvalues and eigenvectors; Determinants; Calculating Jordan forms; Normed linear spaces; Inner product spaces and orthogonality) are very suitable for a first course in linear algebra. The remaining chapters (about two-thirds of the book) contain many applications which are not usually found in linear algebra books: the implicit function theorem; Extremal problems; Matrix valued holomorphic functions; Realization theory; Eigenvalue location problems; Zero location problems… just to name a few.
Linear algebra is needed by many students in many fields of study. Given that and the wonderful way this book was written and organized, I think it can be used by many readers: engineering students, mathematics students, research mathematicians, and researchers in any other field where linear algebra is applied. I strongly recommend this book to anyone interested in “working” linear algebra.
Mihaela Poplicher is an associate professor of mathematics at the University of Cincinnati. Her research interests include functional analysis, harmonic analysis, and complex analysis. She is also interested in the teaching of mathematics. Her email address is Mihaela.Poplicher@uc.edu.