1. Elementary Notions of Logic
Objectives: Master the fundamental tools of mathematical reasoning. Be able to formulate propositions, understand logical connectors (and, or, implication, equivalence), and the different methods of proof (direct, by induction, by contraposition, by contradiction).
2. Sets
Objectives: Acquire the basic set theory language. Understand and manipulate set operations (union, intersection, complement, Cartesian product) and their properties. Be able to work with families of sets.
3. Functions
Objectives: Understand the notion of a function and distinguish between different types of functions (injective, surjective, bijective). Master the composition of functions and the concept of inverse functions. Be able to determine direct and inverse images.
4. Algebraic Structures
Objectives: Understand the fundamental algebraic structures (groups, rings, fields). Recognize the properties of composition laws (associativity, commutativity, identity and inverse elements). Be able to identify substructures.
5. Ring of Polynomials
Objectives: Master operations on polynomials (addition, multiplication, Euclidean division). Understand the concepts of roots and multiplicity. Be able to find the GCD of two polynomials (Euclidean algorithm) and apply Bézout's identity.
6. Rational Fractions
Objectives: Know how to decompose a rational fraction into partial fractions in ℝ(X). Understand the different steps of decomposition based on the nature of the poles. Master this essential technique for integral calculus.
7. Vector Spaces
Objectives: Understand the structure of a vector space and its fundamental properties. Be able to manipulate subspaces, direct sums. Master the concepts of generating set, linear independence, dependence, and basis. Understand the notion of dimension.
8. Linear Applications
Objectives: Recognize and construct linear maps. Determine the kernel and image of a linear map. Understand the notions of endomorphisms and isomorphisms. Be able to compose linear maps.
9. Matrices
Objectives: Master basic matrix operations. Compute determinants and understand their properties. Know how to invert an invertible square matrix. Understand the link between linear maps and matrices. Master change of basis.
10. Linear Systems
Objectives: Solve systems of linear equations using different methods (Cramer's rule, Gaussian elimination). Distinguish between the different cases (unique solution, infinitely many solutions, no solution). Understand the link with linear applications.
11. Matrix Reduction
Objectives: Compute eigenvalues and eigenvectors of a matrix. Understand the concept of diagonalization. Determine whether a matrix is diagonalizable and perform the reduction when possible. Apply these concepts to concrete problems.