N.B. The following order does not necessarily correspond to the order in which these topics will be treated during the course.
0. PROPOSITIONAL LOGIC, SETS, FUNCTIONS AND ALGEBRAIC STRUCTURES (SHORT INTRODUCTION)
Overview of propositional logic. Set theory recalls. Numerical sets and induction principle. Functions. Groups, rings and fields.
1. THE FIELD OF COMPLEX NUMBERS
Roots of -1. Algebraic representation. Gauss plane, Cartesian representation. Trigonometric or polar representation, de Moivre's theorem. Exponential representation. Fundamental theorem of algebra.
3. VECTOR SPACES
Vector spaces on a field. Generator systems. Linear independence. Steinitz exchange lemma. Bases. Dimension. Subspaces and operations.
4. LINEAR APPLICATIONS
Definition, kernel, image. Rank. Invertibility. Matrix of a linear application. Change of bases.
5. ALGEBRA OF MATRICES
Definitions and operations: sum, rescaling and product. Matrices and linear applications. Rank of a matrix. Square matrices. Invertibility.
6. LINEAR SYSTEMS
Definitions. The matrix equation AX = B. Linear systems and linear applications. Homogeneous systems. Resolution of linear systems. Gauss algorithm. Rouché-Capelli theorem. Structure of the solution space.
7. THE DETERMINANT
Definition. Determinant as an alternating multilinear function with value 1 on the identity. Properties of the determinant. First and second Laplace theorem. Binet's theorem. Determinant, rank and invertibility.
Cramer's Rule. Vector product in R^3.
8. EUCLIDEAN SPACES
Scalar products. Norm and distance. Standard scalar product in R^3; scalar product theorem. Orthogonality. Orthonormal bases, Gram-Schmidt orthogonalization. Isometries.
9. EIGENVALUES, EIGENVECTORS AND DIAGONALIZATION
Eigenvalues, eigenvectors and eigenspaces. Characteristic polynomial. Multiplicity. Search for eigenvalues. Endomorphisms and diagonalizability. Diagonalizability by orthogonal matrices. Real spectral theorem.
10. ANALYTICAL GEOMETRY IN THE PLANE AND IN THE SPACE
Affine spaces. Cartesian and parametric equations. Lines in the plane. Lines and planes in R^3. Spheres in R^n.
11. COMPLEMENTARY ARGUMENTS (to be treated depending on the time available and on the needs)
Dual spaces. Singular values decomposition. Discrete dynamic systems.
Quadratic forms. Conics as geometric loci and their classification. Quadrics, definitions and classification. Cayley-Hamilton's theorem. Jordan's canonical form. Hermitian products and complex spectral theorem.
