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.

Assessment method

The evaluation is twofold. One oral exam on theory, without notes, including notably definitions and demonstrations of important results. One written exam based on computation skills obtained from the exercises.

Sources, references and any support material

Lecture notes for theory and exercises.

Language of instruction

Français
Training Study programme Block Credits Mandatory
Bachelor in Mathematics Standard 0 5
Bachelor in Mathematics Standard 1 5