Each part includes a written exam, with solving exercises of increasing difficulty, and an optional oral exam.
Intermediate partial written exams are scheduled during the development of the course. Performance with a sufficient grade on these partial exams will exempt from the written examination.
This part is divided into four sections:
1. Introduction to the elementary language of mathematics
In this part we do the groundwork needed for the correct use of the mathematical language. We analyze and compare different set notations and definitions, and we develop the first abstract reasonings using them.
2.Functions and relations
We introduce the first definitions using examples that allow to overcome the first difficulties we met when the level of abstraction increases, as when we consider the set of equivalence classes.
3.Arithmetics and congruences
Thi is the first training ground for developing reasoning skills. We use this material as an example of the use of elementary mathematics in applications, for example in criptography.
We analyze different techniques for the calculus of the cardinality of finite sets. This setting is appropriate for learning how to model different problems and use the reasonings skills acquired in the previous sections of the program.
In the second part of the course the first notions of linear algebra are introduced and applied to the solution of linear systems and to the representation of motion in plane and space. This concepts are then generalized to higher dimensions.
This part is divided into three sections. In the first section linear systems and their matrix representation are considered. Here the student learn how a mathematical representation can be useful to represent and solve a problem in a concise way. In the second part, definitions and tools of linear algebra (dependence, basis,..) are first introduced in plane and space and then generalized to higher dimensions. The objective is here to expose the student gradually to a higher level of abstraction. In the last part of the course linear transformations and their matrix representation are considered, starting from rotations and symmetries in plane and space and then generalizing to higher dimension. Here we exemplify how certain motion can be reproduced in the computer using their mathematical representation.
In the second part of the course (second semester) concepts and tools of Linear Algebra are introduced.
ARGUMENTS FIRST PART: Connectives and Quantifiers; reasoning strategies. Sets and Functions. Relations, partial and total order relations. Equivalence relations, partitions and equivalence classes. Principle of Mathematical Induction. Elements of Combinatorial Calculus. Integer arithmetic, quotient and remainder, prime numbers and factorization. Maximum common divisor and minimum common multiple. Euclid’s algorithm for the greatest common divisor. Congruences and RSA.
ARGUMENTS SECOND PART : Linear systems. Matrices, sum, product, inverse matrix. Vector spaces of finite dimension on reals, linear dependence and independence, bases and dimension. Scalar product and norm of vectors. Orthogonal projection onto subspaces. Linear transformations. Linear transformations and matrices. Linear transformations and linear systems. Determinants. Diagonalization, eigenvalues and eigenvectors. SVD of a matrix and applications.
For the second module:
Gilbert Strang, Introduction to Linear Algebra (Sixth or Fifth Edition) Wellesley Cambridge Press