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Geometria (EA)
Geometry
Chiara De Fabritiis

Seat Ingegneria
A.A. 2015/2016
Credits 6
Hours 72
Period 1s
Language ENG

Prerequisites
Good knowledge of the contents of the program of mathematics of Liceo Scientifico

Learning outcomes
The course aims to give basic knowledge as regards linear algebra and analytic geometry, in all aspects directly and indirectly associated with the identification on the plane and in the space of geometric shapes.

Program
Vector spaces. Basis of a vector space, coordinates. Dimension of a vector space. Grassman’s theorem. Linear maps. Kernel and image of a linear map. Dimension theorem. Linear systems. Rouche’s theorem. Ladder reduction. Operation on matrices and linear maps. Sum and composition of linear maps. Isomorphisms. Product of matrices. Invertible matrices. Change of basis.. Matrix associated to a linear map with respsct to two basis. Similar matrices. Determinant. Eigenvalues and eigenvectors. Triangolable and diagonalizable endomorphisms. Characteristic polynomial. Algebraic and geometric multiplicity. Necessary and sufficient criterion for diagonalizability of an endomorphism. Scalar products. Cauchy’s inequality.. Congruent matrices. Symmetric and orthogonal endomorphisms.Spectral theorem. Affine and Euclidean geometry. Conics and quadrics

Development of the examination
LEARNING EVALUATION METHODS
The learning evaluation method consists of two parts: - a written exam, with 3 preliminary questions and 3 exercises on topics treated in the classroom lessons, time: 3 hours; - a written and oral exam, consisting on the written exposition of 2 theoretical topics to be completed in 50 minutes and a subsequent discussion on one or more points seen in the course. The written exam is a prerequisite for the oral exam, to take it the student should obtain at least ”appena sufficiente” in the written exam. The oral exam has to be taken within the next exam session in which you passed the written exam and in any case during the accademic year 2015-2016. If the oral exam is not passed, the student should taken again also the written exam .

LEARNING EVALUATION CRITERIA
In the written exam, the student should prove his/her ability to solve the exercises regarding the topics of the course. In the oral exam, the student should prove his/her understanding of the features of the mathematical tools introduced in the lectures. To pass the oral exam, the student should prove to have a general knowledge of the topics, explained in a sufficient correct mathematical language. Top marks will be obtained by showing a deep knowledge of the contents explained with a complete mastery of mathematical language.

LEARNING MEASUREMENT CRITERIA
Final marks are expressed in thirtieths

FINAL MARK ALLOCATION CRITERIA
The written exam marks are ”insufficiente”, ”appena sufficiente”, ”sufficiente”, ”discreto”, ”buono”, ”ottimo”. The final marks takes into account the mark of the written exam, the ampleness and correctness of the answers to the written theoretical questions and the mastery of the subject in the oral questions. Full marks are given to students who took all the proofs completely and correctly and who showed and cleverness in the oral exposition and in the written part of the examination.

Recommended reading
Abate, C. de Fabritiis ”Geometria analitica con elementi di algebra lineare”, third ed., McGrawHill. M. Abate, C. de Fabritiis ”Esercizi di Geometria”, McGraw-Hill

Courses

• Ingegneria Edile-Architettura (Corso di Laurea Magistrale con Riconoscimento Europeo (DM 270/04))