29228 - Geometry and Algebra T

Course Unit Page


This teaching activity contributes to the achievement of the Sustainable Development Goals of the UN 2030 Agenda.

Quality education

Academic Year 2021/2022

Learning outcomes

Knowledge of the main tools of linear algebra (matrices, vector spaces, linear systems, quadratic forms) and of their applications in a geometric environment, granting both the comprehension of the connections among the different parts and the operational ability.

Course contents

Some algebraic structures Groups, rings, fields.

Matrices Initial definitions. Operations. Rank and echelon form. Determinant. Inverse matrix.

Vector spaces Initial definitions. Subspaces. Linear combinations. Linear dependence. Bases and dimension. Linear systems.

Linear applications Linearity. Isomorphisms. Kernel and image. Rank of a matrix. Matrix representation of a linear application. Change of base.

Linear systems Linear systems and their solvability. Methods of resolution. Representations of vector subspaces.

Endomorphisms Eigenvalues and eigenspaces. Matrix similarity. Characteristic polynomial. Diagonalization by similarity.

Euclidean vector spaces Scalar products. Orthogonal operators. Orthogonal complement. Spectral theorem for self-adjoint endomorphisms.

Euclidean spaces (Affine and) Euclidean spaces. Euclidean subspaces. Subspace representations. Parallelism. Orthogonality. Orthogonal transformations. Simplices. Volumes.

Bilinear and quadratic forms Bilinear forms. Matrix representation. Symmetric matrices. Quadratic forms. Canonical forms.

Hyperquadrics Hints to the classification of real conic sections and quadrics.


The detailed and complete teaching program will be published on Insegnamenti On-Line at the end of the lessons.


The topics covered in the course are classical topics in linear algebra and geometry, and can be found in many texts. As examples, some are indicated:

  • Bernardi A., Gimigliano A., Algebra lineare e geometria analitica, CittàStudi Edizoni, De Agostini Scuola 2014.
  • Abeasis S., Elementi di Algebra lineare e geometria, Zanichelli, 1993.
  • Abate M., de Fabritiis C., Geometria analitica con elementi di algebra lineare, McGraw-Hill 2010.
  • Casali M.R., Gagliardi C., Grasselli L., Geometria, Progetto Leonardo, Esculapio 2016.
  • Candilera M., Bertapelle A., Algebra lineare e primi elementi di Geometria, McGraw-Hill 2011.

With regards to the exercises, exercise sheets (with solutions) on the topics covered will be published on  Insegnamenti On-Line. From the same page it will be possible to download "exam-type" exercises. On-line resources will also be reported. If you want the support of a text, what is said for the theoretical part is valid, any book of exercises in geometry and linear algebra is fine. Here there are some titles:

  • Barani A., Grasselli L., Landi C., Algebra Lineare e Geometria, Progetto Leonardo, Esculapio 2005.
  • Gualandri L., Algebra Lineare e Geometria - Quiz risolti d'esame, Esculapio 2007.
  • Abate M., de Fabritiis C., Esercizi di Geometria, McGRaw-Hill 2003.
  • Lipschutz S., Lipson M., Algebra lineare, Collana Schaum's, McGraw-Hill 2001.
  • Parigi G., Palestini A., Manuale di Geometria - Esercizi, Pitagora 2003.
  • Cattabriga A., Mulazzani M., Prove d’esame risolte di Geometria ed Algebra per i corsi di Laurea in Ingegneria, Progetto Leonardo, Esculapio 2019.

Teaching methods

Taught class, dialogue, and activities in small groups.

Assessment methods

The exam consists of a written test and of an oral part.  Both are compulsory and concern the whole program covered in the lectures. It will be possible to take both of them   in presence or remotely (EOL and Zoom for the written test and and Teams for the oral exams). Detailed information on the exams will be sent to the registered students.

The written test lasts two hours and has the aim of testing the ability of the student in solving exercises, giving examples and counter-examples or short argumentations on theoretical issues. During the written test it is allowed, and indeed it is recommended, to make use of books, notes, etc. The use of the scientific calculator is also allowed, but the use of any other electronic device is prohibited.

The student is admitted to the oral interview  if its  written test is at least of 16 over 35. Each written test is valid for three calls, the one of the written test included.

The oral interview  aims to assess the student's knowledge and understanding of the topics covered in the course and his ability to present them with coherence using a specific language and a correct mathematical formalism.

The final result of the exam takes into account the results achieved in both parts: to be sufficient, the grade attributed to both of them must be at least 16 and, in this case, the final grade is approximately the average of the two marks; however the teacher reserves the right to change the assessment in exceptional cases. If the final outcome of the examination is not sufficient, both parts must be repeated. The student has the right to withdraw from both the written and oral tests and to refuse the final grade proposed once (in this case the student will have to repeate the entire exam). 

Entries to the appeals are made on AlmaEsami. Enrollment in the written test is mandatory, please note that registrations close 5 days before the date set for the written exam. Each oral exam will take place starting from the date indicated on AlmEsami, even in the following days, depending on the number of students registered. The oral exam calendar will be published together with the results of the written test.

It is necessary to come to the tests with the university badge.


Teaching tools

All the material are available on Virtuale.

We suggest to visit the sites of Prof. Massimo Ferri  (particularly the Chicken's corner), of Prof. Luciano Gualandri  and of Progetto Matematic@.


The tutor of the course is Dott. Cosimo Flavi e-mail cosimo.flavi2@unibo.it

Office hours

See the website of Alessia Cattabriga