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Linear Algebra

L. Shen, H. Wang, J. Wojdylo
Mercury Learning and Information
Publication Date: 
Number of Pages: 
[Reviewed by
Salim Salem
, on
Every STEM student has to learn linear algebra, it is not only a mathematical beauty but, it, also, has a wealth of applications in, nearly, every subject of science. What should a linear algebra course contain? The answer is always a matter of taste. The following topics are typically included in one way or another in every course: Systems of linear equations, matrices, determinants, linear and euclidian spaces and different types of applications that range from solutions of systems of linear equations, to solutions of systems of linear differential equations and many others whose depending on the author’s taste.
The authors of Linear Algebra consider their book to be a reference for a senior course. It is basically their teaching notes of a linear algebra course. The book, which discusses most of the above topics, contains five chapters, an introduction, an index, and a 10 items bibliography. Each chapter is divided into two parts one theoretical and the second devoted to some practical and computational applications.
Like most French books, this book starts by defining vector spaces as the setting of what’s to come. It defines modules over a ring and considers vector spaces as a special case (no properties of modules are given or used in studying vector spaces). This is followed by a summary of the properties of linear combinations, independent vectors, finite dimensions, and subspaces.  It gives coding as an application of subspaces.
The authors then go to explain linear transformations, their relation to matrices and the algebraic structure of the space of linear transformations without discussing its dimension. As application to this topic the reader finds a little introduction to affine and projective transformations.
A full chapter is devoted to linear operators, in which the authors introduce eigenvalues and eigenvectors, diagonalization, and Jordan forms giving necessary and sufficient conditions to write a matrix in one of these forms. The generalized companion matrix ends the theoretical part of this chapter but this assumes that the reader knows quite a few advanced notions like matrix pencil and the relation of its eigenvalues and the determinant. The chapter ends with some applications in linear programming, geometry and graph theory.
The fourth chapter is concerned with inner products and orthogonality.  There one finds the Gram-Schmidt method to construct an orthonormal basis and a detailed discussion of orthogonal projection and the QR matrix decomposition. Image processing is explained as an application of these topics. 
The final chapter is a summary of matrix decompositions where LU, Cholesky, Jordan form and spectral decompositions are discussed and treated as examples.  The chapter ends with some applications of iterative methods.
The list of topics discussed in this book is quite a long one and it needs more space and explanation than is given therefore some topics are just defined, some theorems like Cayley-Hamilton or Vandermonde determinant are cited without a proof (I assume that a senior student should know a proof of both theorems). Yet, the book skips very important topics like infinite dimensional spaces, the exponential of a matrix, and solutions of systems of linear differential equations.
Moreover, the book is full by typing mistakes that make reading it a tedious task and sometimes make definitions and theorems incomprehensible. It is worth noting that some exercises are solved using Mathematica, MatLab, Maple or Sage.  It is nice to see these solutions but readers may not have access to any of these software packages and may not know how to use them. 
Last but not least, though each chapter is nicely divided into well structured sections, the language and style need to be revised so that the book become self-contained and easier to read.

See the publisher's web page.