# DISCRETE MATHEMATICS

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- Versione italiana
- Academic year
- 2022/2023
- Teacher
- VALERIA RUGGIERO
- Credits
- 6
- Didactic period
- Secondo Semestre
- SSD
- MAT/03

#### Training objectives

- The course is one of the two core courses on Mathematics. The main aim of the course is to teach the language and the basics of linear algebra.

The main topics concern spaces, linear applications, matrix theory, eigenvalues and eigenvectors, Euclidean spaces.

The student must learn:

- the skill of understand a problem stated in the formal language of linear algebra and how to use the basic tools in order to solve it

- the skill related to use the tools from Discrete Mathematics as prerequisite for the courses regarding numerical elaboration and computer science. #### Prerequisites

- The prerequisites for this course are the main topics of the Maths syllabus of the High School, in particular the ones related to analytic geometry, trigonometry, algebraic computation, concepts of function and relation.
#### Course programme

- The course is organized in 48 hours, divided in lectures and exercises, which will be alternated during the frontal lessons.

Recall of topics about sets, relations and functions. Recall of operations and algebraic structures (3 hours).

Reference systems in 2D plane and in 3D space. Geometric vectors and applied vectors; operations among applied vectors (3 hours).

Vector spaces and their properties (6 hours).

Matrices, matrix computation; determinant, matrix rank (7 hours).

Existence of solutions for linear systems (4 hours).

Linear applications and related properties; isomorfism between matrices and linear applications; change of basis (12 hours).

Eigenvalues and eigenvectors, properties and diagonalization criteria (5 hours).

Scalar product, real Euclidean space and related properties (6 hours).

Quadratic form and sign of a quadratic form (2 hours). #### Didactic methods

- Lectures on all the topics of the course are scheduled; during the lessons, the theoretical discussion is supported by a large number of exercises, with the dual aim of simplify the theoretical concepts and to enable the acquisition of the main technical tools for linear algebra problems. In order to support the lectures, further hours of guided exercises are scheduled.
#### Learning assessment procedures

- The aim of the final exam consists in verifying the level of achievement of the described learning outcomes. The examination consists in a written test,

designed to verify the acquisition of the practical and theoretical skills for the numerical solution of the main problems.

It is not possible to consult books or notes during writing. The test is passed if a score greater than or equal to 18/30 is achieved. #### Reference texts

- Appunti di Algebra Lineare e Geometria Analitica- G. Teacher’s handouts

Reference texts

G. Mazzanti-V. Roselli: Appunti di Algebra Lineare e Geometria Analitica, Pitagora Editrice Bologna 1997

G. Mazzanti-V. Roselli: Esercizi di Algebra Lineare e Geometria Analitica, Pitagora Editrice Bologna 1997

Insights: E. Schlesinger, Algebra lineare e Geometria, Zanichelli, 2011.

Geometria e algebra lineare - Vincenzo Giordano, Mondadori. Kindle format.