Algebra 4
Description of the module:
Description of the module:
In this resource, we will know linear forms, bilinear forms, symmetrical bilinear forms, quadratic forms. Their definitions and general properties will be developed, as well as the links that exist between them.
Also notions about Euclidean spaces and hermitic spaces.
Target audience:
Students of 2nd year bachelor maths
Essential prerequisites :
Target audience:
Students of 2nd year bachelor maths
Essential prerequisites :
The basic vocabulary of linear algebra and the notion of finite type space (or finite dimension).
Matrix calculation and basic change formulas.
Objectives :
Objectives :
1) Recognize a bilinear form, a symmetrical bilinear form, a quadratic form.
2) Know how to move from one to another in particular in the case of finite type spaces.
3) Know how to write the matrix associated with a symmetrical bilinear form or a quadratic form in any base, and thus the case of base change.
4) Know how to reduce the quadratic form to the diagonal form where the base will be orthogonal, so the classification of this form from its signature.
5) Know the Euclidean and Hermitian spaces where particularly the symmetrical bilinear form will be defined positive (the scalar product).
Content of the module:
Content of the module:
Chapter 1: Linear forms, Duality
Chapter 2: Bilinear and quadratic forms
Chapter 3: Reduction of quadratic shapes
Chapter 4: Introduction to Euclidean and Hermitian forms