The concepts and methods of Linear Algebra appear in several branches of mathematics, including advanced calculus, differential equations, analysis, combinatorics, and algebra; it also plays a crucial role in some other branches of science, such as computer science and physics. Apart from its wide applications, the beauty of its concepts and problems usually motivates teachers and students to have a course in linear algebra. In fact, in most universities it is treated as a fundamental course. The present book, as its title indicates, introduces the essential concepts of linear algebra and considers some of its applications.

The topics discussed in this book are standard. The book starts with matrix algebra, dealing with matrices and the operations associated with them. The second chapter is a rather extended study of the square matrices of order two, which is suitable for beginners to become ready for the higher dimensional case. The application of matrices to solve linear systems and the computation of the inverse of a matrix is the subject of third chapter.

The author then studies vector spaces and subspaces, and considers linear maps between them, and also proves Jordan's classification theorem of nilpotent transformations on a vector space with finite dimension. Chapter 7 deals with determinants and their properties, as well as techniques to compute them.

Chapter 8, which is about eigenvalue theory, moves toward a proof of Cayley-Hamilton theorem. To this end, the author applies almost all the topics discussed in previous chapters, together with properties of polynomials. Chapter 9 considers diagonalizable matrices, and ends in a complete proof of Jordan's classification theorem, as well as a clean proof of the Cayley-Hamilton theorem.

In the last chapter, the author discusses on bilinear and quadratic forms, and utilizes them to define Euclidean spaces and to study their main geometric properties. The book ends with an appendix with algebraic prerequisites.

In each chapter, the book gives some of the fundamental information in detail and then several solved problems, mainly related to that concept, are introduced. Many of problems have the flavor of the Mathematics Olympiad, with solutions that require intelligent and surprising tricks. On the other hand, the author sometimes gives some applications of the given subjects to other areas of mathematics. This includes, for example, an application of the powers of square matrices of order two in solving the so-called Pell's equation in Number Theory. Also, at the end of each section various problems for practice are included, without solutions.

I think the book is very suitable for students who are going to prepare for competitions and Olympiads. Moreover, the solved problems make it a very good problem book, with nice ideas, for anybody working with linear algebra. I recommend this book for students, and also for classroom use by teachers.

Mehdi Hassani is a faculty member at the Department of Mathematics, Zanjan University, Iran. His fields of interest are Elementary, Analytic and Probabilistic Number Theory.