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Finite-dimensional linear algebra. (English) Zbl 1202.15002

Discrete Mathematics and its Applications. Boca Raton, FL: CRC Press (ISBN 978-1-4398-1563-2/hbk). xxi, 650 p. (2010).
In the preface the author says: “The purpose of this book is to provide a broad and solid foundation for the study of advanced mathematics. A secondary aim is to introduce the reader to many of the interesting applications of linear algebra.” This book is clearly intended for the serious student wishing to have a sound pure-mathematical treatment of linear algebra as well as of non-trivial applications and ancillary subjects. It thus assumes that the student already knows how to solve systems of linear equations and is familiar with the basics of calculus, as well as being used to the formal treatment of pure mathematics. The approach is axiomatic and the proofs are purely algebraic and do not make use of systems of linear equations, as is the case in all “quick and easy” approaches to linear algebra. Vector subspaces, spanning sets, linear dependence, basis and dimension are thus introduced cleanly and rigorously. On the other hand, there is not only theory: the book has many numerical worked examples and – as mentioned above – introduces the student to a host of ancillary applications with numerical examples. Each section contains exercises ranging from easy to challenging, with some “essential” ones containing further theoretical results needed in the text.
Chapter 1 is an optional, concise, non-formal exposition of three ways that linear algebra shows up: linear equations, best approximation and diagonalization (decoupling variables). Chapters 2, 3, 4 and 6 form the core of the book. They treat: Fields and Vector Spaces, Linear Operators, Determinants and Eigenvalues, and Orthogonality and Best Approximation. Special topics are polynomial interpolation and Lagrange basis, approximation of real-valued functions by polynomials, Newton’s method, systems of linear ordinary differential equations, graph theory, coding theory, linear programming and integer programming, the finite element method and Galerkin’s method, Gaussian quadrature and the Helmholtz decomposition. Chapter 5 treats the Jordan canonical form along with the matrix exponential and graphs and eigenvalues, Chapter 7 the spectral theory of symmetric matrices along with optimization and the Hessian and Lagrange multipliers, Chapter 8 singular value decomposition along with the Smith normal form, Chapter 9 matrix factorizations and numerical linear algebra, and Chapter 10 vector analysis along with some Banach and Hilbert space theory. Each chapter begins with the why’s and wherefores of the coming material.
The book is written in an easily readable style. This, together with its in-depth treatment of linear algebra and wide-ranging introduction to applications, makes it a very valuable and useful textbook and reference source which the serious student will enjoy studying and referring to for many years.

MSC:

15-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra
65Fxx Numerical linear algebra
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
41A55 Approximate quadratures
90C05 Linear programming
34A30 Linear ordinary differential equations and systems
65D05 Numerical interpolation
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