Mathematics I
A.Y. 2025/2026
Learning objectives
This course aims to provide the tools necessary to handle the basics of mathematical formalism and method.
Expected learning outcomes
At the end of the course, students will be able to handle elementary set theory and elementary functions. They will know the notions of group, ring and field, and they will be able to apply them to various fundamental examples (natural, integer, rational, real and complex numbers, permutation group, remainder classes). They will be able to solve systems of linear congruences in an algorithmic and non-algorithmical way. They will know the fundamental properties of vector spaces and their bases, linear applications and their matrix representation, which will provide theoretical and algorithmic methods for solving systems of linear equations.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
The course will cover the following topics:
- Basic operations between sets;
- Relations and their fundamental properties: transitivity, reflexivity, symmetry.
- Fundamental sets of numbers: natural, integers and rational numbers.
- Algebraic equalities and inequalities.
- Elementary functions.
- Induction principle.
- Congruences, Chinese Remainder Theorem.
- Groups, homomorphisms between groups. Permutation groups.
- Fields and rings: definitions, examples, fundamental properties.
- Vectors, operations between vectors. Applications might include geometry in space (if time permits).
- Vector spaces: linear dependence, generators, bases, dimension, Grassman formula.
- Matrices: operations between matrices, relationship between matrices and linear systems, Gauss-Jordan method. Relationship with homomorphisms, search for eigenvalues and eigenvectors, diagonalizability.
If time allows, we will also discuss some basic geometry: planes, lines, parallel, orthogonality and interserctions in dimensions 2 and 3.
- Basic operations between sets;
- Relations and their fundamental properties: transitivity, reflexivity, symmetry.
- Fundamental sets of numbers: natural, integers and rational numbers.
- Algebraic equalities and inequalities.
- Elementary functions.
- Induction principle.
- Congruences, Chinese Remainder Theorem.
- Groups, homomorphisms between groups. Permutation groups.
- Fields and rings: definitions, examples, fundamental properties.
- Vectors, operations between vectors. Applications might include geometry in space (if time permits).
- Vector spaces: linear dependence, generators, bases, dimension, Grassman formula.
- Matrices: operations between matrices, relationship between matrices and linear systems, Gauss-Jordan method. Relationship with homomorphisms, search for eigenvalues and eigenvectors, diagonalizability.
If time allows, we will also discuss some basic geometry: planes, lines, parallel, orthogonality and interserctions in dimensions 2 and 3.
Prerequisites for admission
No prerequisites
Teaching methods
Frontal lessons.
Teaching Resources
We will make notes available on the Ariel platform.
Assessment methods and Criteria
There will be a written exam. It will consist of exercises.
MAT/03 - GEOMETRY - University credits: 3
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 6
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 6
Practicals: 36 hours
Lessons: 48 hours
Lessons: 48 hours
Professors:
Lombardi Luigi, Luperi Baglini Lorenzo
Professor(s)
Reception:
Wednesday 15:30-16:30 or by appointment
Math Department in via C. Saldini 50. Office: 1.109