# GEOMETRY

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Versione italiana
2022/2023
Teacher
ALBERTO CALABRI
Credits
9
Didactic period
Secondo Semestre
SSD
MAT/03

#### Training objectives

The main objective of the course is to provide students with the basics of Linear Algebra and Analytical Geometry , fundamental for other scientific teachings . The main knowledge gained will be related to:
- Basic elements on vector spaces
- Fundamental theorems on linear systems
- Diagonalization of matrices
- Analytical representation of lines , planes, spheres , cylinders , cones
- Linear Functions and dimensional theorem.

The main skills will be :
- Solve problems related to vector spaces dependent on parameters ,
- Discuss linear systems of various tipologies containing parameters
- Determine whether a matrix is diagonalizable ,
- Solve problems of analytic geometry of space ,
- Determine the basic elements of a linear function .

#### Prerequisites

Elementary Algebra. Elements of Euclidean geometry. Elements of analytic geometry in the plane. First elements of mathematical logic: definition, theorem, demonstration, role of examples and counterexamples .

#### Course programme

The course includes 90 hours of teaching between lessons and exercises.The topics covered in the course are the following.

Vector spaces ( 15 h ) . Matrices , determinants , linear systems and applications ( 16 h ) . Analytic geometry in space ( 13 h ) . Euclidean spaces (8 h ) . Orthogonal matrices ( 5 h ) . Diagonalization of a matrix . Diagonalization of a symmetric matrix with an orthogonal matrix ( 10 h ) . Quadratic forms . Reduction to diagonal form . Square root of a matrix . Applications of quadratic forms ( 8 h ) . Linear functions . Kernel and image concepts . Dimensional theorem. Fundamentals properties of linear functions . Linear functions and matrices . Eigenvalues and eigenvectors of a linear operator ( 15 h ) .

#### Didactic methods

Lectures to introduce the theoretical concepts . Exercises relating to the application of these concepts.

#### Learning assessment procedures

The goal of the examination is to test the level of achievement of knowledge, skills and abilities related to topics previously mentioned .

Examination is divided into two parts, which take place on different days .
The first part consists of a written test on the application of the concepts introduced (exercises) .
The second part consists of a written test on the theoretical aspects of the course topics, also customized based on the outcome of the previous trial.
The final grade takes account of both tests .
If the student fails to achieve a minimum of 18 to 30 must repeat both tests.

Passing the exam is proof that he has acquired the knowledge and skills specified in the learning objectives of teaching.

#### Reference texts

Giuliano Mazzanti-Valter Roselli
"Appunti di Algebra Lineare, Geometria Analitica, Tensori: Teoria, Esempi,Esercizi svolti, Esercizi proposti"
Pitagora Editrice, Bologna 2013

Giuliano Mazzanti-Valter Roselli
Esercizi di Algebra Lineare e Geometria Analitica
Pitagora Editrice Bologna 1997