81855 - GEOMETRIA 1A

Learning outcomes

At the end of the course the students know some fondamental concepts of linear algebra (matrices, linear systems, vector spaces, linear maps) and can apply them to the solution of problems from analytic geometry.

Course contents

Matrix calculus,  determinants. Systems of linear equations and matrix calculus: Gauss methods for their solutions.  Vector calculus and cartesian geometry: dot product, cross product. Geometry of plane and space, lines, planes and their relative position.  Basic Linear Algebra: Vector spaces, subspaces. Notions of linear dependence and independence. Basis and systems of generators for a vector space. Existence of basis. Dimension of a vector space and of its subspaces. Grassmann relation, direct sum of vector spaces. Linear maps. Kernel and image of a linear map. The vector space of linear maps. Dual of a vector space. Linear maps and matrices. The matrix associated with a linear map. Canonical forms for endomorphisms: Endomorphisms of a vector space, eigenvalues and eigenvectors. Diagonalizability: necessary and sufficient conditions.

Readings/Bibliography

There is no official textbook. The following books, and many others, may be useful:
M.Abate: Geometria (McGraw Hill)  
S.Greco - P.Valabrega :  Lezioni di Geometria. Volume I: algebra lineare. Volume II: geometria analitica.  Ed. Levrotto e Bella, Torino
 E.Sernesi: Geometria 1 (Bollati Boringhieri)

Teaching methods

Lectures at the blackboard and discussion of exercises, office hours.

Assessment methods

The exam consists of a written and an oral examination. The written part will check the ability of the student to solve problems concerning the linear geometry  of three-dimensional space, the solution of systems of linear equations, the study of linear maps and  diagonalization of endomorphisms of vector spaces. The oral part will  verify that the basi concepts and theorems have been understood in a satisfactory way allowing the student to effectively apply them.

Teaching tools

Lectures with exercise sessions. Sheets of exercises will be handed out during the lectures (see http://www.dm.unibo.it/~manaresi/   and  http://www.dm.unibo.it/~ida/  ) , in addition to the ones available in the suggested textbooks. 

In the office hours students will be coached individually.

Links to further information

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

Office hours

See the website of Mirella Manaresi

See the website of Monica Idà