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Geometria (CA)
Geometry
Amedeo Altavilla

Seat Ingegneria
A.A. 2016/2017
Credits 9
Hours 72
Period I
Language ENG

Prerequisites
Good knowledge of the math program of high school ‘Liceo Scientifico’. Complex numbers.

Learning outcomes
KNOWLEDGE AND UNDERSTANDING:
The course will teach to students basic notions of linear algebra and analytic geometry, for the comprehension and analysis of engineering problems . In particular the course will concern the notions of vector space, linear map, symmetric bilinear forms, scalar product, the geometry of lines and planes, quadric surfaces.
CAPACITY TO APPLY KNOWLEDGE AND UNDERSTANDING:
Students will be able to solve exercises of linear algebra on matrices, linear maps, bilinear forms, to solve linear systems and to solve problems of analytic geometry in the space.
TRANSVERSAL SKILLS:
The study and the exercises proposed will consolidate the “transverse skills” obtained by the students, such as the ability of critically analyze a problem and the ability of use a correct, precise and formal language. The ability to learn in autonomy and to develop the study will be strengthened, too.

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
Written and oral 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 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
M. Abate, C. de Fabritiis ”Geometria analitica con elementi di algebra lineare”, III ed., McGrawHill. M. Abate, C. de Fabritiis ”Esercizi di Geometria”, McGraw-Hill.

Courses

• Ingegneria Civile e Ambientale (Corso di Laurea Triennale (DM 270/04))