Linear Algebra III Description

Key Elements


MATH 203


BS Mathematics





Number of Teaching Hours


Number of Tutoring Sessions


Number of Laboratory Sessions




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.


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.