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.

Bilinearand quadratic forms.
Bilinear forms. Matrix representation. Symmetric matrices. Quadratic forms. Canonical forms.

Hyperquadrics.
Hints to the classification of realconic 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. 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.

During the lectures period, an "in itinere" test takes place, whose structure is identical to the one of the final test. For the theory "in itinere" part the 5.5 points threshold DOES NOT apply, and negative scores are changed to zero.

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

The score of the "in itinere" test is valid only up to the beginning of the second semester lectures and acts on the final score only if favourable:

P=note of the in itinere test, F=note of final test.

The final note D 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. For exams taking place after the beginning of the second semester lectures D = F. The student is admitted to the oral part immediately following the written part if the final note D is at least of 15 points. Each final written test is valid exclusively for the oral part of the same "appello" (i.e. immediately after the written part).

Applications for the exam (NOT required for the "in itinere" tests) must be carried out by AlmaEsami.

Please come to all tests with your university card.

Teaching tools

One can download the exam testsof the Academic Year 2010-2011 and of the Academic Year2011-2012: 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 ofProf. Luciano Gualandriand ofProgetto Matematic@.

Links to further information

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

Office hours

See the website of Massimo Ferri