#### Objective

1. Determine the eigenvectors of an endomorphism or of a matrix.
2. Know and use the conditions necessary and sufficient of an endomorphism or a matrix to be upper triangular .
3. Reduce a matrix to an upper triangular form by induction or by using Jordan canonical form.
4. Know the definition of the dual space V of a given vector space V.
5. Determine the orthogonal of a subset , know and use the theorem of dimensions.
6. Determine the dual basis B in V of a given basis B of a vector space V.
7. Define the transpose of a linear mapping . Know and use its properties.
8. Define a bilinear form and a symmetric bilinear form of a vector space.
9. Know and find the matrix representation of a bilinear form or of a quadratic form related to a symmetric bilinear form.
10. Know the invariants of a quadratic form.
11. Know Sylvester’s theorem.
12. Know Gauss’method.
13. Know a non degenerate and a definite positive quadratic form .
14. Know an orthogonal and an orthonormal basis
15. Construct an orthonormal basis using Gauss’s method.
16. Define an inner product space .
17. Know and use Cauchy-Schwartz’s theorem
18. Construct an orthonormal basis using Gram-Shmidt’s theorem.
19. Know the adjoint of a linear mapping , the orthogonal operator and the orthogonal
matrice.
20. Know and use the fact that : a real matrix is orthogonally diagonalizable if and only if
It is symmetric.
21. Study the geometry of an orthogonal operator in R2.
22. Study and use the diagonalization of a quadratic form using Gram-Shmidt’s theorem
23. Know the conic section.

#### Content

Dual space: Basic properties, dual basis, orthogonality, the transpose of a linear transformation, the double dual of a vector space.
Upper triangular form: Induction method, Jordan canonical form.
Bilinear forms and Quadratic forms: matrix representation, symmetric bilinear forms, orthogonality, Sylvester’ theorem, reduction of a quadratic form.
Inner product space: Gram Shmidt process, the adjoint of a linear operator, orthogonal
operator, application to the geometry in IR2 .
Orthogonal diagonalization: diagonalization of a symmetric real matrix, diagonalization of
quadratic forms, conic section.