## DISCRETE STRUCTURES AND LINEAR ALGEBRA MATEMATICA DISCRETA

A.Y. Credits
2015/2016 6
Lecturer Email Office hours for students
Raffaella Servadei Tuesday 14-16 or by appointment

### Assigned to the Degree Course

Date Time Classroom / Location

### Learning Objectives

Aim of the course is to give to the students some basic tools and topics in algebra and linear algebra.

### Program

01. Sets:
01.01 Sets and their representation.
01.02 Operations on sets: union, intersection, complement, difference and Cartesian product.
01.03 Relations. Equivalence relations and order relations.

02. Algebraic structures:
02.01 Binary operations and their properties. Identity element and inverse element with respect to a binary operation. Algebraic structures: preliminaries.
02.02 Semigroups: definition and preliminaries. Subsemigroups.
02.03 Monoids: definition and preliminaries. Identity element and its unicity*. Submonoids.
02.04 Groups: definition and preliminaries. Inverse element and its properties*. Abelian groups. Subgroups. Cancellation rule*. Necessary and sufficient condition for a subset to be a subgroup of a group*.
02.05 Rings: definition and preliminaries. Properties of rings*. Subrings. Commutative rings and rings with identity. Zero divisors. Integral domains. Necessary and sufficient condition for a commutative ring to be an integral domain*.
02.06 Fields: definition and preliminaries.
02.07 The polynomial ring. Division algorithm.
02.08 Ring of integers modulo p.

03. Vector spaces:
03.01 Vectors and operations on them: sum, difference, scalar multiplication, scalar product and vector product.
03.02 Vector spaces: definition and properties*.
03.03 Vector subspaces. Necessary and sufficient condition for a subset to a subspace*.
03.04 Linear combination of vectors. Subspaces spanned by a set of vectors.
03.05 Linear dependence and linear independence of vectors.
03.06 Basis and dimension of a vector space. Uniqueness of the representation of a vector as linear combination of the elements of a basis*.
03.07 Operations on subspaces: sum and intersection. Grassmann Theorem.
03.08 Direct sum of vector subspaces. Necessary and sufficient condition for a vector space to be direct sum of its subspaces*.

04. Complex numbers:
04.01 Algebraic definition of complex number and operations with complex numbers.
04.02 Gauss plane.
04.03 Conjugate and modulus of a complex number.
04.04 Trigonometric representation of a complex number and De Moivre's Theorem.
04.05 Roots of complex numbers.
04.06 Complex exponential representation and Euler's formula.
04.07 Complex zeros of algebraic equations.
04.08 The fundamental theorem of algebra.

05. Matrices:
05.01 Preliminaries and operations with matrices: sum, scalar multiplication, product and their properties. Vector spaces of matrices (m,n).
05.02 Square matrices and diagonal matrices.
05.03 Inverse matrix and its unicity*. Inverse matrix of the product*.
05.04 The transpose of a matrix and its properties*. The transpose of the inverse matrix*.
05.05 Symmetric and antisymmetric matrices.
05.06 Matrix associated with a set of vectors with respect to a basis.
05.07 The determinant of a square matrix and its properties.
05.08 Sarrus rule and Laplace Theorem for the determinant.
05.09 Construction of the inverse matrix.
05.10 Binet Theorem and its consequences*.
05.11 Orthogonal matrices and their properties*.
05.12 Rank of a matrix. Kronecker Theorem.
05.13 Elementary transformations of a matrix and diagonal canonical form.
05.14 Eigenvalues and eigenfunctions of a matrix. Characteristic polynomial and characteristic equation.

06. Linear systems:
06.01 Preliminaries and methods of solution.
06.02 Matrix equation.
06.03 Homogeneous and non-homogeneous linear systems.
06.04 Cramer Theorem.
06.05 Rouché-Capelli Theorem.
06.06 Linear systems depending on parameters.

07. Linear maps:
07.01 What is a linear map. Necessary and sufficient condition for a map to be linear*. Image of the zero vector by a linear map*. Operations with linear maps: sum, scalar multiplication and composition.
07.02 Invertible linear maps.
07.03 Kernel and image of a linear map and their properties*.
07.04 Rank-nullity theorem*.
07.05 Matrix associated with a linear map.
07.06 First and second theorem of equivalence for linear maps.
07.07 Isomorphic vector spaces. Vector spaces with finite dimension n are isomorphic*.
07.08 Eigenvalues and eigenfunctions of an endomorphism. Necessary and sufficient condition for a scalar to be an eigenvalue of an endomorphism*. Eigenspace related to an eigenvalue and its dimension.
07.09 Diagonalizable matrices and endomorphisms. Diagonalizability criteria. Sufficient condition for an endomorphism to be diagonalizable.

* : this means that the proof is required.

### Bridging Courses

There are no mandatory prerequisites.

It is worth noticing that the topics covered by this course will be used in Procedural and Logic Programming, Calculus, Digital Signal and Image Processing, Modeling and Verification of Software Systems.

It is recommended to take the exam of Discrete Structures and Linear Algebra during the first year of the Laurea Degree Program in Applied Computer Science.

### Learning Achievements (Dublin Descriptors)

Knowledge and understanding:
At the end of the course the student will learn the basic notions of algebra and linear.

Applying knowledge and understanding:
At the end of the course the student will learn the methodologies of algebra and linear algebra and will be able to apply them to the study of various problems.

Making judgements:
At the end of the course the student will be able to apply the techniques of algebra and linear algebra in order to solve new problems, also coming from real-world applications.

Communications skills:
At the end of the course the student will have the ability to express the fundamental notions of algebra and linear algebra using a rigorous terminology.

Learning skills:
During the course the student will learn the ability to study the notions of algebra and linear algebra, also in order to use it in solving different kind of problems.

### Teaching Material

The teaching material prepared by the lecturer in addition to recommended textbooks (such as for instance slides, lecture notes, exercises, bibliography) and communications from the lecturer specific to the course can be found inside the Moodle platform › blended.uniurb.it

### Teaching, Attendance, Course Books and Assessment

Teaching

Theorical and practical lessons.

Attendance

Although strongly recommended, course attendance is not mandatory.

Course books

Abate - De Fabritiis, Geometria analitica con elementi di algebra lineare, McGraw Hill
Abate - De Fabritiis, Esercizi di geometria, Mc Graw Hill
Lang, Linear algebra, Springer-Verlag
Lang, Introduction to linear algebra, Springer-Verlag
Lang, Algebra, Springer-Verlag

Assessment

The exam of Discrete Structures and Linear Algebra consists of a written exam and an oral one, both of them mandatory.
The written exam, to carry out in two hours, consists of exercises related to the topics of the course. The written exam is passed if the mark is, at least, 15/30. During the written exam it is not allowed to use textbooks, workbooks or notes. Moreover, it is not allowed to use scientific calculators and mobile phones, under penalty of disqualification.
The oral exam consists of a discussion related to the topics of the course. The oral exam can be taken only if the written one has been passed. If so, the oral exam can be taken only in the same call in which the written exam has been passed or in the other calls of the same session.
The final mark of Discrete Structures and Linear Algebra is the average of the marks of the written exam and the oral one.

Disabilità e DSA

Le studentesse e gli studenti che hanno registrato la certificazione di disabilità o la certificazione di DSA presso l'Ufficio Inclusione e diritto allo studio, possono chiedere di utilizzare le mappe concettuali (per parole chiave) durante la prova di esame.

A tal fine, è necessario inviare le mappe, due settimane prima dell’appello di esame, alla o al docente del corso, che ne verificherà la coerenza con le indicazioni delle linee guida di ateneo e potrà chiederne la modifica.

### Notes

The course is offered also on-line inside the Moodle platform > elearning.uniurb.it

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