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Discrete mathematics

A.Y. 2020/2021

Learning objectives

The course has the main purposes of intoducing the algebraic language and the notions of vector spaces and linear applications and of analyzing the problem of resolving linear systems of equations (even from an algorithm point of view).

Expected learning outcomes

The student should be able to understant the formal language of the abstract algebra, to discuss the resolution of linear systems, to recognise the vector spaces and the linear applications. Moreover, he should be able to work with the matrices, to associate them to linear systems and to discuss their diagonalization.

**Lesson period:**
Second semester

**Assessment methods:** Esame

**Assessment result:** voto verbalizzato in trentesimi

Course syllabus and organization

### Single session

Responsible

Lesson period

Second semester

**Course syllabus**

1) Basic algebraic tools and algorithms

Integers: induction; division; Euclidean algorithm; prime numbers; integer factorization.

Polynomials with real coefficients. Roots. Irreducible polynomials. Factorization of polynomials.

Linear systems; Gauss-Jordan method.

Matrices and their algebra.

2) An outline of abstract algebra

Sets and relations: equivalence relations, partial orderings,maps. Congruence: the integers mod n.

Algebraic structures and their homomorphisms: groups, rings (fields and polynomial rings).

3) Linear Algebra

Vector spaces. Bases. Linear maps and matrices; the rank of a matrix. Determinants of square matrices. Inverse matrices: existence and computing. Cramer and Rouché-Capelli theorems. Eigenvalues and eigenvectors.

The course will be supported by practical exercises to improve comprehension of the several subjects discussed during lectures.

[Program for not attending students with reference to descriptor 1 and 2]:

1) Basic algebraic tools and algorithms

Integers: induction; division; Euclidean algorithm; prime numbers; integer factorization.

Polynomials with real coefficients. Roots. Irreducible polynomials. Factorization of polynomials.

Linear systems; Gauss-Jordan method.

Matrices and their algebra.

2) An outline of abstract algebra

Sets and relations: equivalence relations, partial orderings,maps. Congruence: the integers mod n.

Algebraic structures and their homomorphisms: groups, rings (fields and polynomial rings).

3) Linear Algebra

Vector spaces. Bases. Linear maps and matrices; the rank of a matrix. Determinants of square matrices. Inverse matrices: existence and computing. Cramer and Rouché-Capelli theorems. Eigenvalues and eigenvectors.

Integers: induction; division; Euclidean algorithm; prime numbers; integer factorization.

Polynomials with real coefficients. Roots. Irreducible polynomials. Factorization of polynomials.

Linear systems; Gauss-Jordan method.

Matrices and their algebra.

2) An outline of abstract algebra

Sets and relations: equivalence relations, partial orderings,maps. Congruence: the integers mod n.

Algebraic structures and their homomorphisms: groups, rings (fields and polynomial rings).

3) Linear Algebra

Vector spaces. Bases. Linear maps and matrices; the rank of a matrix. Determinants of square matrices. Inverse matrices: existence and computing. Cramer and Rouché-Capelli theorems. Eigenvalues and eigenvectors.

The course will be supported by practical exercises to improve comprehension of the several subjects discussed during lectures.

[Program for not attending students with reference to descriptor 1 and 2]:

1) Basic algebraic tools and algorithms

Integers: induction; division; Euclidean algorithm; prime numbers; integer factorization.

Polynomials with real coefficients. Roots. Irreducible polynomials. Factorization of polynomials.

Linear systems; Gauss-Jordan method.

Matrices and their algebra.

2) An outline of abstract algebra

Sets and relations: equivalence relations, partial orderings,maps. Congruence: the integers mod n.

Algebraic structures and their homomorphisms: groups, rings (fields and polynomial rings).

3) Linear Algebra

Vector spaces. Bases. Linear maps and matrices; the rank of a matrix. Determinants of square matrices. Inverse matrices: existence and computing. Cramer and Rouché-Capelli theorems. Eigenvalues and eigenvectors.

**Prerequisites for admission**

Basic knowledge of mathematics, like solving equations and polynomials algebra.

**Teaching methods**

Frontal lectures about theory and classes of exercises.

Tutoring .

Tutoring .

**Teaching Resources**

Ariel web page of the course.

Suggested books:

M. Bianchi, A. Gillio - Introduzione alla matematica discreta - McGraw Hill (seconda edizione 2005).

A. Alzati, M. Bianchi, M. Cariboni - Matematica discreta - Esercizi - Pearson Education - (2006).

Suggested books:

M. Bianchi, A. Gillio - Introduzione alla matematica discreta - McGraw Hill (seconda edizione 2005).

A. Alzati, M. Bianchi, M. Cariboni - Matematica discreta - Esercizi - Pearson Education - (2006).

**Assessment methods and Criteria**

The final examination consists of two parts: a written exam and an oral exam.

- During the written exam, the student must solve some exercises with the aim of assessing the student's ability to solve problems about the contents of the course. The duration of the written exam will be proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves, however, the duration will be probably two hours. In place of a single written exam given during the first examination session, the student may choose instead to take one written exam in the middle of the course. If the student passes this exams, he/she has to solve less exercise in the written test at tthe end of course. The outcomes of these tests will be available in the SIFA service through the UNIMIA portal.

- The oral exam can be taken only if the written component has been successfully passed. In the oral exam, the student will be required to illustrate results presented during the course and will be required to be able to discuss the written part of the exam, in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.

The complete final examination is passed if both the written and oral exams are successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.

- During the written exam, the student must solve some exercises with the aim of assessing the student's ability to solve problems about the contents of the course. The duration of the written exam will be proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves, however, the duration will be probably two hours. In place of a single written exam given during the first examination session, the student may choose instead to take one written exam in the middle of the course. If the student passes this exams, he/she has to solve less exercise in the written test at tthe end of course. The outcomes of these tests will be available in the SIFA service through the UNIMIA portal.

- The oral exam can be taken only if the written component has been successfully passed. In the oral exam, the student will be required to illustrate results presented during the course and will be required to be able to discuss the written part of the exam, in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.

The complete final examination is passed if both the written and oral exams are successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.

MAT/01 - MATHEMATICAL LOGIC

MAT/02 - ALGEBRA

MAT/03 - GEOMETRY

MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS

MAT/05 - MATHEMATICAL ANALYSIS

MAT/06 - PROBABILITY AND STATISTICS

MAT/07 - MATHEMATICAL PHYSICS

MAT/08 - NUMERICAL ANALYSIS

MAT/09 - OPERATIONS RESEARCH

MAT/02 - ALGEBRA

MAT/03 - GEOMETRY

MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS

MAT/05 - MATHEMATICAL ANALYSIS

MAT/06 - PROBABILITY AND STATISTICS

MAT/07 - MATHEMATICAL PHYSICS

MAT/08 - NUMERICAL ANALYSIS

MAT/09 - OPERATIONS RESEARCH

Practicals: 24 hours

Lessons: 32 hours

Lessons: 32 hours

Professor(s)

Reception:

Thursday 4:30pm-6pm, write an email for an appointment

Math Department in via C. Saldini 50. Office: 1.007