Linear Algebra with Applications: two volumes

Linear Algebra is the science of systems of linear equations, related key tools mainly vectors and matrices, and their applications. Since concepts and techniques from linear algebra have common use in many branches of science, introductory courses about it seem necessary for a wide range of students, including high school and university. The book under review is aimed to use in an introductory course. Considering its educational purposes, the book contains many worked and solved examples, and also a remarkable number of exercises, which emphasis on the computational side of the subject. The text is about real linear algebra, based on elementary high school knowledge in algebra, set theory, and geometry. More precisely, concepts from Calculus are not needed. Hence, the book is easy to read for introductory students and their instructors.

Chapter 1 contains techniques and concepts related to systems of linear equations. In Chapter 2, the author studies matrix algebra. This is the key tool used throughout the book.  In Chapter 3, the concepts of determinants and diagonalization are introduced. This concept is used to determine the solvability of systems of linear equations. Chapter 4 introduces vector geometry, which is the geometrical study of vectors and related topics. These chapters cover a one-semester introductory course for beginners and contain several applications of the introduced topics, including network flow, electrical networks, chemical reactions, directed graphs, input-output economic models, Markov chains, linear recurrence, linear dynamical systems, systems of differential equations, and computer graphics. Such a wide range of applications and related topics is remarkable and interesting for anyone who seeks to learn the subject quickly.

Chapters 5-9 contain a second-semester course that covers several topics including vector spaces, linear transforms, orthogonally, change of basis, inner product spaces, and canonical forms. These chapters also consider several applications, including linear codes over finite fields, constrained optimization, and statistical principal component analysis.

Although the book emphasis on the computational examples and exercises, but detailed proof of results, as more as enough examples and exercises with abstract nature, also is given. Hence, the book can be used both for mathematics students focusing on proofs and for applied purposes focusing on applications, which a remarkable number of them addressed in the book. For the exercises, selected solutions are available at the end of the book or in the student solution manual. A complete solution manual is available for instructors. The text of the book is self-contained and therefore is also suitable for self-study. I recommend using this book in, or at least parallel to, an introductory course in linear algebra for instructors and students. The book also is very useful to give fast information about needed topics from linear algebra in other parts of mathematics and sciences, or in several related applications.  

Mehdi Hassani is an Associate Professor in the Department of Mathematics at the University of Zanjan