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Geometria (EDI)
Geometry
Mario Marietti

Seat Ingegneria
A.A. 2016/2017
Credits 6
Hours 48
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. On completion of the course, the student will be able to understand the tools of linear algebra and analytic geometry that are essential for the basic scientific knowledge and the engineering applications.
CAPACITY TO APPLY KNOWLEDGE AND UNDERSTANDING:
On completion of the course, students will be able to apply their knowledge and understanding skills to the analysis and modeling of engineering problems, by consciously using the acquired methods of linear algebra and analytic geometry. Students will develop the ability to apply these methods also by means of exercises that require the acquired tools to solve.
TRANSVERSAL SKILLS:
On completion of the course, students will be able to set and solve problems through deductive logic. These skills are fundamental in scientific and technological disciplines. In addition, students will improve their ability to learn and their independence of judgment, as well as their ability to communicate effectively, thanks to the specific language of the basic courses.

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. Linear maps. Matrices associated with a linear map. Kernel, Image, and their dimensions. Scalar product. Cauchy-Schwarz inequality. Fourier coefficient. Orthogonal and orthonormal bases. Gram-Schmidt process. Change of orthonormal bases. 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. Plane geometry: points, lines, direction vectors. Distance. Circles. Space geometry: points, planes, lines, direction vectors. Distance. Vector product.

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 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
In 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.

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 Edile (Corso di Laurea Triennale (DM 270/04))