29228 - Geometry and Algebra T

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

THEORETICAL PART

[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. 

PRACTICAL PART

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/programmacorso2022.shtml

Readings/Bibliography

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

Teaching methods

Taught class.

Assessment methods

Due to the Covid-19 epidemic, the following assessment methods could be changed. Please visit the web page http://www.dm.unibo.it/~frosini/modalitaesame2022.shtml or contact me by email for details (patrizio.frosini@unibo.it).

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1) The exam will consist of an oral examination. During the oral exam, the student will be asked to answer theoretical questions and solve exercises. The reference program will be the one carried out in the last academic year (the program carried out in the 2021/2022 academic year is available on the web page http://www.dm.unibo.it/~frosini/programmi/programma_svolto_GA_2021_2022.pdf ). Questions could be asked on any topic of the program. Each student will obviously have the right to withdraw from the oral examination.

2) Admission to the oral exam will be subject to passing an online test consisting of a 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. Any student can refrain from answering by choosing the "I DON'T KNOW" option. Each correct choice will be awarded 2 points, each incorrect choice 0 points, and the choice "I DON'T KNOW" 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 randomly will be 36, the same that will be obtained by always answering "I DON'T KNOW".) The students who have obtained at least 48 points are admitted to the oral examination. This score choice has been determined on the basis of the technical characteristics of EOL-ESAMI ONLINE. The duration of the questionnaire test will ordinarily be one hour, but can be individually extended in the presence of documented personal needs (e.g., SLD). In the latter case, the student must inform me well in advance by email of the request for increased time. I underline that the typology of the questions, their difficulty and the structure of this new questionnaire is the same as previously used for the old face-to-face questionnaires. The only novelty is the administration via EOL.

If during the test of the questionnaire a long disconnection should occur, I reserve the right to ask the student to resume the test on another date. Any student who has to leave his/her station (for example, to go to the bathroom) will obviously be able to do so, without however this entailing a change in the time of delivery of the questionnaire.

3) The score obtained in the questionnaire will not contribute to the evaluation of the exam and will be used only for admission to the oral exam.

4) Registration for the online test consisting of the questionnaire must be done on AlmaEsami.

5) The oral exams will take place starting from the date communicated on AlmaEsami, according to a calendar that will be published on the web page http://www.dm.unibo.it/~frosini/DIDING.shtml. The outcome of the exam will be recorded within five days of taking it. In the event that a student admitted to the oral exam does not pass this exam or does not intend to take it, he/she must retake the test of the questionnaire in order to be admitted to a subsequent oral exam.

6) The questionnaire will be administered via EOL and its completion will be checked via the Zoom platform. The oral exams will take place face-to-face unless the University decides otherwise. In the case that the oral exam takes place online because of the epidemic, it is necessary that I can see both the student and what he/she writes and therefore the optimal solution would be a white board or vertical white sheets on which to write with markers. Alternatively, a tablet can be used, taking care to view both what the student writes and his/her image taken from a webcam. If this is not possible, the student can use a sheet of paper on a table, showing me from time to time what he/she is writing. However, it is important that I can see not only what the student writes, but also what is around him/her.

7) This new examination method obviously cannot give the same verification guarantees as the face-to-face one and will require collaboration between teacher and student. I am forced to specify that any attempt to break the rules will be severely sanctioned.

8) I do not know how long this exam method will remain in effect. It all depends on the decisions of our university in relation to the evolution of the epidemic.

9) I invite students not to worry about the new exam method: the oral exams will have the same difficulty seen in the past.

10) I invite you to frequently read the web page http://www.dm.unibo.it/~frosini/DIDING.shtml, where I will put any communications for students. Any student with technological difficulties is kindly requested to contact me. I remain available for any request for information relating to topics not illustrated here.

See also the web page http://www.dm.unibo.it/~frosini/modalitaesame2022.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

http://www.dm.unibo.it/~frosini/

Office hours

See the website of Patrizio Frosini