29228 - Geometry and Algebra T

Academic Year 2023/2024

  • Teaching Mode: Traditional lectures
  • Campus: Bologna
  • Corso: First cycle degree programme (L) in Computer Engineering (cod. 9254)

Learning outcomes

Knowledge of the main tools in linear algebra (matrices, vector spaces, linear systems, eigenvalues, quadratic forms) and their application in geometry, ensuring both the understanding of the links between the different parts of the theory, and operational capability.

Course contents


[Equations and linear systems] Some algebraic structures (groups, rings, fields). Standard operations on K^n. Linear systems.

[Matrices] Main definitions. Operations. Linear systems and matrices.

[Vector spaces] Main definitions. Vector subspaces. Linear combinations. Sum space. Row space and column space of a matrix.

[Bases] Linear dependence. Bases and dimension. Rank of a matrix. Linear systems.

[Linear maps] Linearity. Isomorphisms. Kernel and image of a linear map. 

[Matrix representation of a linear map] Linear maps, bases, matrixes.

[Determinants] Permutations. Determinant and its main properties. Laplace expansion. Inverse matrix. Determinant of an endomorphism. Rank of a matrix. Linear systems.

[Representation of a vector subspace] Rank, kernel, image. Cartesian and parametric representations.

[Eigenvalues and eigenvectors] Eigenvalues and eigenspaces of an endomorphism. Similar matrices. Characteristic polynomial. Diagonalization of matrices. 

[Euclidean vector spaces] Inner products and norms induced by inner products. Orthogonality. Orthogonal bases and orthonormal bases. Isometries. Orthogonal complement. Wedge product.

[Euclidean spaces] Euclidean subspaces. Representations of Euclidean subspaces. Parallelism and orthogonality in R^3. 


Computation of determinants and ranks of matrices. Discussion and solution of linear systems. Computation of matrices associated with linear maps. Computation of equations for vector subspaces. Computation of eigenvalues and eigenvectors. Diagonalization of matrices. Exercises on parallelism and orthogonality in R^3. Computation of angles between lines.


Further details at the web page http://www.dm.unibo.it/~frosini/programmi/programmacorso2023.shtml


A. Gimigliano, A. Bernardi, "Algebra lineare e geometria analitica", CittàStudiEdizioni, 2018 (second edition).

Teaching methods

Taught class.

Assessment methods

The exam consists of a compulsory online questionnaire on EOL (Esami OnLine), a compulsory written test in presence and an optional oral test in presence (which can be made compulsory at the discretion of the teacher). All tests cover the entire program developed in class.

QUESTIONNAIRE: Admission to the written test will be subject to passing a mandatory theoretical questionnaire held on EOL-ESAMI ONLINE, consisting of 9 groups of 4 statements (therefore 36 statements in total). Each group will refer to a topic covered in the program. For each statement the student will have to say whether it is a true or false statement. It will be possible to abstain from the answer by choosing the "I DON'T KNOW" option. Each correct choice will be awarded 2 points, each incorrect choice 0 points, the "I DON'T KNOW" choice 1 point. The total score can therefore vary between 0 and 72. (As you can see, it will always be convenient to choose one of the three options, but the expected value by answering at random will be 36, the same that will be obtained by always answering "I DON'T KNOW".) They will be admitted in the final exam and in the eventual oral exam those who have obtained at least 48 points (in practice it is a question of correctly evaluating at least one third of the statements). This choice regarding the scores was determined on the basis of the technical characteristics of EOL-EXAMS ONLINE. Students who, for personal reasons, wish to report the P score obtained in a T grade expressed out of thirty can use this formula: T=5P/6-30 (for example, the score P=48 corresponds to 10/30). The duration of the questionnaire test will normally be one hour, but can be individually extended in the presence of documented personal needs (e.g., DSA). In the latter case, the student must inform me well in advance by email of the request for increased time. If a long disconnection occurs during the test of the questionnaire, I reserve the right to ask the student to take the test again on another date. Any student who has to leave his/her workstation (for example, to go to the bathroom) will obviously be able to do so, without however involving a change in the time of delivery of the questionnaire.

WRITTEN TEST: The written test usually consists of nine exercises that must be solved in person. The final exam mark is the mark obtained in the written test. 

ORAL EXAMINATION: Each student admitted to the written test can request to be admitted to a supplementary oral test. Except in exceptional cases, the final mark obtained with the oral test can differ from the score of the written test by at most 8 points. The oral exams will take place starting from the date written on AlmaEsami, according to a schedule that will be communicated via email and/or on Virtual (https://virtuale.unibo.it/).

VERBALIZATION: The outcome of the exam must be recorded within five days of obtaining the vote. The scores obtained in the questionnaires, in the written tests, and in the oral tests are valid only for the session in which they are achieved.

The verbalization is ordinarily done in the presence of the student, during the oral exams of the session in which the written exam was taken. Only in the case of serious personal impediments will it be allowed to record the vote obtained in a different way.

REGISTRATION: Registration for the exam sessions is done on AlmaEsami [https://almaesami.unibo.it/almaesami/welcome.htm], starting about two weeks before the questionnaire test. Unregistered students will not be able to take the exam.

It is necessary to show up for the written test and for the eventual oral test with the university card and an identity document with photo.

IMPORTANT NOTE: Any student who finds errors in the correction of his/her exam papers (for example due to typos that I didn't realize) is invited to highlight them when handing in the corrected papers, during the oral exams. If the error is actually present and affects the mark attributed to the student's exam, this will be increased accordingly (without decreasing the marks obtained by the other students). This increase will be attributed only upon request of the student concerned.

Teaching tools

Instructional material: some instructional material will be available at the web page http://www.dm.unibo.it/~frosini/ .

Links to further information


Office hours

See the website of Patrizio Frosini