OVERVIEW

The course provides an introduction to linear algebra and analytic geometry. In particular, it teaches algorithms for solving systems of linear equations, offers an overview of matrix theory and vector spaces, and addresses problems in plane and spatial analytic geometry. It is a first-year course, and the concepts and skills acquired will be useful 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. Attendance at lectures and exercises is strongly recommended.
Working students and students with certified specific learning disorders (SLD), disabilities, or other special educational needs are encouraged to contact the instructor at the beginning of the course to agree on teaching and assessment methods that, while respecting the learning objectives, take into account individual learning styles

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. 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).

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

https://corsi.unige.it/en/corsi/11924/studenti-orario

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.

All relevant information is available on the University website: https://unige.it/disabilita-dsa.

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.

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