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Geometria (MECC) (M/Z)
Geometry
Mario Marietti

Seat Ingegneria
A.A. 2016/2017
Credits 9
Hours 72
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 aims to provide the student with a clear understanding of the basic ideas of linear algebra and analytic geometry and also with the capability of understanding more advanced topics in this area of mathematics. This will equip the student with the necessary tools pertaining to the basic scientific background and the engineering applications
CAPACITY TO APPLY KNOWLEDGE AND UNDERSTANDING:
At the end of the course, the students will be able to apply the acquired knowledge and understanding to the analysis and modelling of engineering problems, by using properly the methods of linear algebra and of analytic geometry taught during the course. The ability to apply these methods will be acquired by the student also by means of problem sessions which require their use
TRANSVERSAL SKILLS:
On completion of the course, the students will be able to set up and solve problems by logical deductive reasoning, which is essential in all scientific and technological disciplines. In addition, the students will improve the ability to develop independent thought and learning capabilities. Oral presentations of the topics taught in the course, with the language proper of the basic disciplines of the degree course will help developing communication skills

Program
The space of the mxn matrices: sum and product by scalars. The transpose. Square, symmetric, skew-symmetric matrices. Product of matrices. Invertible matrices. The determinant and its properties. Laplace's Theorem. Binet's Theorem. The inverse of an invertible matrix. Rank and independence of columns (rows). Gauss elimination. Linear systems. Cramer's Theorem. Rouché-Capelli Theorem. Linear systems with parameters. Ladder reduction. Vector spaces and vector subspaces. Generators of a vector space. Linear independence of vectors. Bases, coordinates, and dimension. Vector subspaces of Rn: bases, dimension, equations. Change of bases and coordinates. Grassmann Formula. Affine subspaces. Linear maps. Matrices associated with a linear map. Kernel, Image, and their dimensions. Isomorphisms. Scalar product. Cauchy-Schwarz inequality. Projections. Fourier coefficient. Orthogonal and orthonormal bases. Gram-Schmidt process. Change of orthonormal bases. Orthogonal matrices. Endomorphism and change of bases: similar matrices. Diagonalizable endomorphisms and diagonalizable matrices. Eigenvectors and eigenvalues. Characteristic polynomial. Algebraic and geometric multiplicity. Criteria for diagonalizability. Symmetric endomorphisms. Spectral theorem. Orthogonal endomorphisms. Plane geometry: points, lines, direction vectors. Distance. Circles. Sheaves of lines. Change of cartesian coordinates. Conics and their classification. Space geometry: points, planes, lines, direction vectors. Distance. Spheres. Vector product. Change of cartesian coordinates.

Development of the examination
LEARNING EVALUATION METHODS
here will be two examinations: - a written examination, consisting in solving some exercises, - an oral examination, consisting in the discussion of some of the topics (part of the exposition could be asked to be written down). In order to be admitted to the oral examination, the candidate must obtain a positive mark (18 or higher) in the written examination.

LEARNING EVALUATION CRITERIA
In order to pass the exam, students must show in the examinations that they have adequately understood the topics of the course and are able to apply the acquired knowledge and understanding by using properly the methods of linear algebra and analytic geometry taught during the course.

LEARNING MEASUREMENT CRITERIA
n the exams, the teacher evaluate how well the students have understood the topics of the course and are able to apply the acquired knowledge and understanding. The passing grade (eighteen/thirtieths) is given to students that solve the exercises in a sufficient way and prove their knowledge of the fundamental topics of the course. The maximum grade (thirty/thirtieths) is obtained by proving a deep knowledge of the topics of the course.

FINAL MARK ALLOCATION CRITERIA
After the written examination, the papers are marked (a number between 0 and 30). In order to be admitted to the oral examination, the candidate must obtain a positive mark (18 or higher) in the written examination. The final grade of the exam is given after the oral examination (it takes into account both examinations). Candidates passing the exam have a final grade between 18 and 30 cum laude. A final grade of 30 cum laude is awarded to the candidates that have shown exceptional skill in both examinations.

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

Courses

• Ingegneria Meccanica (Corso di Laurea Triennale (DM 270/04))