17990 - Geometry and Algebra (L-Z)

Learning outcomes

The first aim of the course is to give the student a basic knowledge if Linear Algebra. A student should be able, after the course, to decide whether a system of linear equation has solutions or not, to use matrices, to know the operation among them, to know the basics of vector spaces and euclidean spaces.

As for Analytic Geometry, the aim is to acquire the ability to study points, lines and planes in 3-dimensional space, determine parallelism, orthogonality and elementary relations among them.

Course contents

Set Theory:

Operations with sets, partitions, applications. Algebraic structures.

Analytic Geometry:

Cartesian coordinates on the line, the plane and 3-space. Geometric loci, equations (cartesian or parametric) of lines and planes in 3-space. Vectors. Plane conics. Planes and lines in 3-space, equations, parallelism, ortogonality.

Linear Algebra:

Vector spaces, linear dependence, generators and bases. Dimension and subspaces.
Linear systems . Matrices - Row reduction - rank of a matrix - Theorem of Rouché-Capelli . Determinant -linear transforms - Base change.
Eigenvalues and eigenvectors - Diagonalizing matrices.
Metric spaces, scalar product- Norm - Orthonormal Bases

Readings/Bibliography

1) P. MAROSCIA:   INTRODUZIONE ALLA GEOMETRIA E ALL'ALGEBRA LINEARE, ZANICHELLI,  BOLOGNA 2000

2) Notes of the course.

3) M.Abate, C. de Fabritiis: Esercizi di Geometria ,  McGrow-Hill. (Book of exercises)

(All texts in Italian)

Assessment methods

Final exam is made of two parts:

Written exam: a 2 hours assignment, usually with one test about linear algebra and one about Geometry. A mark of 15/30 is enough to pass to the oral exam.

Oral exam: About 20min, small exercises and theory questions.

Teaching tools

On the web site:

http://elearning.ing.unibo.it/index_s.php

one can find notes of the course (in Italian). Moreover on the web site:

http://www.dm.unibo.it/matematica/

Links to further information

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

Office hours

See the website of Alessandro Gimigliano