This book is a clean and concise introduction to the basic theory of matrices. It also contains many interesting applications of matrices to different aspects of mathematics and also other areas of science and engineering. With forty percent new material, this second edition is significantly different from the first edition. The book mixes together algebra, analysis, complexity theory and numerical analysis. As such, this book will provide many scientists, not just mathematicians, with a useful and reliable reference. It is intended for advanced undergraduate and graduate students with either applied or theoretical goals. This book is based on a course given by the author at the École Normale Supérieure de Lyon.

Based on the author's lectures, this book introduces nonstandard analysis as a radically different way of viewing mathematical concepts and constructions, a source of new ideas, objects and proofs and a wellspring of powerful new principles of reasoning.

A modern introduction to three areas of number theory: quadratic forms, Dirichlet's density theorem and modular forms. "... Accessible to graduate or even undergraduate students, yet even the advanced mathematician will enjoy reading it." - American Scientist.