The algebra course has two main objectives: first to introduce the bases of linear algebra and matrix computations which are necessary for their subsequent curriculum as mathematician or physicists, and second to train them in the rigourous expression of intuitive and abstract concepts.
To be able to understand, explain, and illustrate a rigorous proof of linear algebra.
The course contains elements of linear and multilinear algebra. The framework is that of finite dimensional vector spaces and introduces the notions of linear independence, dual vectors space, matrix computations, eigen structure, metric, inner product. It covers unitary and Hermitian operators and projections. These concepts are applied to the ractical solution of systems of linear equations (Gauss, LU, QR and least-squares).
1.Linear forms and dual spaces 2. .Multilinearity and determinants 3. Metric spaces and untary linear applications 4. Hermitian forms 5. Matricial norms 6. . Projections
The exercices are given in parallel with the course : they aim to explaining, illustratring and precising the course.
The exam is oral, in two parts : a) a large topic, prepared 20 minutes, with notes and syllabus, including a complete proof b) a list of small questions to be answered without preparation, on the whole matter.
Syllabus