MATH 203

BS Mathematics

4

6

30

30

0

## Content

#### 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.

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