The course proposes to learn fundamental concepts of linear algebra and their representations or applications in analytical geometry. The concepts covered in this course are central to many disciplines in mathematics and physics, including functional analysis, classical and quantum mechanics, differential geometry, relativity, dynamical systems, field theory (electromagnetism, etc.) and numerical computation.
The main objective is to establish several essential elementary notions of linear algebra, and its applications in geometry, as well as several central theorems and results for the rest of the curriculum.
The course successively addresses the following fundamental notions of linear algebra: vector spaces, duality, multilinearity, determinant, Hermitian forms, unitarity. Each chapter begins with the algebraic structure before giving a representation or an application in geometry (affine spaces, lines and planes, parallelism, contravariant and covariant coordinates, tensors, vector product, volume, orthogonality, length, etc.). A last chapter on vector analysis in 3- dimensional Euclidean space closes the course by mixing several concepts seen previously.
Oral examination on the theoretical part, including the restitution of definitions and the demonstration of theorems or important results of the course. A list will be drawn up at the end of the course to facilitate the study. The theoretical exam is not open ended. Written examination of exercises, aiming at assessing computational skills (determinant calculus, exercises in analytical geometry, linear algebra and vector analysis).
Two syllabi are available at the reproduction service, one for the theoretical course, the other for the exercises.