Academic Year 2023-2024

GEOMETRY 1

Teachers

Stefano Urbinati
Sebastiano Thei
Course Year
1
Unit Credits
12
Teaching Period
Annuity
Course Type
Basic
Prerequisites. The only requirement is a basic knowledge of algebra.
Teaching Methods. Theoretical lessons (70% of the time) alternated with sessions of problem-solving (30% of the time). There will be a tutor that will help students with solving exercises.
Verification of Learning. Written exam consisting of exercises to test the abilities, and an oral exam with theoretical questions to check the knowledge of the subject.

It will be possible to replace the written exam by two partial tests, one half-way through the course and one at the end of the course.

More Information. The textbook covers the theory of this course. Moreover, the instructor will provide notes, which will be available on the e-learning webpage for the class.
Objectives
https://www.uniud.it/en/education/offer/courses/scientific-area/mathematics-computer-sciences-multimedia-physics/bachelor-degree-courses-undergraduate-180-ects/mathematics
Contents
Solving a system of linear equations. Gauss elimination method.

Matrix calculus, invertible matrices. Rank of a matrix. Algebraic properties of matrices.

Vector spaces, subspaces, linear dependence, bases. Dimension of a vector space.

Sums of vector spaces, intersection of vector spaces.

Linear maps. Kernel and image of a linear map. Matrix of a linear map. Matrix associated with a change of basis. Duality

Determinant of a matrix. Eigenvalues and eigenvectors of a linear map. Diagonalizable matrices. Page-Rank algorithm. Jordan’s theory

The space of geometric vectors: inner product and its properties, the norm of a vector, Schwarz inequality.

Quadratic forms. Symmetric bilinear forms. Spectral theorem for real symmetric matrices.

Affine spaces and subvarieties. Affine coordinates. Affine transformations. Euclidean space. Isometries. Parallel, incident and skew subvarieties. Distance, angles. Volume of parallelepipeds: explicit formulas.

Texts
– Edoardo Sernesi – Geometria. Vol. 1 – Bollati Boringhieri 1989

– A. Bernardi, A. Gimigliano – Algebra lineare e geometria analitica

(Seconda edizione) – 2018 – ISBN:

9788825174243

– Notes from the instructor