Objectives of Programme's Learning Outcomes

• Understand "elementary" mathematics profoundly
• Use vector spaces, linear applications and the techniques associated with them
• Understand and use the naive set theory
• Understand basic algebraic structures
• Manipulate previously acquired knowledge that appears in a question
• Give examples and counterexamples for definitions, properties, theorems, etc.
• Understand and produce strict mathematical reasoning
• Write clearly and concisely
• Use mathematical vocabulary and formalism appropriately
• Make sense of formal expressions
• Rely on a picture to illustrate a concept, rationale, etc.
• Solve new problems
• Abstract and manipulate theories and use these to solve problems
• Adapt an argument to a similar situation
• Use knowledge from different fields to address issues

Learning Outcomes of UE

Structure results in linear algebra: reduction of endomorphisms and spectral theory in Euclidean spaces.
The aim of this course is to develop the algebraic theory of endomorphism algebras of finite dimensional vector spaces, possibly endowed with a definite symmetric bilinear form.

UE Content: description and pedagogical relevance

Diagonalisation, eigenvalue, eigenvector, characteristic polynomial, minimal polynomial, Cayley-Hamilton theorem, Jordan-Chevalley decomposition.
Duality, bilinear symmetric form, orthogonality, non-degeneracy, transpose and adjoint endomorphism, automorphism, orthogonal basis, definite form.
Euclidean space, norm, orthonormal basis, Gram-Schmidt process, spectral theorem.

Prior Experience

"Algèbre linéaire et géométrie I" course.