CODE 98169 2024/2025 6 cfu anno 1 INGEGNERIA GESTIONALE 10716 (L-9) - GENOVA MAT/03 GENOVA 2° Semester AULAWEB

## OVERVIEW

The course provides an introduction to linear algebra and analytic geometry. In particular, it teaches algorithms to find the solutions of a system of linear equations, provides an overview of basic matrix theory, teaches about vector spaces and deals with problems from analitic geometry in the plane and space. It is a course for first year students, whose concepts and expertise will be helpfull for subsequent courses.

## AIMS AND CONTENT

### LEARNING OUTCOMES

The course aims at providing the basic concepts and tools of linear algebra and analytic geometry. At the end of the course the student will be able to: - give correct definitions of the objects and properties studied, using the appropriate mathematical formalism; - recognize in concrete examples the geometrical objects and the algebraic properties studied; - describe the set of solutions of systems of linear equations; - solve exercises of plane and space geometry involving points, lines, planes, angles, distances, scalar products, orthogonal projections, conics, quadrics; - give explicit examples of objects that satisfy the geometrical or algebraic properties studied; - apply the notions and procedures studied in order to solve problems, also of new types and of an abstract nature.

### AIMS AND LEARNING OUTCOMES

The first goal of the course is to teach how to solve systems of linear equations over real and complex numbers, making use of the theory of matrices. Inspired by physics, we will study further the geometry of vectors and their basic properties and operations. In particular, vectors will lead us to vector spaces and matrices to linear maps, making an entrance in the realm of linear algebra. In this course special attention will be paid to symmetric and orthogonal matrices, to the interconnection between linear operators and matrices, to diagonalization techniques and their applications to the geometry of vectors, conics and quadrics.

In short the course aims to provide the basic concepts of linear algebra and analytic geometry, to develope a "scientific" approach to studying and solving problems. The student is expected to learn how to understand the text of a problem, carry out solutions in a reasoned and autonomous way, by making use of the methods provided in the course, and finally provide clear and precise conclusions.

### PREREQUISITES

Basic knowledge of arithmetics, algebra, trigonometry and set theory.

### TEACHING METHODS

The goal of the lectures is to present the theoretical part of the course, as well as providing solutions to problems, whose aim is to help explain better the theory. There will be additional hours (tutorato), devoted to discussions suggested by the professor and providing answers to students questions related to the course.

### SYLLABUS/CONTENT

Sets and maps. Complex numbers and polynomials. Linear systems and gaussian elimination. Matrices, determinants, rank. Points, lines and planes: cartesian and parametric equations, parallelism, angles, distances, orthogonal projections. Vector spaces. Vectors in geometry. Subspaces, bases, dimension. Linear maps. Matrices related to a linear map. Eigenvalues, eigenvectors. The diagonal form of a matrix. The Spectral theorem. Quadratic forms.

•    A. Bernardi, A. Gimigliano - "Algebra Lineare e Geometria Analitica", Città Studi Edizioni.
•    E. Carlini, M.V. Catalisano, F. Odetti, A. Oneto, M.E. Serpico - "Geometria per ingegneria", Editore Esculapio (Bologna), 2011.
•    M. V. Catalisano, A. Perelli - "Appunti di Geometria e calcolo numerico" (http://www.diptem.unige.it/catalisano/AppuntiGeometria.pdf )
•    S. Greco, P. Valabrega - "Algebra lineare", Levrotto & Bella, 2009.
•    S. Greco, P. Valabrega - "Geometria analitica", Levrotto & Bella, 2009.
•    F. Odetti, M. Raimondo – "Elementi di algebra lineare e geometria analitica" – ECIG, 2002.
•    J. Hefferon - "Linear Algebra" (https://hefferon.net/linearalgebra/).
•    I. Lankham, B. Nachtergaele, A. Schilling - "Linear Algebra" (https://www.math.ucdavis.edu/~anne/linear_algebra/mat67_course_notes.pdf).
•    D. Cherney, T. Denton, R. Thomas, A. Waldron - "Linear Algebra" (https://www.math.ucdavis.edu/~linear/linear-guest.pdf).

## LESSONS

### LESSONS START

In accordance with the manifesto.

### Class schedule

The timetable for this course is available here: Portale EasyAcademy

## EXAMS

### EXAM DESCRIPTION

Written test that consists in solving some problems similar to those seen during the lectures. There might be a possible oral test. More details will be communicated on Aulaweb.

Students with DSA certification ("specific learning disabilities"), disability or other special equcational needs are advised to contact the professor at the beginning of the course and agree on the teaching and examination methods that are in compliance with the main teaching objectives and takes into account individual learning arrangements and provides appropriate compensatory tools.

### ASSESSMENT METHODS

The exam aims to verify whether the student has acquired the required skills and knows further how to use and express them in correct terms. In particular, it will asses the student's ability to solve problems related to the main topics of the course, provide adequate explanations on the procedures and express clear conclusions.