Mathematics Ii
A.Y. 2025/2026
Learning objectives
The aim of the course is to introduce the basic concepts of algebra, the notions of vector space and linear map and to analyze the problem of solvability of systems of linear equations (also from an algorithmic point of view).
Expected learning outcomes
By the end of the course, students will be able to understand the formal language of algebra, will have gained familiarity with the basic properties of the ring of integers and the field of complex numbers, and will be able to recognize vector spaces and linear maps between them. They will also be able to factor polynomials, discuss the solvability of systems of linear equations, perform computations with matrices, associate them to systems of linear equations, and discuss their diagonalizability.
Lesson period: Second 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
Second semester
Course syllabus
1. Basic Algebra
Sets, relations between sets and their composition: equivalence relations, order relations; functions.
Number sets: natural, integer, rational, and real numbers.
Algebraic structures and homomorphisms: monoids, groups, rings.
2. Integer Numbers
Representation of integers; unary representation; positional representation in base n.
Principle of induction.
Division among integers and the Euclidean algorithm for computing the greatest common divisor.
Prime numbers and prime factorization.
Congruence relations. Modular arithmetic.
3. Complex Numbers
Algebraic, trigonometric, and exponential representations.
Operations with complex numbers, roots of unity.
Fundamental Theorem of Algebra, factorization of polynomials.
4. Polynomials
Fields.
Polynomials with real coefficients and operations on polynomials; polynomial rings.
Roots and their multiplicity.
Irreducible polynomials; factorization.
5. Linear Algebra
Systems of m linear equations in n unknowns. Solving by row reduction (Gauss-Jordan method).
Matrices and operations on matrices.
Vector spaces. Bases.
Linear maps and matrices; rank of a matrix.
Determinant of a square matrix and its properties. Inverse of a square matrix: existence and computation.
Cramer's and Rouché-Capelli theorems.
Eigenvalues and eigenvectors.
Sets, relations between sets and their composition: equivalence relations, order relations; functions.
Number sets: natural, integer, rational, and real numbers.
Algebraic structures and homomorphisms: monoids, groups, rings.
2. Integer Numbers
Representation of integers; unary representation; positional representation in base n.
Principle of induction.
Division among integers and the Euclidean algorithm for computing the greatest common divisor.
Prime numbers and prime factorization.
Congruence relations. Modular arithmetic.
3. Complex Numbers
Algebraic, trigonometric, and exponential representations.
Operations with complex numbers, roots of unity.
Fundamental Theorem of Algebra, factorization of polynomials.
4. Polynomials
Fields.
Polynomials with real coefficients and operations on polynomials; polynomial rings.
Roots and their multiplicity.
Irreducible polynomials; factorization.
5. Linear Algebra
Systems of m linear equations in n unknowns. Solving by row reduction (Gauss-Jordan method).
Matrices and operations on matrices.
Vector spaces. Bases.
Linear maps and matrices; rank of a matrix.
Determinant of a square matrix and its properties. Inverse of a square matrix: existence and computation.
Cramer's and Rouché-Capelli theorems.
Eigenvalues and eigenvectors.
Prerequisites for admission
Mathematical knowledge as outlined in the national curriculum for upper secondary schools (in particular: number sets, calculations with literal expressions, operations with polynomials, first- and second-degree equations and inequalities, the notion of a function).
Teaching methods
Lectures and exercise sessions.
Teaching Resources
Written notes covering all the topics addressed during the course will be made available on the Ariel website associated with the course.
Additional reference textbooks for further study and exercises include the following:
C. Delizia, P. Longobardi, M. Maj, C. Nicotera, Matematica discreta, McGraw-Hill
A. Alzati, M. Bianchi, M. Cariboni, Matematica discreta - Esercizi, Pearson Education
Additional reference textbooks for further study and exercises include the following:
C. Delizia, P. Longobardi, M. Maj, C. Nicotera, Matematica discreta, McGraw-Hill
A. Alzati, M. Bianchi, M. Cariboni, Matematica discreta - Esercizi, Pearson Education
Assessment methods and Criteria
The exam consists of a written test and an oral test.
The written test involves solving exercises related to the topics covered in the course.
The oral test consists of an interview aimed at assessing the student's knowledge, including theoretical knowledge, understanding, and ability to present the topics covered in the course in a logical manner. At the discretion of the examination board, the student may also be asked to discuss their written test and solve additional exercises during the oral test.
Failure to pass either the written or the oral tests results in failure of the exam.
The final grade takes into account the results of both the written and the oral tests. The final grade is expressed on a scale of thirty points and is communicated and explained to the student at the end of the oral test.
The written test involves solving exercises related to the topics covered in the course.
The oral test consists of an interview aimed at assessing the student's knowledge, including theoretical knowledge, understanding, and ability to present the topics covered in the course in a logical manner. At the discretion of the examination board, the student may also be asked to discuss their written test and solve additional exercises during the oral test.
Failure to pass either the written or the oral tests results in failure of the exam.
The final grade takes into account the results of both the written and the oral tests. The final grade is expressed on a scale of thirty points and is communicated and explained to the student at the end of the oral test.
MAT/03 - GEOMETRY - University credits: 3
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 3
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 3
Practicals: 24 hours
Lessons: 32 hours
Lessons: 32 hours
Professors:
Carai Luca, Colombo Giulio
Professor(s)
Reception:
By appointment (to be scheduled via email)
Department of Mathematics, via C. Saldini 50, second floor, office 2090