Mathématiques 2

Course content

There are two parts.

Part 1 (Q1):
1) Basic probability concepts (probability space definition, elementary, uniform and conditional probability, independence);
2) Random variables (definition, discrete r.v., continuous r.v., expected value, variance);
3) Random models (particular cases of discrete and continuous random variables);
4) Multivariate models (joint probability, independence, expected value and moments in the multivariate case, conditional expectation, law of iterated moments);
5) Limit theorems (law of large numbers, central limit theorem).

Part 2 (Q2):

1) Number spaces and matrices (linear combinations, matrix factorisations);
2) Vector spaces and linear applications (linear independence, basis, kernel, column space, change of basis, linear applications);
3) Euclidian spaces (distance and scalar product, clustering, orthonormal basis, Gram-Schmidt procedure, matrix transpose and symmetric matrices);
4) Least squares (motivation and definition , solution to the least squares problem, curve fitting);
5) Eigenvalues (motivation, definition, computation, properties, commuting matrices, Rayleigh quotient, applications);
6) The singular value decomposition (definition, properties, best rank k approximation, applications);
7) Sequences (definition, examples, convergence, recurrence, fixed points);
8) Series (definition, examples, power series, exponential, geometric and binomial series, practical manipulation of power series);

Objectives (and/or specific learning outcomes)

Part 1 (Q1):
At the end of this teaching unit, a student will be able to :
1) calculate probabilities in a discrete and continuous univariate environment;
2) use Bayes theorem, law of total probabilities to calculate probabilities of dependent sets;
3) calculate the expected value and variance for discrete and continuous random variables;
4) assign a probability law to a real world random event, and calculate their associated probabilities and moments;
5) calculate probabilities in a discrete and continuous multivariate environment;
6) calculate the expected value and variance for discrete and continuous multivariate random variables;
7) use the of iterated expectations (tower rule) and apply it to stochastic sums;
8) identify where the usage of the law of large numbers and central limit theorem is relevant.

Part 2 (Q2):
At the end of this teaching unit, a student will be able to :
1) understand the use of number spaces in science and engineering;
2) verify linear independence and compute a basis of the fundamental subspaces of a matrix;
3) understand the use of a distance in Rⁿ, e.g. for data clustering;
4) compute a least solution to an incompatible system and to apply this to curve fitting;
5) compute eigenvalues and eigenvectors of a small matrix;
6) understand the use of singular values in science and engineering;
7) determine convergence or divergence of a sequence;
8) use power series;

Prerequisites and Corequisites

Required and Corequired knowledge and skills

Part 1 & 2 (Q1 & Q2): General mathematics (fractions, real numbers, functions, derivatives, integrals)

Required and corequired courses

Courses requiring this course

Teaching methods and learning activities

Theoretical courses and exercises.

Contribution to the teaching profile

(only available in French)
1. Acquérir un savoir et faire preuve de polyvalence dans le domaine des sciences
1.1. S’approprier et maitriser les concepts fondamentaux en biologie ainsi que les bases nécessaires en chimie, physique et mathématique.
1.3. Analyser, synthétiser et relier les connaissances
1.4. Adopter un raisonnement logique et structuré pour résoudre un problème, réel ou fictif, en utilisant des savoirs et des savoir-faire acquis pendant la formation
1.5. Assimiler rapidement de nouveaux concepts
1.6. Utiliser un langage précis et spécifique au domaine
2. Adopter et Maîtriser une démarche scientifique
2.5. Mettre en oeuvre un protocole : savoir observer, mesurer et analyser des données.

References, bibliography, and recommended reading

Part 1 (Q1): syllabus for sale at PUB under the course code MATHF315 and available as pdf on UV (moodle).
Part 2 (Q2): syllabus for sale at PUB and available as pdf on UV (moodle).

Course notes

Syllabus
Université virtuelle

Other information

Contacts

Part 1 (Q1): Jennifer Alonso García, mail: jennifer.alonso.garcia@ulb.be, bureau: campus Plaine, bâtiment NO, local 2.O9.116
Part 2 (Q2): Ignace Loris, mail/Teams: Ignace.Loris@ulb.be , bureau: campus Plaine, bâtiment NO, local 2.O7.107

Campus

Plaine

Evaluation

Method(s) of evaluation

written examination

written examination

Open question with short answer
Open question with developed answer
Closed question with multiple choices (MCQ)
Visual question
Closed question True or False (T/F)

Part 1 (Q1): a written exam in January. In some rare special cases (force majeure, open session, ...) the written exam may be replaced by an oral exam.
Part 2 (Q2): a written exam in June. In some rare special cases (force majeure, open session, ...) the written exam may be replaced by an oral exam.

Mark calculation method (including weighting of intermediary marks)

Part 1 (Q1) and part 2 (Q2) count for 50% each of the final mark.

Partial marks >=10/20 of 2023-24 will be transferred to 2024-25.

Language(s) of evaluation

french