Subject: LINEAR ALGEBRA (A.A. 2022/2023)
Unit Algebra lineare
Verifica preparazione iniziale
To give the basic knowledge and operative methods of discrete mathematics and linear algebra, which turn out to be of use in theoretical computer science and its applications.
For a detailed description of the content and of the learning goals, see the specific sections below.
The prerequisites are the knowledge of High School Mathematics.
[1,5 CFU] LOGIC of propositions, elementary SET THEORY, relations, functions.
[1,5 CFU] COMBINATORICS: partial permutations, permutations, combinations, binomial coefficient. Counting techniques.
[1 CFU] ALGEBRAIC STRUCTURES: groups, rings, domains, fields. INTEGER ARITHMETIC: prime factors, gcd and lcm, euclidean algorithm, Bézout identity.
[1 CFU] MODULAR ARITHMETIC: congruence modulo n, invertible and zero-divisors elements, Euler’s phi function, Euler theorem, congruence modulo a prime number. Chinese remainder theoerem.
[1,5 CFU] MATRICES: operations, determinant, inverse matrix, elementary row operations and echelon matrices, rank and its computation.
[1 CFU] LINEAR SYSTEMS. Rouché-Capelli, Cramer, Gauss-Jordan.
[1 CFU] Abstract VECTOR SPACES: definition, vector subspaces, linear combinations, generators, linear dependence and independence, bases and dimension. Parametric and cartesian equations for vector subpaces.
[0,5 CFU] DIAGONALIZATION of matrices: similar matrices, eigenvalues, eigenvectors, eigenspaces, characteristic polynomial, algebraic and geometric multiplicity, diagonalizable matrices.
Teaching activities in the classroom: lessons, with theoretical results and exercises; tutoring, as a training for the two intermediate written tests. Attendance is not mandatory, but it is strongly recommended. The language of the course is Italian.
Final written test, and elective oral test. The written test (duration: 2 hours) requires the resolution of some exercises covering all the program of the course, similar to the ones presented during the lectures. In order to pas the written test one has to get a score of at leat 18/30 in the written test. The final written test can be substituted by two partial written test during the course, about the first part and the second part of the course. In order to do the second partial written test, the student has to pass the first one. If the student does not pass one of the partial written tests, he has to undergo the global written test. After passing the written part of the exam, the student can undergo the oral part, which is not mandatory. The maximum score without doing the oral exam is 23/30. Oral tests of the admitted candidates take place in the days after the assessment of the written test. It is typically twenty minutes long. It tests the knowledge of the contents of the course, as well as their connections and applications.
At the end of the course, the students should be able to:
- perform mathematical computations correctly;
- know the formulas and basic algorithms of discrete mathematics and linear algebra;
- apply correctly the techniques and basic algorithms of linear algebra and analytic geometry;
- express propositions and predicates with the correct mathematical formalism;
- define precisely the objects of discrete mathematics and linear algebra;
- explain clearly basic theorems about discrete mathematics and linear algebra;
- discuss with deductive reasoning the arguments of discrete mathematics and linear algebra;
- design a model for a counting problem through discrete mathematics;
- identify the appropriate techniques to solve problems in discrete mathematics and linear algebra;
- develop autonomously a technique for easy problems of discrete mathematics.
Dispense fornite dal docente (disponibili sul portale Moodle di ateneo).
I seguenti libri di testo sono utili per la consultazione e l'approfondimento personale. Nessuno di essi costituisce materiale obbligatorio di studio. Ciascuno di essi è sufficiente per la preparazione individuale di tutta la parte di matematica discreta, o di tutta la parte di algebra lineare.
Per la parte di matematica discreta:
- C. DELIZIA, P. LONGOBARDI, M. MAJ, C. NICOTERA, Matematica discreta, McGraw-Hill, 2009.
- K.H. ROSEN, Discrete Mathematics and Its Applications, Fifth edition, McGraw-Hill, 2003.
Per la parte di algebra lineare:
- R. FIORESI, M. MORIGI, Introduzione all'algebra lineare, CEA casa editrice ambrosiana, 2021.
- G. CATINO, S. MONGODI, Esercizi svolti di Geometria e Algebra Lineare, Società Editrice Esculapio, 2020.