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Geometria (EL)
Geometry
Chiara De Fabritiis

Seat Ingegneria
A.A. 2016/2017
Credits 6
Hours 48
Period I
Language ENG

Prerequisites
Good knowledge of the contents of the program of mathematics of Liceo Scientifico . Complex numbers.

Learning outcomes
KNOWLEDGE AND UNDERSTANDING:
The course gives the students the basic knowledges in geometry and in the field of complex numbers, using in particular tools in linear algebra and analytic geometry such as vector spaces, linear applications and their representations in terms of vectors and matrices. These knowledges cooperate with the ones learned in the courses of Mathematical Analysis I and II so that the student acquires the ability to understand criteria, procedures and limits of application of mathematical methods to real problems.
CAPACITY TO APPLY KNOWLEDGE AND UNDERSTANDING:
The student must develop the ability to apply the tools of linear algebra and analytic geometry such as vector spaces, linear applications and their representations in terms of vectors and matrices in order to formalize, identify and solve problems in Electronics Engineering.
TRANSVERSAL SKILLS:
Solving problems and exercises, that will be worked out both during practice hours and at home, and the preparation for the oral part of the exam will improve judgment independence in general, communication skills, autonomous learning abilities and the capability of the student to draw conclusions.

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.

Development of the examination
LEARNING EVALUATION METHODS
The learning evaluation method consists of two parts: - a written exam, with preliminary questions and exercises on topics treated in the classroom lessons; - a written and oral exam, consisting on the written exposition of theoretical topics 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 2016-2017. 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 Elettronica (Corso di Laurea Triennale (DM 270/04))