Mathematical Methods for Digital Communication
A.Y. 2024/2025
Learning objectives
The aim of this course is to introduce the algebraic language and the basic notions of vector spaces and linear applications. These theoretical arguments are applied to the problem of solving linear systems of equations (even from an algorithm point of view).
Expected learning outcomes
The student should be able to understand and to utilize the formal language of abstract algebra, vector spaces and linear applications. Moreover, he should be able to work with matrices, to associate them to linear systems and to use them to discuss their solutions.
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
Part 1. Linear algebra.
Systems of linear equations. Gauss-Jordan solution method. Matrices and their algebras. Vector spaces and subspaces (sketch). Bases. Determinants and their properties. Invertible matrices. Inverse matrix. Rank of a matrix; matrices and linear applications. Cramer's and Rouché-Capelli's Theorems. Eigenvalues and eigenspaces. (Each topic is accompanied by examples and exercises.)
Parte 2. Discrete mathematics.
Combinatorics (sketch). Equivalence relations and order relations; examples and applications. Transitive closure of a relation, and how to compute it. Injective, surjective, and bijective functions. Kernel of a function. Permutations. Sign of a permutation and factorisation into cycles Groups (sketch). Integers and divisibility. The Euclidean algorithm. Prime numbers and factorisation. Diophantine equations (sketch). Modular arithmetic (sketch). Polynomials and divisibility. Division of polynomials. Irreducible polynomials and factorisation. Rings (sketch).
Systems of linear equations. Gauss-Jordan solution method. Matrices and their algebras. Vector spaces and subspaces (sketch). Bases. Determinants and their properties. Invertible matrices. Inverse matrix. Rank of a matrix; matrices and linear applications. Cramer's and Rouché-Capelli's Theorems. Eigenvalues and eigenspaces. (Each topic is accompanied by examples and exercises.)
Parte 2. Discrete mathematics.
Combinatorics (sketch). Equivalence relations and order relations; examples and applications. Transitive closure of a relation, and how to compute it. Injective, surjective, and bijective functions. Kernel of a function. Permutations. Sign of a permutation and factorisation into cycles Groups (sketch). Integers and divisibility. The Euclidean algorithm. Prime numbers and factorisation. Diophantine equations (sketch). Modular arithmetic (sketch). Polynomials and divisibility. Division of polynomials. Irreducible polynomials and factorisation. Rings (sketch).
Prerequisites for admission
Mathematics at high school level.
Teaching methods
Lectures and exercises.
Teaching Resources
For the Linear algebra part of the course some useful books are:
R. Fioresi, M. Morigi, Introduction to Linear Algebra, CEA, 2019.
M. Bianchi e A. Gillio, Introduzione alla matematica discreta, McGraw-Hill, 2005.
Textbook for the second part of the course:
G. Piacentini Cattaneo, Matematica discreta e applicazioni, Zanichelli, 2008.
R. Fioresi, M. Morigi, Introduction to Linear Algebra, CEA, 2019.
M. Bianchi e A. Gillio, Introduzione alla matematica discreta, McGraw-Hill, 2005.
Textbook for the second part of the course:
G. Piacentini Cattaneo, Matematica discreta e applicazioni, Zanichelli, 2008.
Assessment methods and Criteria
The final examination consists of a written exam. During the written exam, the student will solve some exercises in the format of open-ended questions, with the aim of assessing the student's ability to solve problems about the topics of the course. The duration of the written exam will be proportional to the number of assigned exercises, and is normally of two hours. Two midterm exams are offered that can replace the first exam. The exams' outcomes (marks given using the numerical range 0-30) will be available in the SIFA service through the UNIMIA portal.
MAT/01 - MATHEMATICAL LOGIC
MAT/02 - ALGEBRA
MAT/03 - GEOMETRY
MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS
MAT/05 - MATHEMATICAL ANALYSIS
MAT/06 - PROBABILITY AND STATISTICS
MAT/07 - MATHEMATICAL PHYSICS
MAT/08 - NUMERICAL ANALYSIS
MAT/09 - OPERATIONS RESEARCH
MAT/02 - ALGEBRA
MAT/03 - GEOMETRY
MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS
MAT/05 - MATHEMATICAL ANALYSIS
MAT/06 - PROBABILITY AND STATISTICS
MAT/07 - MATHEMATICAL PHYSICS
MAT/08 - NUMERICAL ANALYSIS
MAT/09 - OPERATIONS RESEARCH
Lessons: 48 hours
Professors:
Marra Vincenzo, Van Geemen Lambertus
Professor(s)
Reception:
By appointment
Dipartimento di Matematica "Federigo Enriques", via Cesare Saldini 50, room 2048