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

01. Sets: 
01.01 Sets and their representation. 
01.02 Operations on sets.
01.03 Equivalence relations and order relations.

02. Algebraic structures: 
02.01 Binary operations and their properties.
02.02 Semigroups and monoids.
02.04 Groups.
02.05 Rings.
02.06 Fields.
02.07 Polynomial rings.
02.08 Rings of integers modulo p.

03. Complex numbers: 
03.01 Algebraic definition of complex number and operations with complex numbers. 
03.02 Gauss plane.
03.03 Trigonometric representation of a complex number and De Moivre's formula. 
03.04 Complex exponential representation and Euler's formula.
03.05 Complex zeros of algebraic equations.
03.06 The fundamental theorem of algebra.

04. Vector spaces: 
04.01 Vectors and operations on them. 
04.02 Vector spaces.
04.03 Linear combination of vectors. Generators, linear independence, bases.
04.04 Operations on subspaces: sum and intersection. Grassmann Theorem. 

05. Matrices:
05.01 Preliminaries and operations with matrices. Square matrices.
05.02 Invertible matrices.
05.03 The determinant of a square matrix and its properties.
05.04 Sarrus rule and Laplace Theorem for the determinant. 
05.05 Construction of the inverse matrix of an invertible matrix.
05.06 Binet Theorem.
05.07 Rank of a matrix.
05.08 Eigenvalues and eigenfunctions.

06. Linear systems: 
06.01 Preliminaries and methods of solution. 
06.02 Matrix associated with a linear system.
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 Dfinition of linear map.
07.02 Invertible linear maps.
07.03 Kernel and image of a linear map. 
07.04 Rank-nullity theorem. 
07.05 Isomorphic vector spaces.
07.06 Diagonalizable matrices and endomorphisms. Diagonalizability criteria.

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.

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 abstract 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 abstract 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 abstract algebra and linear algebra using a rigorous terminology.

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

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

The teaching material and specific communications from the lecturer can be found, together with other supporting activities, inside the Moodle platform.

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

Bramanti - Pagani - Salsa, Analisi matematica 1 con elementi di geometria e algebra lineare, Zanichelli
Lang, Linear algebra, Springer-Verlag
Lang, Introduction to linear algebra, Springer-Verlag
Lang, Algebra, Springer-Verlag

Assessment

La prova orale consiste in un colloquio sugli argomenti del programma del corso. L'esame è considerato superato se entrambe le prove risultano sufficienti. A quel punto la valutazione dell'orale può portare a una variazione di massimo 12 punti in più o in meno rispetto alla valutazione dello scritto.

The exam of Discrete Structures and Linear Algebra consists of a written exam and an oral one, both of them mandatory. 
The written exam consists of exercises related to the topics of the course. The written exam is passed if the mark is, at least, 18/30. During the written exam it is not allowed to use textbooks, notes, calculators or mobile phones, under penalty of disqualification. 
The oral exam consists of a discussion related to the topics of the course. The exam is considered passed if both parts are sufficient. At that point, the evaluation of the oral exam can cause a variation of maximum plus or minus 12 points with respect to the evaluation of the written exam.

Disability and Specific Learning Disorders (SLD)

Students who have registered their disability certification or SLD certification with the Inclusion and Right to Study Office can request to use conceptual maps (for keywords) during exams.

To this end, it is necessary to send the maps, two weeks before the exam date, to the course instructor, who will verify their compliance with the university guidelines and may request modifications.