
Course unit
MATHEMATICAL METHODS
SSO2043117, A.A. 2014/15
Information concerning the students who enrolled in A.Y. 2013/14
ECTS: details
Type 
ScientificDisciplinary 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 
Hours of Individual study 
Shifts 
Lecture 
12.0 
108 
192.0 
No turn 
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 CayleyHamilton 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 semidefinite 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.
Preypreyer 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, antihermitian and unitary matrices. Householder matrices. Positive definite and semidefinite matrices. The HamiltonCayley 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 ndimensional 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 σalgebra of Lebesgue measurable sets. Measurable and integrable functions.Definition of the integral of a function on a measurable set. Properties of the integral. FubiniTonelli'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 onethird 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 onethird 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, .

NOBLE B., DANIEL J.W., Applied Linear Algebra. : PrenticeHall inc., 1988. terza edizione

STRANG G, Algebra Lineare e sue applicazioni. : Liguori, .

N. FUSCO, P. MARCELLINI, C. SBORDONE, Analisi due. : Liguori, .

E. Gregorio, l.Salce, Algebra Lineare. : Libreria Progetto, .

Michiel Bertsch, Roberta Dal Passo, Lorenzo Giacomelli, Analisi Matematica. : McGraw Hill, .


