Algebra

Duration

30h Th, 20h Pr

Number of credits

Lecturer

Michel Rigo

Language(s) of instruction

French language

Organisation and examination

Teaching in the second semester

Units courses prerequisite and corequisite

Prerequisite or corequisite units are presented within each program

Learning unit contents

This course is dedicated to the study of finite dimensional linear algebra. A large part of the course is devoted to linear maps (kernel, range, theorem of dimension,...), eigenvectors and eigenvalues of an endomorphism, a fine study of diagonalization. The notion of dual of a vector space is introduced. We also consider normal, hermitian and unitary matrices and their applications. Finally, polynomials and rational functions are also studied: Gauss' lemma, the fundamental theorem of algebra, Viète's formulas, the ring of polynomials over an arbitrary field and the corresponding ideals (in particular, the notion of principal ideal domain), Descartes' rule.

Learning outcomes of the learning unit

At the end of this course, the student should have mastered the rigor of mathematical reasoning and a strong ability to grasp abstract structures and concepts arising in linear algebra. He/she will be able to give arguments about his/her assertions. The student will have at his/her disposal a set of deeply understood theoretical results for which he/she will be able to give a proof. He/she will be able to arrange several results from the course to solve an exercise. The student will make use of matrix representations of linear maps, diagonalization (in particular for normal, hermitian and unitary matrices). Moreover, he/she will easily work with polynomial and rational functions (for instance, finding GCD, asymptotic behavior, decomposing into simple functions,...). In particular, the student will be able to adapt the learned techniques to other contexts appearing in mathematics or physical sciences: geometrical loci, extrema of a function of several variables, applying the theory of diagonalization to solve systems of differential equations, Markov chains, in combinatorics (for instance, give estimate on the number of paths of length n in a graph) or in statistics (like in principal components analysis), computing n-th power of a matrix, ...

Prerequisite knowledge and skills

Perfect knowledge from the course "mathématiques générales" (general mathematics) and "geometry" is expected, in particular complex numbers, systems of linear equations, matrix computations including determinants and the notion of rank. The knowledge of vector (or linear) spaces is assumed to be well-known. Being trained to abstraction and mathematical reasoning is an advantage. Corresponding courses : MATH0009-4, MATH0009-5 and MATH0476-1.

Planned learning activities and teaching methods

The practical sessions are mainly dedicated to solve exercises corresponding to the theory considered during the lecture sessions. These sessions are also useful to obtain extra informations or enlightenments on the concepts presented during the lecture sessions.
Moreover, the preparation of lists of exercices for the next practical session will be systematically asked .

Mode of delivery (face-to-face ; distance-learning)

The theoretical lectures are given on a three hours a week basis. The schedule for lectures and practical sessions as well will be communicated at the beginning of the academic year. Theoretical lectures using "blackboard and chalk" interacting with students and recorded using "podcast" (students have later on access to recorded courses). During exercises sessions, students are facing exercises that must be solved.

Recommended or required readings

Lecture notes are available (in french) and can be downloaded from http://www.discmath.ulg.ac.be/

Assessment methods and criteria

The final examination consists of two parts: a written one and an oral one. The written part is essentially devoted to the resolution of problems and exercises. The oral part is devoted to the theory (mainly proofs of theorems) but also includes direct applications of the theory.

Work placement(s)

Organizational remarks

Some useful informations are given on http://www.discmath.ulg.ac.be/ In particular, one has access to the log of the year and also the ones of previous years.

Contacts

M. Rigo
Institut de Mathématique (B37) -
Grande Traverse 12 -
Sart Tilman, 4000 Liège
Tél. : (04) 366.94.87 -
E-mail : M.Rigo@ulg.ac.be

Items online