00674 - Mathematics

Learning outcomes

At the end of the course the student becomes acquainted with the basic concepts and tools of mathematical analysis of real functions of a real variable; the student  will also become acquainted with the basic concepts of matrix theory and linear algebra. In particular, the student will be able to: - compute limits, derivatives and Taylor polynomials of a function - study and plot functions - compute integrals and generalized integrals of a function - solve systems of linear equations by using vectors and matrices - recognize linearly independent sets and subspaces - compute scalar products, norms, orthogonal projections - diagonalize square matrices.

Course contents

The course will be delivered jointly by Francesco Regonati and Luca Vincenzo Ballestra. The contents of the course are listed below, with the indication of the corresponding lecturer.

1. Basic notions about sets and mappings between sets. Real numbers; operations, order, absolute value; least upper bound. Radicals; powers and logarithms. Trigonometric functions. Equations and inequations.
Real functions of one real variable; bounded, monotone functions. Affine linear and quadratic functions, straight lines and parabolas. Power, exponential, logarithmic, trigonometric and inverse trigonometric functions. (Regonati)

2. Neighbourhoods of a point. Limit points of a set. General definition of a limit. Limits of monotone functions. Limits and arithmetic operations, indeterminate forms; limits and order; change of variables. Comparison and asymptotic analysis.
Some special limits. The number "e". (Regonati)

3. Continuity, asymptotes. Continuous functions on a compact set and on an interval. Derivative and tangent line. Derivatives of elementary functions. Angular points, vertical tangent lines. Algebra of derivatives. Derivative of a composite function. Maxima and minima. Stationary points. Theorems on derivable functions. Monotonicity test. De l'Hopital theorem. Second-order derivative, concavity and convexity. (Ballestra)

4. Studying and plotting functions. Differential and linear approximation. Taylor-MacLaurin polynomials with Peano's remainder. (Ballestra)

5. The integral as a limit of sums; classes of integrable functions. Properties of the integral. The fundamental theorem of integral calculus. Computation of indefinite and definite integrals; basic integrals; integration by parts; integration by substitution. Integration of discontinuous functions. Generalized integrals; integration of unbounded functions and of functions defined on unbounded intervals. Integral function. (Regonati)

6. The standard n-dimensional vector space R^n; geometric interpretations. Subspaces. Finitely generated vector spaces; linear independence, bases and dimension; coordinates. Finite sets of vectors, rank; elementary operations.
Linear mappings between vector spaces; composition, inversion. Matrices; product of matrices; associativity, noncommutativity; transposition; inversion; powers. Rank of a matrix. Equivalence between the algebra of linear mappings and the algebra of matrices. (Regonati)

7. Determinant of a square matrix; Laplace expansions; characteristic properties; Binet theorem. Applications to bases recognition, coordinates computation, rank computation and inversion of matrices.
The standard n-dimensional euclidean vector space R^n; geometric interpretations. Finitely generated euclidean vector spaces; orthogonal projections; orthogonal and orthonormal bases, coordinates computation. Orthogonal matrices. (Regonati)

8. Systems of linear equations; vector and matrix representation; geometric interpretations. Systems of n linear equations in n unknowns with exactly one solution; characterization; elimination; Cramer rule. Systems of m linear equations in n unknowns; Rouchè-Capelli theorem; a resolution method; structure of the solution set.
Real eigenvalues and eigenvectors of a real square matrix. Characterization of the eigenvalues of a matrix, characteristic polynomial. Linear independence of eigenvectors associated to distinct eigenvalues. Diagonalizable matrices; reconstruction from their eigenvectors and eigenvalues. Symmetric real matrices; complete reducibility in R of their characteristic polynomials; orthogonality of eigenvectors associated to distinct eigenvalues. Spectral theorem. (Regonati)

Readings/Bibliography

For lectures delivered by Francesco Regonati: lecture notes will be published weekly on his web page.

For lectures delivered by Luca Vincenzo Ballestra: Ballestra Luca Vincenzo; Matematica per l'economia - Elementi di teoria ed esercizi; Maggioli editore 2015

Teaching methods

In order to explain all mathematical steps and logic relations, lessons will be taught at the blackboard.

Assessment methods

The learning outcomes are verified through a written exam and, in case of a positive result of the written exam, an optional oral exam, at the student request.

The written exam takes two hours and consists of some exercises and a couple of questions about theory. The use of books or drafts is not permitted. Non-programmable pocket calculators are allowed.

The written exam can be subivided into two partial written tests, each taking one hour and a half, one at the end of the first subcycle and the other in the winter session, that will refer respectively to parts 1,2,6,7 and 3,4,5,8.

Teaching tools

The lectures will consist of theory, examples and exercises. During the course some teaching materials will be provided by the lecturers as integration/completion of the textbook.

Finally, the course will be preceded by a crash-course, such to provide students with the necessary basic mathematical knowledge. All the students are strongly encouraged to attend it.

Office hours

See the website of Francesco Regonati

See the website of Luca Vincenzo Ballestra