# Unit GEOMETRY

Course
Civil and environmental engineering
Study-unit Code
GP004388
Curriculum
In all curricula
Teacher
Federico Alberto Rossi
Teachers
• Federico Alberto Rossi
Hours
• 48 ore - Federico Alberto Rossi
CFU
6
Course Regulation
Coorte 2023
Offered
2023/24
Learning activities
Base
Area
Matematica, informatica e statistica
MAT/03
Type of study-unit
Obbligatorio (Required)
Type of learning activities
Attività formativa monodisciplinare
Language of instruction
Italian
Contents
1. Vector Spaces
2. Matrices
3. Linear Systems
4. Linear Applications
5. Diagonalizability of matrices
6. Euclidean geometry of plane and space
7. Real Projective Planes Conics and Quadrics
Reference texts
Recommended Textbooks:
1. M. Abate, C. de Fabritiis "Geometria Analitica" McGraw-Hill
2. M. Abate, C. de Fabritiis "Esercizi di Geometria e Algebra lineare" McGraw-Hill
3. A. Basile "Algebra Lineare e Geometria Cartesiana" COM edizioni

Other Textbooks:
1. A. Basile, L. Stramaccia "Algebra lineare e geometria" COM Publishing and Communicaiton
2. E. Schlesinger "Algebra lineare e geometria" Zanichelli
3. L. Mauri E. Schlesinger "Esercizi di algebra lineare e geometria" Zanichelli
4. G. Catino, S. Mongodi "Esercizi svolti di geometria e algebra lineare" Esculapio
5. S. Lang "Algebra Lineare" Bollati-Boringhieri
Educational objectives
The main objective of teaching is to provide students with knowledge in the area of linear algebra, Euclidea geometry, curves and surfaces, so that they can use mathematical tools in their future studies. Particular focus is given to comprehension of arguments and rigor in the presentation of ideas and reasoning.

Knowledge and Understanding:
Mathematical understanding of the proposed topics and knowledge of both the theory carried out and fundamental examples. Mode of testing
knowledge: Written examination.

Skills:
Be able to read and understand, independently, basic Linear Algebra texts. Connect similar arguments, find examples and counterexamples. Be able to understand and solve problems and exercises that are unfamiliar but clearly related to what has been done in theory and in lecture. Mode of testing skills: Written exam.

Autonomy of judgment:
The exposition of content and arguments will be carried out in a way that enhances the student's ability to recognize rigorous demonstrations, identify fallacious reasoning, and adopt optimal strategies for solving problems and exercises.

Communication skills:
The presentation of topics will be carried out in a way that will enable the acquisition of a good ability to communicate problems, ideas and solutions, both in written and oral form.
Prerequisites
Set theory. Functions and Applications. Equivalence relations and partitions. Binary operations. Complex numbers. Polynomials, division, roots and reducibility. Basic algebraic calculations.
Teaching methods
The course is organized in classroom lectures on all course topics. Part of each lecture will be devoted to solving problems and exercises.
Other information
Attendance is strongly recommended.

For information on support services for students with disabilities and/or learning disability ("DSA") visit the university web page https://www.unipg.it/disabilita-e-dsa .
Learning verification modality
The exams are structured into several tests, as follows.

1) Theory test (multiple-choice test): n multiple-choice questions. Evaluation is done by assigning the following scores: +3 for a right answer, -1 for a wrong answer, 0 for a question left unanswered. A score of at least 3n/2 (i.e., 50%) must be obtained to pass the test.

2) Written exam, in which you have to solve some exercises (such as those done in the lectures and tutorials) in 120 minutes, justifying all the steps thoroughly. A score of not less than 15/30 is required to pass the test.

The theory test and the written exam are held on the same day, one after the other. Consultation of books and notes is not allowed during the conduct of exam.

The final grade (on a scale of 30) will be the weighted sum of the theoretical and written test grades, with weights of 1 and 3, respectively. The examination is passed if the final mark is not less than 18.

An optional oral test may take place at the request of the lecturer or the student.

For information on support services for students with disabilities and/or learning disability ("DSA") visit the university page: https://www.unipg.it/disabilita-e-dsa
Extended program
1. Vector spaces: linear dependence, bases. Euclidean scalar product, vector product and geometric interpretation.
2. Matrices: operations, rank, invertibility, determinant. Elementary transformations and reduction to scale.
3. Systems of linear equations: basic results and Rouché-Capelli and Cramer theorems.
4. Linear applications: associated matrix, properties.
5. Diagonalizability of matrices: eigenvalues, eigenvectors, algebraic and geometric multiplicity. Spectral Theorem.
6. Affine and Euclidean geometry in the plane and space: affine linear subspaces.
7. Real Projective Plane. Conics and Quadrics: Projective Plane and Homogeneous Coordinates. Representation of lines and planes in homogeneous coordinates. Complexification of the real plane. Classification of conics.
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