OVERVIEW

The course is designed to provide the basic concepts and tools of linear algebra and analytic geometry. It is a first-semester, first-year course in which concepts that will be used in many subsequent courses are introduced.

AIMS AND CONTENT

LEARNING OUTCOMES

This teaching unit aims to provide the basic concepts and technical tools related to complex numbers, linear algebra, and analytic geometry.

AIMS AND LEARNING OUTCOMES

The main goal of the course is to provide the basic concepts of linear algebra and analytic geometry, aiming at the development of a scientific approach to these topics and of the necessary tools to solve problems.

The course introduces methods for solving systems of linear equations over real and complex numbers, using the theory of matrices. The geometry of vectors is studied through their basic properties and operations, leading naturally to vector spaces, linear maps and their associated matrices.

Special attention is paid to symmetric and orthogonal matrices, to the relation between linear operators and matrices, to diagonalization techniques, to the Spectral theorem, and to applications to the geometry of vectors, conics and basic quadrics.

Students are expected to gain the skills of understanding the text of a problem, selecting the appropriate tools among those introduced in the course, solving the problem using suitable arguments, and expressing the results and conclusions in a clear and precise way.

PREREQUISITES

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

TEACHING METHODS

Lectures will be devoted to developing the theoretic part of the course, as well as to solving problems aimed to a better understanding of the theory. There will be additional hours  devoted to discussion of exercises suggested by the lecturer.

SYLLABUS/CONTENT

Basics on sets and functions. Complex numbers and polynomials. Systems of linear equations and Gauss' algorithm. Matrices, determinants and rank.

Cartesian systems of coordinates, points, lines and planes: cartesian and parametric equations, parallelism, angles, distances and orthogonal projections. Free and applied vectors, their geometrical representation, scalar and cross product, their basic geometric properties and their significance.

Vector spaces, subspaces, bases and dimension. Linear maps/operators and associated matrices. Change of basis, with particular attention to orthonormal changes of basis. Translations and rotations and their matrix representation.

Eigenvalues, eigenvectors, diagonalization of matrices and the Spectral theorem, with particular attention to symmetric and orthogonal matrices and their geometric significance.

Quadratic forms and their applications to circles, spheres, conics and basic quadrics.

RECOMMENDED READING/BIBLIOGRAPHY

• A. Bernardi, A. Gimigliano: Algebra Lineare e Geometria Analitica, Città Studi Edizioni
• E. Sernesi: Geometria vol. 1, Bollati-Boringhieri
• D. Gallarati: Appunti di Geometria, Di Stefano Editore-Genova
• M.R. Casali, C. Gagliardi, L. Grasselli: Geometria, Società Editrice Esculapio
• F. Odetti, M. Raimondo: Elementi di Algebra Lineare e Geometria Analitica, ECIG Universitas
• M. Abate: Algebra Lineare, McGraw-Hill
• L. Robbiano: Algebra lineare per tutti, Springer
• F. Bottacin: Esercizi di algebra lineare e geometria, Società Editrice Esculapio
• V. Bertella, A. Damiano: Esercizi su spazi vettoriali e applicazioni lineari, Società Editrice Esculapio
• L. Mauri, E. Schlesinger: Esercizi di algebra lineare e geometria, Zanichelli

Material provided by the lecturer, available on the AulaWeb webpage of the course.

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

In accordance with the manifesto. All class schedules are posted on the EasyAcademy portal.

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

Written test consisting of some exercises to be solved of the type seen during the course and possible oral test. Details will be communicated on Aulaweb.

ASSESSMENT METHODS

The written exam is intended to verify the student's capacity to solve problems, apply the main algorithms in the course, and show a good understanding of the main theoretical concepts developed during the semester, such as main theorems and definitions. The oral exam aims to verify the student's understanding of the basic concepts, definitions, and properties, seen during the course.

FURTHER INFORMATION

Ask the professor for other information not included in the teaching schedule.

Compensatory and dispensatory measures Disability/Invalidity/Specific Learning Disorder

Dispensatory measures and compensatory tools are intended to enable students to achieve the same learning objectives as their fellow students, not to facilitate the examination.

The use of compensatory tools and the application of dispensatory measures must be authorised in advance by the teacher in agreement with the Referee.

To take advantage of the adaptations during the examination, fill in the Adaptation request form; the request will be automatically sent by the system to the teacher in charge of the teaching, to the Contact Person of your School/Area/Department and in copy to the Sector; you will also receive a copy of the request sent by e-mail.

The adjustments available to students are as follows:

• Additional time (+30% DSA)
• Additional time (+50% disability/invalidity)
• Additional time during oral exams to organise the answer
• Calculator (programmable and graphing calculators are not allowed)
• Conceptual Maps
• Tables and/or Forms
• Take the exam in written form
• Take the exam in oral form
• Tutor reader (for written tests only)
• Tutor-writer (for written tests only)

Your request for adaptations must be submitted at least 7 working days before the scheduled exam date.

All information for students with disabilities and DSA is available on the webpage: Services for students with disabilities or DSA | UniGe | University of Genoa

Reference for inclusion: Sergio Di Domizio - sergio.didomizio@unige.it