29228 - Geometry and Algebra T

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 Virtuale at the end of the lessons.

Readings/Bibliography

The topics covered in the course are classical topics in linear algebra and geometry, and can be found in many texts. mainly, I will  mainly follow the text

                 Savo A., Geometria, Hoepli Editore 2021

but any book on Linear Algebra is fine. As examples, some other 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. Comprehension quizzes administered through Wooclap will be used during the lessons. Asynchronously, the Perusall platform will be used for the shared and commented viewing of introductory videos or in-depth analysis of the topics.

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.

The written test lasts two  and a  half 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, consultation of paper resources such as books, notes, etc. is permitted and indeed recommended. The use of the scientific calculator is also permitted, 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 33. The written test and the oral one have to be taken in the same call. The oral schedule will be published with the results of the written test

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. 

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.

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 site of Prof. Massimo Ferri  particularly the Chicken's corner. 

 

Office hours

See the website of Alessia Cattabriga