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Geometria (MECC) (A/L)
Geometry
Agnese Ilaria Telloni

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

Prerequisites
Contents of precourses

Learning outcomes
Basic linear algebra geometry notions relevant for engineering students.

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 Theorem. Binet Theorem. The inverse of an invertible matrix. Rank and independence of columns (rows). Gauss elimination. Linear systems. Cramer 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. Standard scalar product. Cauchy-Schwarz and triangle inequalities. 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 and skew-symmetric endomorphisms. Spectral theorem. Orthogonal endomorphisms. Plane geometry: points, lines, direction vectors, midpoint of a line segment. Mutual positions of lines. Distance. Circles. Sheaves of lines. Change of cartesian coordinates. Conics and their classification. Space geometry: points, planes, lines, direction vectors. Mutual positions of points, lines, and planes. Distance. Spheres. Vector product. Area of the parallelogram and triangle. Sheaves of planes. Sheaves and stars of lines. Mixed product. Volume of the parallelepiped and tetrahedron. Change of cartesian coordinates.

Development of the examination
LEARNING EVALUATION METHODS
The learning level will be evaluated by a written exam, composed by 4 exercises and lasting 2 hours and a half, and by an oral exam. Those who have reached a minimum score of 18/30 will be admitted to the oral exam. The written and the oral part of the exam must be taken in the same exam session.

LEARNING EVALUATION CRITERIA
In order to successfully deal with the exam, the student has to: - know the fundamental methods and concepts of Geometry - show the ability to apply correctly the acquired knowledge in the given exercises - be able to link, mix and elaborate the acquired knowledge in order to create new strategies.

LEARNING MEASUREMENT CRITERIA
The learning level will be determined by these basic parameters: - knowledge of methods and concepts of Geometry - ability to explain with appropriate language the discussed topics - ability to mix and actively manipulate the acquired knowledge.

FINAL MARK ALLOCATION CRITERIA
Both the written and oral exam are evaluated in thirtieths. The final mark will be determined by the average between the two scores.The minimum sufficient score is assigned to the student who has the basic knowledge of the topics discussed, and the ability to solve simple algebric and geometric problems. The maximum score is assigned to the student who shows the ability to correctly explain the newly-acquired knowledge and to actively mix, link and operate with it, in order to create new strategies for new problems.

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. A. Cavicchioli, F. Spaggiari, “Primo modulo di Geometria“, Pitagora. A. Cavicchioli, F. Spaggiari

Courses

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