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.
 
Algebraic equations . 
 
Eigenvalues.
Eigenvalues and eigenspaces. Matrix similarity. Characteristic polynomial. Diagonalization by similarity.
 
Euclidean vector spaces.
Scalar products. Orthogonality. Orthogonal sets. 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. 12.3, 12.4, 12.6, Prop. 12.8, 12.10, Def. 12.16, 12.17, 12.19, Teor. 12.37, Prop. 12.35 (in this order).

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, Bologna, 2010 (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. This exercises will be marked, if the student asks to sit anyway for the oral part; of course they 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.

During the lectures period, two "in itinere" tests take place: one consists of the only theory section, the other of the only exercise sheet. The score of the two tests is expressed in the same way as for the final test; i.e. each admits a maximum score of 18 (over 15). For the theory "in itinere" test the 5.5 points threshold DOES NOT apply, and negative scores are changed to zero.

Whatever the score obtained in an "in itinere" test (and even in case of no show), one may participate to the other, and of course can participate to the final test, which is compulsory.

The not too simple formula, by which we take the "in itinere" tests into account, has been studied in order to give maximum guarantee to the student. Here it is:

P = (2*note_of_the_better_"in_itinere"_test+ note_of_the_worse_"in_itinere"_test)*2/3;

a note expressed in thirtieths (which can reach 36, anyway) comes out of it;

F=note_of_final_test.

The final note which is recorded (or is used as reference for a possible oral part) is the closest integer to:

max{F, (P+F)/2}

in the sense, however, that we register as 30 the final notes 30, 31, 32, and as honors ("lode") the final notes 33, 34, 35, 36. Approximation is carried out at the last passage. The note can be recorded, if in the final test at least 5.5 points in the theory section have been obtained (and at least 18 from the whole computation).

ATTENTION: one may sit for the oral part even with final note less than 18 or with theory score less than 5.5. In such case, anyway, the failure - if any - will be recorded. The student who, with passing final note, sits for the oral part, automatically renounces to the exemption from the oral part itself, i.e. renounces to the mere recording without oral.

The notes obtained in the "in itinere" tests and the passing notes of the final test have 12 months validity.

Applications for the exam (NOT required for the "in itinere" tests, nor for the oral part) must be carried out by AlmaEsami.
Please come to all tests with your university card or booklet.

Teaching tools

One can download the exam tests of the Academic Year 2008-2009 and of the Academic Year 2009-2010: 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 e 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