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Geometria (CA)
Geometry
Maria Chiara Brambilla

Seat Ingegneria
A.A. 2015/2016
Credits 9
Hours 72
Period I
Language ENG

Prerequisites
none.

Learning outcomes
Students must be able to use the tools of analytic geometry and linear algebra and to apply them to the solving of scientific and technological problems

Program
Vector spaces. Basis and dimension of a vector space, coordinates. Grassmann's theorem. Linear maps. Kernel and image of a linear map. Dimension theorem. Linear systems. Rouche'-Capelli's theorem. Ladder reduction. Affine geometry. Equations of lines and planes. Mutual position of points, lines and planes; incidence and parallelism conditions. 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 respect 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. Euclidean geometry. Conics and quadrics.

Development of the examination
LEARNING EVALUATION METHODS
The learning evaluation is carried out by two exams: - a practical examination, which consists of solving exercises and problems related to the topics explained in the course. The test must be completed in 3 hours; - a theoretical examination, consisting in a discussion, written and oral,of the topics of the course. In particular the knowledge and the understanding of all definitions, theorems and proofs explained in the classes will be tested. The practical exam is preliminary to the theoretical one. It is necessary to pass the practical exam in order to do the theoretical one. The two exams must be passed in the same exam session. If the student fails the theoretical exam, he/she must repeat also the practical one.

LEARNING EVALUATION CRITERIA
In order to pass the learning evaluation, the student must demonstrate that he/she has understood the basic concepts of linear algebra and geometry explained in the course. In particular in the practical test the student must show that he/she is able to apply independently the learned techniques in solving exercises and problems. In the theoretical exam the student must be able to expose the theoretical contents with the correct language and accuracy.

LEARNING MEASUREMENT CRITERIA
Each of the tests is graded on a scale from 0 to 30. The final grade will be decided starting from the two test grades.

FINAL MARK ALLOCATION CRITERIA
The final grade will be positive only if in both of the tests the students gets the passing grade (18/30). The maximal grade is reached if the student proves a knowledge and a thorough understanding of the course content. The maximal grade with honors is reserved to the students who passed both of the tests in a complet and correct way, showing special independence and excellence.

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

Courses

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