Fiches de cours

Viazovska Maryna

English

Summary

The purpose of this course is to introduce the basic notions of linear algebra and to prove rigorously the main results of the subject.

Content

- Linear forms, dual space, bilinear forms, sesquilinear forms, symmetric and hermitian matrices, Sylvester's theorem.

- Inner products: orthonormal bases, orthogonal projections, orthogonal and unitary matrices.

- Systems of linear differential equations with constant coefficients.

-Basics of multilinear algebra

Keywords

inner product, bilinearity, orthogonality, scalar product, spectral theorem

Linear algebra I

Learning Outcomes

By the end of the course, the student must be able to:
• Give an example to illustrate the basic concepts of the course
• State all definitions and theorems from the course
• Reconstruct proofs from the course
• Apply techniques from the course to various problems in mathematics and physics
• Compute basechange for linear maps, bilinear forms, sesquilinear forms; Gram matrix of a bilinear or sesquilinear form, Sylvester basis for a symmetric form, orthonormal basis for a given symmetric or symplectic form, orthogonal projection on a vector subspace, singular values of a linear map, Jordan normal form of a matrix, exponential of a matrix.
• Formulate main ideas of the course
• Synthesize major results of the course to give a `big picture' of the material and its potential applications
• Create new proof of correct statements in linear algebra
• Design counterexamples for wrong statements in linear algebra

Transversal skills

• Use a work methodology appropriate to the task.
• Assess one's own level of skill acquisition, and plan their on-going learning goals.
• Continue to work through difficulties or initial failure to find optimal solutions.
• Access and evaluate appropriate sources of information.

Teaching methods

Ex cathedra course, exercises in classroom

Expected student activities

Understanding the course notes, solving the exercices

Written exam

Supervision

 Office hours Yes Assistants Yes Forum No

Resources

Bibliography

- R. Cairoli, Algèbre linéaire, PressesPolytechniques Universitaires Romandes, 2e édition 1999.

- K. Hoffman, R. Kunze, Linear Algebra,Prentice-Hall, second edition, 1971.

- R. Dalang, A. Chabouni, Algèbre linéaire, PressesPolytechniques Universitaires Romandes, 2e édition, 2004.

Dans les plans d'études

• Semestre
Printemps
• Forme de l'examen
Ecrit
• Coefficient
6
• Matière examinée
• Cours
3 Heure(s) hebdo x 14 semaines
• Exercices
3 Heure(s) hebdo x 14 semaines

Semaine de référence

LuMaMeJeVe
8-9
9-10
10-11
11-12
12-13
13-14
14-15
15-16
16-17
17-18
18-19
19-20
20-21
21-22
En construction

Cours
Exercice, TP
Projet, autre

légende

• Semestre d'automne
• Session d'hiver
• Semestre de printemps
• Session d'été
• Cours en français
• Cours en anglais
• Cours en allemand