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

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

Prerequisites
none

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
There 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 above examinations that they have adequately understood the topics of the course.

LEARNING MEASUREMENT CRITERIA
Candidates passing the exam have a final grade between 18 and 30 cum laude.

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). 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

Courses

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