When I first looked at *Linear Algebra: A Course for Physicists and Engineers*, I was surprised:

- A 450 page book with only six chapters!
- A mathematics book with no theorems!!

In fact, this book is addressed to physicists and engineers who need linear algebra as a tool and who won’t delve deep into its theoretical side.

Unlike most books, the book starts by developing the notion of vectors, building on very useful examples drawn from our usual day to day life. It then goes on to define all the vector operations: addition, multiplication by scalars, and dot products. The authors then introduce the notions of orthogonality, independence, and subspaces.

In the second chapter the authors discuss “Matrices.” They define them and study their most important properties and give practical methods to compute the product of two matrices or the inverse of a matrix. The chapter also contains a discussion of the relation between matrices and linear transformations.

The third chapter is concerned with the determinant of a square matrix. Again the authors choose to define determinants in an inductive manner and using this definition they produce the usual definition and all their fundamental properties. The fourth chapter introduces eigenvectors and eigenvalues, giving their properties and how we compute them and explaining their importance in working with matrices and linear applications.

The last two chapters (about 125 pages) are dedicated to applications of the above mentioned ideas. These applications cover a large number of topics: statistics, systems of differential equations, linear sequences, exponentiation of a matrix, series of matrices etc…

The book is written in a lovely style: it is easy to read, it is self-contained and assumes no mathematical knowledge beyond high school level. It also contains a huge number of examples showing how linear algebra can be used in other mathematical, physical and engineering domains and even in social science.

The authors use nice and easy to learn techniques to explain matrices and determinant computations: they introduce “elementary matrices” and use them to calculate the inverse of a square matrix and to reduce a matrix to its diagonal form whenever this is possible. Similarly, their axiomatic definition of “determinant” renders its computation more comprehensible and easy to follow when the order of the matrix is small.

It is true that the book does not explicitly contain theorems and proofs, but implicitly it is full of them. They are written in a simple language that stays away from the sometimes difficult mathematical jargon. Yet this comes at the expense of less rigor. In the treatment of “linear operator,” for example, it is not clear what the authors mean by “operator.”

All in all, the book is one of the nicest elementary books on linear algebra. I recommend it not only for physicists and engineers but also for all students who need linear algebra as a tool.

Salim Salem is Professor of Mathematics at the Saint-Joseph University of Beirut.