# Fiches de cours 2018-2019

## Advanced linear algebra II

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.

- Adjoint operator, self-adjoint and normal operators, spectral theorem, singular values.

- 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
Advanced linear algebra II
• 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