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

Theory

 

Some algebraic structures.

Groups, rings, fields.

Matrices

Initial definitions. Operations. 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.

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

Euclidean vector spaces.
Scalar products. Orthogonal operators. Orthogonal complement.

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. In particular:
Cap. 10: only sect. 4. Cap. 12: Def. 12.17, 12.18, 12.20, Prop. 12.24, 12.30, 12.46, Oss. 12.47, Def. 12.54, 12.55, 12.57, Teor. 12.88, Prop. 12.86 (in this order).
ATTENTION: for the numbering of the 2010 edition, please go to http://www.dm.unibo.it/~ferri/hm/progt.htm

 

Exercises

Computation of determinants and ranks of matrices. Discussion and resolution of linear systems. Determining and representing linear transformations. Determining equations of linear and affine subspaces. Passage between different representations. Computation of eigenvalues and eigenvectors. Matrix diagonalization. Resolution of problems concerning parallelism and orthogonality. Representation and study of bilinear and quadratic forms. Classification of conic sections.

Readings/Bibliography

· Casali M.R., Gagliardi C., Grasselli L., "Geometria", Progetto Leonardo, Ed. Esculapio, Bologna, 2016 (official textbook of the course).

ATTENTION - Harriot-Descartes Theorem, not present in the 2000 edition, and partially uncorrect in the 2002 one, can be downloaded here.

As for exercises, of course the first thing to do is to download the exam tests and try to solve them without help, then compare solutions. If one wishes to have the support of a textbook, any exercise book of geometry and linear algebra will do. Of course, one has to watch out for notation differences. I indicate the titles of three exercise books written by colleagues.

· A. Barani, L. Grasselli, C. Landi, "Algebra lineare e Geometria - Quiz ed esercizi commentati e risolti", Progetto Leonardo, Bologna, 2005.

· L. Gualandri, "Algebra lineare e Geometria – Esercizi e quiz risolti e d'esame", Progetto Leonardo, Bologna, 2007.

· G. Parigi, A. Palestini, "Manuale di Geometria, Esercizi", Pitagora Editrice Bologna, 2003.

Teaching methods

Traditional lecture.

Assessment methods

The exam consists of a compulsory written test ("prova finale", final test) and of an oral part. Both concern the whole program covered in the lectures.

The written test is composed by two sections: a theory form with nine multiple choice questions, and an exercise sheet. The theory form must be filled during the first hour, in total absence of helps, whereas during the second hour, dedicated to the exercises, books, lecture notes, computing tools are allowed and even recommended. The theory forms are gathered at the end of the first hour all together.

ATTENTION: the test is considered below standards, if the theory section scores less than 5.5 points. In this case (which is referred to as N.C., "Non Classificato", in the note list) the exercise section will not be marked. The exercises can be marked during office hours.

If the threshold of 5.5 points is reached or exceeded, the note of the final test (here denoted by F) is simply the sum of the scores of the two sections.

The student is admitted to the oral part immediately following the written part if its final note is at least of 15 points. Each final written test is valid for three calls, the one of the written test included.

Applications for the exam must be carried out by AlmaEsami [https://almaesami.unibo.it/almaesami/welcome.htm] .

Please come to all tests with your university card.

Teaching tools

One can download the exam tests of the Academic Year 2015-2016 and of the Academic Year 2016-2017: they are an essential part of the course. Here are also some hints (1 and 2) to applications, projected during the course. Projected pictures: first part, second part and third part.

The hardcopy of the exam tests is available at the copy center of the Engineering Faculty.

We suggest to visit the sites of Prof. Luciano Gualandri and of Progetto Matematic@.

Links to further information

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

Office hours

See the website of Massimo Ferri