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

The purpose of this course is to develop further the techniques learned in the previous module of the course. In particular, the aim is to present the following subjects:

1. Characteristic spaces, Lemma of kernels and Cayley-Hamilton Theorem; nilpotent endomorphisms and their triangulation by completing the bases and by Jordan decomposition. More generally the triangulation problem for endomorphisms.
2. Bilinear applications: the dual vector space, dual basis, transposed linear application; bilinear forms with applications and properties, the matrix of a bilinear form and properties (symmetric bilinear forms, nondegenerate bilinear forms), the congruence of matrices. Defining character of a symmetric bilinear form on a real vector space, Euclidean vector spaces, norms and angles, orthogonality, orthogonal projections; orthonormal bases and Gram-Schmidt theorem, orthogonal complement of a subspace.
3. Endomorphisms between Euclidean vector spaces: isometries and their characterizations, orthogonal and special orthogonal matrices, properties, description of plane isometries; self-adjoint endomorphisms and their characterizations, symmetrical matrices, real spectral theorem, diagonalization of self-adjoint endomorphisms by means of orthonormal bases. Signature of a real symmetric matrix, Sylvester's inertia theorem.
4. Conics and quadrics: definition of a quadric, the matrix associated with a quadric or a quadratic form. Classification of quadrics by affine transformations, geometric properties of conics and quadrics, bundles of conics.
5. Affine and projective spaces: definition of the affine space over a field, the Euclidean affine space, and their properties. Lines, planes, hyperplanes in an affine space, Euclid's five postulates; affine transformations, coordinate systems and coordinate changes. Projective spaces: motivations, models of projective spaces. Straight lines, planes, hyperplanes in a projective space, Euclid's fifth postulate does not apply in projective geometry. Homogeneous coordinates, equations of lines and planes, affine maps, points at infinity. Projectivity: definition, properties, characterization of fixed points, fixed lines, lines through fixed points using linear algebra.

At the end of the course the students will be able to:

1. Determine the Jordan canonical form of an endomprhism.
2. Work with bilinear forms.
3. Classify projective and affine conics and quadrics.
4. Understand the basics of Euclidean/projective geometry in dimension two and three.

TEACHING METHODS

Standard Frontal Lesson and Tutoring.

SYLLABUS/CONTENT

1. The Jordan canonical form of an endomorphism.

2. The scalar product, Euclidean vector spaces, Gram-Schmidt algorithm and orthogonalization. Orthogonal automorphisms. Orthogonal projections. 

3. Diagonalization of real symmetric matrices. 

4. Analytic geometry in two and three dimensions: free and applied vectors, the scalar product, the vector and mixed product, coordinate systems, planes and lines in the plane and in space.        Introduction to curves and surfaces.

5. Real quadratic forms. Affine classification of conics and quadrics.

6. (only for Mathematics and SMID students) Affine and projective space. Projective plane and real projetive line. Affinity and projectivity. Projective classification of conics.

RECOMMENDED READING/BIBLIOGRAPHY

• M.E. Rossi, Algebra lineare, Dispense disponibili nella pagina del corso.
• F. Odetti - M. Raimondo, Elementi di Algebra Lineare e Geometria Analitica, ECIG Universitas.
• Marco Abate, Algebra Lineare , ed. McGraw-Hill.
• E. Sernesi, Geometria vol 1, ed Bollati-Boringhieri.
• A.Bernardi, A.Gimigliano, Algebra Lineare e Geometria Analitica, Citta'Studi Edizioni. 

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

From February 23rd, 2026.

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

FURTHER INFORMATION

The students with disabilities or specific learning disorders (DSA) can ask for special conditions if they inserted their certification on the website of Ateneo on the page servizionline.unige.it in the section “Studenti”. The documents will be verified by Settore servizi per l’inclusione degli studenti con disabilità e con DSA dell’Ateneo, as indicated by the federal site at the following link: STATISTICA MATEMATICA E TRATTAMENTO INFORMATICO DEI DATI 8766 | Studenti con disabilità e/o DSA | UniGe | Università di Genova | Corsi di Studio UniGe

Afterwards, and much in advance of the exam the student should send an e-mail to the professor of the course, inserting in "cc" the docente Referente di Scuola per l'inclusione degli studenti con disabilità e con DSA and il Settore indicated above.The e-mail should specify the following:

•            The name of the course.

•            The date of the exam.

•            The surname, name and university card number of the student.

•            The compensatory tools and measures asked by the students, that agree with the written rules.

The referente will confirm to the professor, whether the student has the right to make the request and the conditions allowed to be applied. The professor will reply by communicating to the students whether the conditions asked can be satisfied.

The requests must be sent with at least 7 days prior to each exam, so that the professor has enough time to evaluate them. In particular, if one is allowed to use notes, that should be much shorter than the actual notes of the course, if they are sent with delay there might not be time to read them and change them to a more appropriate form, when necessary.

For further information on this one can consult the following document: Linee guida per la richiesta di servizi, di strumenti compensativi e/o di misure dispensative e di ausili specifici .