Learning outcomes

The course introduces fundamental notions and tools of linear algebra, before giving their representation and applications to analytic geometry. The elements given here are crucial for many disciplines of mathematics and physics, including functional analysis, differential geometry, classical and quantum mechanics, relativity, field theory, dynamical systems, to name but a few.

Goals

To give fundamental tools in Linear algebra and basic applications in analytic geometry;

Content

Each chapter introduces a specific topic of linear algebra, before giving the representation or application in geometry. For instance, the student will therefore discover the links between vector and affine spaces, duality and covariant coordinates, multinearity and wedge product, determinant and volume of an automorphism, metric and orthogonality, length and its invariance under isometries. One last chapter on vector calculus and the geometry of curves and surface in 3D euclidean space closes the lecture.

Table of contents

  1. Affine and vector spaces (in geometry : lines and planes, parallelism, contravariant coordinates)
  2. Duality in vector and metric spaces (in geometry: covariant coordinates and dual vectors)
  3. Multilinearity: tensor product, permutations, (anti-)symetrisation, wedge product of vectors
  4. Determinant, invariance and volume generated by a linear transform
  5. Hermitic forms (geometry: scalar product, metric, orthogonality, length and angle)
  6. Orthogonal and unitary transforms, isometries
  7. Vector calculus (curves and surfaces in 3-D euclidean space, fundamental forms, curvilinear orthogonal coordinates, gradient, divergence, curl and laplacian).

Exercices

Crucial for developping computational skills, for both mathematicians and physicists.

Teaching methods

A classical course, blackboard and chalk, with many examples of applications in both mathematics and physics to motivate the students and open their minds to more advanced topics they will encounter later on.

Assessment method

Theoretical skills (theory lectures) and computational skills (tutorial sessions) are assessed separately.

The assessment consists of two separate written examinations, covering the theoretical and practical components (problem-solving exercises), respectively.

The theoretical part of the assessment includes recalling definitions, answering questions on understanding and interpretation, establishing connections between the different topics covered in the course, and, of course, proving theorems.

The theoretical examination is closed-book. There is no oral examination.

The final grade is the average of the grades obtained for the two components.

Sources, references and any support material

Lecture notes for theory and exercises.

Language of instruction

French
Training Block Credits Mandatory
Bachelor in Mathematics 1 5 Yes