First cycle
degree courses
Second cycle
degree courses
Single cycle
degree courses
School of Science
Course unit
SSO2043117, A.A. 2014/15

Information concerning the students who enrolled in A.Y. 2013/14

Information on the course unit
Degree course First cycle degree in
SS1451, Degree course structure A.Y. 2009/10, A.Y. 2014/15
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Degree course track METODOLOGICO [001PD]
Number of ECTS credits allocated 12.0
Type of assessment Mark
Course unit English denomination MATHEMATICAL METHODS
Website of the academic structure
Department of reference Department of Statistical Sciences
Mandatory attendance No
Language of instruction Italian
Single Course unit The Course unit can be attended under the option Single Course unit attendance
Optional Course unit The Course unit can be chosen as Optional Course unit

Teacher in charge GIULIA TREU MAT/05
Other lecturers PAOLA MANNUCCI MAT/05

Course unit code Course unit name Teacher in charge Degree course code

ECTS: details
Type Scientific-Disciplinary Sector Credits allocated
Educational activities in elective or integrative disciplines MAT/02 Algebra 4.0
Educational activities in elective or integrative disciplines MAT/05 Mathematical Analysis 8.0

Course unit organization
Period First semester
Year 2nd Year
Teaching method frontal

Type of hours Credits Teaching
Hours of
Individual study
Lecture 12.0 108 192.0 No turn

Start of activities 01/10/2014
End of activities 24/01/2015
Show course schedule 2015/16 Reg.2009 weekly timetable
2015/16 Reg.2009 single teaching timetable

Examination board
Board From To Members of the board
4 Commissione a.a. 2015/2016 01/10/2015 30/09/2020 TREU GIULIA (Presidente)
MANNUCCI PAOLA (Membro Effettivo)
PARMEGGIANI GEMMA (Membro Effettivo)
VITTONE DAVIDE (Membro Effettivo)
3 Commissione a.a. 2014/2015 01/10/2014 30/09/2015 TREU GIULIA (Presidente)
MANNUCCI PAOLA (Membro Effettivo)
PARMEGGIANI GEMMA (Membro Effettivo)
SALCE LUIGI (Membro Effettivo)

Prerequisites: The topics of the courses of Linear Algebra 1 and Calculus 1 and 2.
Target skills and knowledge: The course is divided in two parts: Linear Algebra and Mathematical analysis.
The part devoted to Linear Algebra has the aim to provide a basic understanding of the theory of the eigenvalues and eigenvectors of real and complex matrices, as well as some useful tools such as the Cayley-Hamilton theorem and the Gerschgorin circle theorem for a rough estimate of the eigenvalues. In addition to the problems of the diagonalization and triangolarization of a matrix, we will treat the case of normal matrices, reaching to the spectral theorem. We will also provide various characterizations of the positive definite and semi-definite matrices. The theoretical treatment will be accompanied by several numerical exercises, in order to make the student actually capable of working with matrices.
In the part devoted to Mathematical Analysis we will deal with multivariate differential and integral calculus, sequences and series of functions. The students will learn both the theoretical foundations and the practical skills that will let them to solve applicative problems.
Examination methods: Written examination consisting in two parts.
Linear Algebra: two numerical exercises and a theoretical question.
Mathematical Analysis: an exercise on the continuity and differentiabilty of functions, an exercise on minima and maxima, an exercise and integration and an exercice on sequences or series of functions.
Assessment criteria: The two parts of the exam are evaluated separately and each of them is weighted differently. The weight for the Algebra part is 1/3 e the weight for the Analysis part is 2/3. To obtain a positive evaluation one has to get at least 16/30 in both parts.
Every question of each exercise contributes for a certain specified amount to the highest grade of 33/30 (corresponding to 30 cum laude).
The correctness, accuracy and completeness of the answers given to the different exercises are the criteria for the evaluation.
Course unit contents: Linear Algebra.
Prey-preyer linearized model. Eigenvalues, eigenvectors, eigensystems of real and complex matrices. Characteristic polynomial and its properties. The spectrum of a matrix. Similar matrices and their characteristic polynomials. Algebraic and geometric multiplicity of the eigenvalues. Indipendence of the different eigenspaces.
Diagonalizable matrices, unitary triangularization of complex matrices and the Schur theorem. Normal matrices. Spectral theorem: multiplicative and additive versions. Hermitian, anti-hermitian and unitary matrices. Householder matrices. Positive definite and semi-definite matrices. The Hamilton-Cayley theorem and the Gerschgorin circle theorem.

Mathematical Anlysis.
Sequences and series of functions. Pointwise and uniform convergence of sequences of real functions. The uniform limit of a sequence of continuous functions. Pointwise, uniform and total convergence of series of real functions. Power series, convergence radius. Taylor's series. Analitic functions. Differential calculus for real functions of n real variables. Basics of topology in the n-dimensional euclidean space. Open, closed, compact and connected sets. Limits. Algebra of limits. Continuity of a function at a point and in set. Continuity of composed fuction. Weierstrass theorem and connection theorem. Partial and directional derivatives. Higher order derivatives, Hessian matrix, Schwartz theorem. Differential of a function at a point. Chain rule. The problem of finding maxima and minima of a function. First order necessary condition for unconstrained problems. Sufficient conditions. Implicit function theorem. The problem of finding maxima and minima of a function: the case with constraints. Lagrange multipliers theorem. Lebesgue measure theory. The σ-algebra of Lebesgue measurable sets. Measurable and integrable functions.Definition of the integral of a function on a measurable set. Properties of the integral. Fubini-Tonelli's theorem (reduction formula) and the change of variables formula.
Planned learning activities and teaching methods: Linear Algebra. There will be 36 hours of lectures, of which about one-third dedicated to the performance of numerical and theoretical exercises. The students will be asked to solve some homework problems.
Mathematical Analysis. There will be 72 hours of lectures, at least one-third of them dedicated to numerical and theoretical exercises.
Additional notes about suggested reading: Linear Algebra. The program of the course is covered by Chapters 5 and 6 of the book "Linear Algebra" by E. Gregorio and L. Salce, Ed Libreria Progetto, Padova, 2012 (3rd ed.) and by the material in the network at the site of the former Facoltà of Scienze Statistiche.

Mathematical Analysis. In the MOODLE page of Statistical Science, in the spase devoted to the course, there are the lecture notes, the exercises of past examinations and other staff. The theacher will provide the password to the access to the theaching staff.
Textbooks (and optional supplementary readings)
  • P. Marcellini e C. Sbordone, Esercitazioni di Matematica, II vol. Parti prima e seconda. --: Liguori, --. Cerca nel catalogo
  • NOBLE B., DANIEL J.W., Applied Linear Algebra. --: Prentice-Hall inc., 1988. terza edizione Cerca nel catalogo
  • STRANG G, Algebra Lineare e sue applicazioni. --: Liguori, --. Cerca nel catalogo
  • N. FUSCO, P. MARCELLINI, C. SBORDONE, Analisi due. --: Liguori, --. Cerca nel catalogo
  • E. Gregorio, l.Salce, Algebra Lineare. --: Libreria Progetto, --. Cerca nel catalogo
  • Michiel Bertsch, Roberta Dal Passo, Lorenzo Giacomelli, Analisi Matematica. --: McGraw Hill, --. Cerca nel catalogo