29228 - Geometry and Algebra T

Course Unit Page

Academic Year 2018/2019

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/programmacorso2018.shtml


A. Gimigliano, A. Bernardi, "Algebra lineare e geometria analitica", CittàStudiEdizioni, 2014.

Teaching methods

Taught class.

Assessment methods

The exam consists of a compulsory written test ("final exam") and an optional oral exam (from which the exemption can be suspended in the opinion of the teacher). Both embrace the entire program of the course.

The written test consists of two parts: a theory sheet with 36 questions and a worksheet. The theory sheet must be completed during the first hour in total absence of aids, while during the second hour, intended for the exercises, it is permitted and indeed recommended to use books and notes. Calculators are also allowed, but only for elementary numerical calculation. The theory sheets are collected together at the end of the first hour. On the worksheet only the final results of the calculations must be transcribed in the spaces indicated.
To be admitted to the oral examination it is necessary to have obtained at least 6 points both in the theory questionnaire and in the exercises. The tests that do not reach these thresholds are reported in the list of grades as N.A., i.e. Not Admitted to the oral test (however optional). The final mark of the written tests is simply the sum of the scores obtained in the theory sheet and in the exercises, rounded up to the upper level. Except in special cases, the final grade obtained with the oral test may differ from the total score of the written test of at most 8 points.

IMPORTANT NOTE: if the mark obtained in the questionnaire is less than 6 the part related to the exercises will not be corrected. The exercises will be corrected upon request during the office hours.
If the threshold of 6 points is reached or exceeded both in the theoretical questionnaire and in the exercises, the final mark of the written test is simply the sum of the scores achieved in its two parts, rounded up to the upper level. This sum of scores is recorded as a final mark (or used as a reference mark for a possible oral) in the sense that the scores 30, 31, 32 become the mark 30, and the scores 33, 34, 35, 36 become the mark 30L. If more than 18, this definitive mark can be directly recorded. The recording procedure is usually done in the presence of the student, during the oral tests of any exam session of the academic year in which the written test was held. Only in the case of serious personal impediments it will be allowed to record the mark in different ways.

Please note that if at least 6 points have been obtained both in the theoretical questionnaire and in the exercises it is allowed to take the oral exam even with a mark lower than 18. In this case, however, any negative result will be recorded in the minutes.

Any mark equal to or greater than 18 obtained in the final exam will remain valid throughout the academic year. Any student who has obtained a grade equal to or greater than 18 in the written test of a round of exams has the possibility to take an optional oral examination in the same round of exams (or even in another round of exams of the same academic year). Marks below 18 obtained by taking at least 6 in both parts of the written test entitle the student to take the optional oral examination only within the same round of exams of the written test. Based on the outcome of the oral test the mark can be increased. Except in special cases, the final grade obtained in the oral exam may differ from the total score of the written test of at most 8 points.

Students can sign up for written exams on AlmaEsami [https://almaesami.unibo.it/almaesami/welcome.htm].
During written exams students are requested to show their university ID card and a photo ID.

IMPORTANT NOTE: Any student who finds errors in the correction of her/his exam (for example due to typos in the test) is invited to inform me about that. If the error is actually present and affects the grade 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 by the interested student.

See also the web page http://www.dm.unibo.it/~frosini/modalitaesame2018.shtml

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