AIMS AND CONTENT

LEARNING OUTCOMES

The aim of the course is to introduce students to the basic elements of linear algebra, affine and Euclidean geometry. These topics are part of the fundamentals of the study of modern mathematics and will be used in all subsequent courses. In addition, it is not a secondary objective to show students a theory that is strongly motivated by concrete problems, and that can be treated in a comprehensive and rigorous manner.

AIMS AND LEARNING OUTCOMES

During the first semestrer the following will be treated:

1. Complex numbers: definition and motivation, operations with complex numbers, properties of complex numbers, equations with complex numbers, in particular resolution of polynomial equations of degree 2 and calculation of complex roots.

2. Matrices: definition and operations with matrices, properties (powers, zero divisors, nilpotent matrices, invertibility), determinant of a square matrix, methods for calculating the determinant, in particular Laplace's theorem and elementary oprations. Rank of a matrix, properties, methods for calculating the rank (Kronecker's theorem).

3. Linear systems: resolution of systems of linear equations with the Gauss method, results about existence and quantity of solutions using the properties ofthe matrix associated to the linear system (Cramer, Rouché-Capelli)

4. Geometry in the plane and in the three-dimensional space: vectors and operations with vectors (sum, product, scalar product, vector product); lines and their equations in the plane and in the three-dimensional space; position of a line with respect to another line or to a plane, position of a plane with respect to another plane; orthogonal projection of a point on a line or on a plane, of a line on a plane, and their symmetric. Distance of a point from a plane or from a line. Circles, spheres, cones, cylinders and their equations in the three dimensional space.

5. Vector space: definition of vector space over a field, examples, properties; subspaces, indipendent vectors, generators, bases and dimension of a vector space. Operations with subspaces: intersection, sum, Grassmann's theorem.

6. Linear functions: definition, properties, kernel and image of a linear function, dimension theorem; matrix of a linear function with respect to a base of the domain and one of the codomain, relation between the properties of the matrix and the properties of the linear function. Isomorphism between the space of linear functions between two vector spaces and a space of matrices. Similitude of matrices and properties of similar matrices.

At the end of the first semester of this course the student will be able to make calculations with complex numbers and solve polynomial equations with them. He/she will be able to determine if a linear system has solutions, how many solutions it has, and the precise form of the solutions. He/she will be able to determine fundamental geometrical properties of planes and lines (directional vector, reciprocal position), and correctly write the equation of a plane, of a line, of a sphere, oa circle verifying some further properties. He/she will in measure to study a matrix and determine its fundamental properties (rank, determinant, invertibility). He/she will be able to calculate the dimension of a vector space, to find a base of it and to establish if a family of vectors verifies properties like linear independence and the space spannes by them, and to study some fundamental properties of linear functions (kernel, image, invertibility) making use of matrices and linear systems. He/she will be able to correctly use the language of linear algebra and establish if a given statement in linear algebra is correct making use of a proof or of counterexamples.

TEACHING METHODS

Traditional

SYLLABUS/CONTENT

1. Complex numbers

2. Matrices

3. Linear systems

4. Geometry in a three-dimensional space

5. Vector spaces

6. Linear functions

7. Diagonalizability and trigonability

8. Euclidean vector spaces

9. Isometries and self-adjoint endomorphisms

10. Conics and quadrics

11. Affine and projective spaces

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.

F. Odetti, M. Raimondo: Elementi di Algebra Lineare e Geometria Analitica, ECIG Universitas.

M. Abate: Algebra Lineare, McGraw-Hill.

C. Ciliberto, Algebra Lineare, Bollati-Boringhieri

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

According to the academic calendar

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

The exam has a written part and an oral part.
The written exam is considered as passed if the student obtains an evaluation at least equal to 18/30. To participare to the written exam the student has to perform the inscription to the exam on the UNIGE website https://servizionline.unige.it/studenti/esami/prenotazione at least two days before the exam. 

During the year there will be two intermediate exams (one at the end of the first semester, the other at the end of the second semester) that, if passed, replace the written exam. The first intermediate exam il considered as passed if the student obtains an evaluation at least equal to 16/30. To take part to the second intermediate exam, the student needs to pass the first intermediate exam. The two intermediate exams are considered as passed if the student obtains an evaluation at least equal to 16/30 in both intermediate exams, and the average of the two evaluations is at least equal to 18/30.

The oral exam takes place during the same exam session of the written exam, and the student that gets to the oral exam after having passed the two intermediate exams may choose to carry out the oral exam either during the exam session of June (the first session) or during the exam session on July (the second session). The final evaluation will be the average of the evaluations of the written and the oral parts of the exam. If the oral exam is considered as not sufficient, the commission may consider to cancel the result of the written exam as well.

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

ASSESSMENT METHODS

The written exam consists in the resolution of some exercices about the whole program of the course. The exam lasts three hours. During the exam the student will not have the possibility to use any books, notes, or electronic devices such as calculators, tablet, cell phones or smartwatch, but we suggest the student to prepare an A4 paper handwritten form with the formulas and the results that he/she may consider to be useful for the exam. To take part to the written exam, the student needs to perform the inscription on the UNIGE website not later than two days before the exam. The written exam will be consdiered as passed it the obtained evaluation is at least 18/30. Only for very special cases the commission will take into consideration the possibility to lower this threshold.

There will be moreover two intermediate exams that replace the written exam. The first one will take place at the end of the first semester, and will consist in the resolution of exercices concerning the program of the first semester of the course. The second intermediate exam will take place at the end of the second semester, and it will consist in the resolution of exercices concerning the program of only the second semester of the course. Both intermediate exams last three hours. During the exam the student will not have the possibility to use any books, notes, or electronic devices such as calculators, tablet, cell phones or smartwatch, but we suggest the student to prepare an A4 paper handwritten form with the formulas and the results that he/she may consider to be useful for the exam. To take part to the intermediate exams, the student needs to perform the inscription via specific inscription forms that will be available on the AulaWeb page of the course. The first intermediate exam is considered as passed if the obtained evaluation is at least 16/30, and one may participate to the second intermediate exam only in this case. Both intermediate exams are considered as passed, and in this case they will replace the written exam, if both evaluations are at least 16/30, and the average of the two evaluations is at least 18/30.

The oral exam requires the knowledge and the ability to present the definitions, the statements and the proofs that have been treated alla long the course, the ability to give examples that illustrate the main notions of the course, and the ability to establish if a given statement is true or false by means of proofs or counterexamples. In order to determine if the student is able to use the instrument of Linear Algebra, the teacher will furthermore propose the resolution of some exercices. The oral exam will take place during the same exam session of the written exam, or during the exam sessions of June or July (the first and the second exam sessions) for the student that get to the oral exam after having passed the two intermediate exams  If the oral exam is considered as not sufficient, the commission may consider to cancel the result of the written exam as well.

Exam schedule