  The main purpose of this book is to propose an introduction to the modern
tools of algebraic complexity. To remain as simple as possible while providing
meaningful examples, we chose to focus on effective linear algebra; this is
certainly one of the best possible choices to give an idea of the main problems
in algebraic complexity. The contents of the book are the following: classical
results of linear algebra and basic algorithms in linear algebra, straight-line
programs as a model of computation, with an emphasis on Strassen's method known
as "elimination of divisions", a discussion on various notions of complexity, a
presentation of the general algorithmic strategy "divide and conquer", a first
important example: the fast multiplication of polynomials, the very heart of
the book: the fast multiplication of matrices, with a discussion of derived
fast algorithms for various problems of linear algebra. The particular case of
the computation of the determinant is then discussed, and the last chapter
deals with the difficult computation of the permanent (Valiant's conjecture).
This book gives an introduction to a fascinating subject at a reasonable level.
. . for French-speaking people.
