Geometry

Learning objectives

The main objective of the course is to provide students with the tools to understand and apply the concepts of linear algebra, Euclidean geometry and conic theory in various disciplinary contexts.
Through classroom lessons led by the teacher and independent and conscious study, the student is expected to be able to address the topics described in the program in order to:
• Acquire mathematical knowledge as a personal asset that can be used at any time in one's educational journey.
• Be able to interpret complex engineering solutions using linear algebra theory.
• Understand the practical applications of geometry in engineering.
• Have the ability to apply geometric principles in the design and resolution of engineering problems.
• Have the ability to independently assess one's own knowledge and skills.
• Have the ability to communicate your ideas effectively and argue accurately.

Expected learning outcomes

At the end of the course, the student will be able to:
1. Understand and apply concepts of vector spaces, bases and dimensions.
2. Solve linear systems using algebraic and geometric methods.
3. Calculate and interpret eigenvalues ​​and eigenvectors.
4. Use linear transformations and matrices to describe geometric properties.
5. Represent and analyze Euclidean and affine geometries in n-dimensional spaces.
These topics provide a fundamental foundation for developing problem-solving skills in engineering courses.

Course content

Matrices and linear systems. Matrices: notable matrices; operations between matrices; determinant of a matrix and related properties; Laplace's theorem; inverse matrix; rank of a matrix.

Systems of linear equations: linear systems and associated matrices; step linear systems; Cramer linear systems; the Rouché-Capelli theorem; Gaussian reduction.

Vector spaces. Fundamental models of vector space. Vector subspaces. Matrices and vectors. Linear dependence and independence of a set of vectors. Bases and dimension of a vector space. Linear applications: kernel and image of a linear application; dimensional equation; associated matrices; changes of basis. Diagonalization of matrices: eigenvalues ​​and eigenvectors of an endomorphism; spectral theorem and similarity diagonalization of a square matrix. Standard scalar product and standard n-dimensional Euclidean vector space: quadratic forms; scalar product and norm of a vector; angle between vectors; orthogonality; orthonormal bases; vector product and related properties.

Euclidean spaces. Reference systems. Real Euclidean plane: representations of a line; reciprocal positions between lines; bundles of lines; parallelism and orthogonality; Euclidean distance. Real Euclidean space of dimension three: representations of a line and a plane; reciprocal positions between planes, between lines and between lines and planes; parallelism and orthogonality, distances.

Theory of conics. Homogeneous and non-homogeneous equations; associated matrices. Degenerate and non-degenerate conics. Classification of non-degenerate conics. Center of a conic. Diameters. Asymptotes. Axes and vertices. Euclidean canonical equations of conics. Equilateral hyperbolas, empty ellipses and circles. Foci and directrices. Introduction to conic sheaves.

Prerequisites

It is important to be familiar with the following concepts: basic concepts in set theory and associated symbols; knowledge of fundamentals of geometric polynomials and vectors; understanding of basic algebraic structures, including the concepts of groups and fields.

Bibliography

Abate, M., & de Fabritiis C. (2010). Analytic geometry with elements of linear algebra. Milan: McGraw-Hill.
Bernardi A., & Gimigliano A. (2014). Linear algebra and analytic geometry. Turin: CittàStudi Edizoni, De Agostini School.
Cavicchioli, A., & Spaggiari, F. (2002). First module of geometry. Bologna: Pitagora Editrice.
Cavicchioli, A., & Spaggiari, F. (2004). Second module of geometry. Bologna: Pitagora Editrice.
Casali, MR, Gagliardi, C., & Grasselli, L. (2010). Geometry (new ed.). Bologna: Aesculapius.
Cattabriga A., & Mulazzani M. (2019). Solved exams in Geometry and Algebra for the Degree courses in Engineering, Progetto Leonardo. Bologna: Aesculapius.
Gualandri L. (2007). Linear Algebra and Geometry-Quiz solved exam. Bolonia: Esculapio.
Lang, S. (2012). Introduction to linear algebra (2nd ed.). Turin, Italy: Boringhieri joint stock company.
Other texts:
Landi, G., & Zampini, A. (2018). Linear Algebra and Analytic Geometry for Physical Sciences. Springers. doi: 10.1007/978-3-319-78361-1
Neri, F. (2016). Linear algebra for computational sciences and engineering. Cham: Springer. doi: 10.1007/978-3-319-40341-0
Penney, R. C. (2008). Linear algebra: ideas and applications. New York, NY: Wiley-Interscience.
Scovenna, M., Citterio, MG, & Moretti, A. (2001). In-depth notebook. Linear algebra. CEDAM School
Ruini-F, B., & Spaggiari, F. (2002). Geometry exercises. Bologna: Pitagora Publishing.
Sernesi, E. (1989). Geometry I. Turin: Boringhieri.

Teaching methods and tools

The course includes both theoretical lessons and exercises guided by the teacher. The teaching material is structured in 8 chapters that will be made available by the teacher on the platform before the start of the lessons, allowing students to have the study material in advance.
Each chapter includes:
Theoretical reminders with guided examples and solved exercises.
Suggested exercises, organized by topic and difficulty level.

During the theoretical lessons, the teacher will provide slides to illustrate the topics in a clear and understandable way to the students. These lessons will be integrated with practical exercises to put into practice what has been learned.

Geometry exercises are structured so that students can apply the theory learned during the lessons, both working in groups and independently. The teacher will provide the related solutions to facilitate the learning process.

At the end of the course, the teacher will provide further study material, including sample exam papers, in order to consolidate the knowledge acquired.

Assessment methods and criteria

The exam includes a written test and an oral interview. The oral interview focuses on the theoretical topics of the program and is aimed at assessing the level of knowledge and understanding of these topics. It is mandatory to obtain a sufficient grade in both parts, both in the written test and in the oral interview. The final grade will be the average of the grades obtained in the two parts.