Unit MATHEMATICS I AND GEOMETRY

GEOMETRY

Code	A002894
CFU	6
Teachers	Massimo Giulietti (Codocenza)
Hours	54 ore (Codocenza) - Massimo Giulietti
Learning activities	Base
Area	Matematica, informatica e statistica
Sector	MATH-02/B
Type of study-unit	Obbligatorio (Required)
Language of instruction	English
Contents	This course introduces the fundamental concepts of linear algebra, emphasizing both theoretical foundations and practical applications. Topics include systems of linear equations, matrix algebra, vector spaces, linear transformations, determinants, eigenvalues, eigenvectors, inner products, orthogonality, and least squares solutions. Applications in data science, engineering, and modeling (e.g., Markov chains, data fitting) are integrated to connect theory to real-world problems.
Reference texts	Lay, D. C., Lay, S. R., & McDonald, J. J. (2022). Linear Algebra and Its Applications (6th ed.). Pearson
Educational objectives	Upon completion of this course, students will be able to understand and apply the fundamental concepts of linear algebra, mastering both the theoretical frameworks and the computational methodologies necessary to address complex problems in various scientific fields. Specifically, students will acquire the ability to solve systems of linear equations through Gaussian elimination and matrix algebra, as well as the skill to interpret linear transformations as tools for operating on vector spaces. The course also aims to provide the proficiency required to calculate determinants, eigenvalues, and eigenvectors, thereby enabling students to perform matrix diagonalization and analyze dynamic models such as Markov chains. Finally, students will be able to utilize the properties of inner products and orthogonality to apply the Gram-Schmidt process and the least-squares method, with a particular focus on their practical implementation using computational software for data analysis and data fitting.
Prerequisites	Basic Calculus: A solid understanding of algebraic calculation, elementary functions (polynomial, exponential, logarithmic), and fundamental concepts of differential calculus (such as derivatives and function graphs). Analytic Geometry: Familiarity with the basics of the Cartesian plane, equations of lines and planes, and the concept of coordinates. Logical-Mathematical Skills: Aptitude for deductive reasoning and mathematical formalization.
Teaching methods	Lectures and tutorials
Learning verification modality	The student must pass two written exams: the first theoretical in nature and the second practical. Passing the first exam is a necessary condition to be admitted to the second.
Extended program	The course curriculum covers systems of linear equations and Gaussian elimination, moving on to matrix algebra, including fundamental operations such as addition, scalar multiplication, and matrix multiplication, as well as finding matrix inverses and solving systems. The study continues with vectors in $\mathbb{R}^n$, linear combinations, span, linear independence, and bases, extending to the concepts of vector spaces, subspaces, linear transformations, and their matrix representations. The course also covers the definition and properties of determinants, Cramer's rule, the theory of eigenvalues and eigenvectors, the calculation of the characteristic polynomial, matrix diagonalization, and related applications such as Markov chains. Finally, it addresses inner products, norms, orthogonality, the Gram-Schmidt process, the least squares method, and its applications in data fitting.
Obiettivi Agenda 2030 per lo sviluppo sostenibile	A Linear Algebra course contributes transversally to achieving the goals of the 2030 Agenda by providing the mathematical and computational foundations to analyze complex systems, optimize resources, and model real-world phenomena

MATHEMATICS I

Code	A002893
CFU	6
Teachers	Laura Angeloni (Codocenza)
Hours	54 ore (Codocenza) - Laura Angeloni
Learning activities	Base
Area	Matematica, informatica e statistica
Sector	MATH-03/A
Type of study-unit	Obbligatorio (Required)
Language of instruction	ENGLISH
Contents	The purpose of the course is to present the fundamental issues of basic calculus.
Reference texts	The teacher will advise some text at the beginning of the course. Among them: 1. "Calculus for Scientists and Engineers", Martin Brokate, Pammy Manchanda, Abul Hasan Siddiqi, Springer, 2019. 2. "Mathematical Analysis 1", Claudio Canuto, Anita Tabacco, Pearson, 2021. 3. "Calculus for Business, Economics, Life Sciences, and Social Sciences", Raymond Barnett, Michael Ziegler, Karl Byleen, Christopher Stocker, Pearson ed. 2019. 3. "Calculus: Early Transcendentals", James Stewart, Daniel Clegg, Saleem Watson, Cengage Learning, 2020. Moreover, in the UniStudium webpage of the course, slides on the main topics of the course and on exercises will be available.
Educational objectives	The purpose of the course is to furnish the main concepts of mathematical analysis and to support the competence in calculus skills, fundamental tools that contribute to the future management engineer. The main knowledge (descriptor Dublin 1) will be acquired: • knowledge of the concept of function and of calculating the limits of functions together with the basic concepts of topology; • knowledge of the differentiability of functions of one variable and all those concepts that enable the student to carry out the study of function; • knowledge of the notion of integral, main results and integral calculus. The main skills acquired (ability to apply their knowledge, descriptor Dublin 2, and to take with independent judgment the appropriate approach, Dublin descriptor 3) will be: • ability to solve equations, inequalities, limits, derivatives, integrals; • ability to develop an argument that leads the student to identify the methods of solving the problem; • ability to identify a common logical-deductive methodology in various topics to enable it to identify the approach to be followed.
Prerequisites	General notions about sets theory, equations and inequalities of first and second degree, elementary functions.
Teaching methods	The course is organized as follows: 1) Lectures on all topics of the course. 2) Classroom exercises.
Other information	Attendance is recommended for all lessons.
Learning verification modality	The verification of the educational objectives of the course (test) includes a written and an oral test. The written test will be held on the dates set out on the calendar of the CdS. The written test, of about 2,5 hours, consists in solving some problems regarding the main topics of the course and some multiple choice theoretical questions. The test has the aim to verify: i) the ability to understand the problems proposed during the course, ii) the ability to correctly apply the theoretical knowledge (descriptor Dublin 2), iii) the ability to formulate the appropriate approach for the solution of the problems (descriptor Dublin 3), iv) the ability to suitably and efficaciously communicate in written form (descriptor Dublin 4). The oral examination consists of a discussion no longer than 15 minutes aimed to verify: i) the level of knowledge about the theoretical contents of the course (descriptor Dublin 1), ii) the level of expertise in exposing their own logical-mathematical abilities (descriptor Dublin 2), iii) the independence of judgment (descriptor Dublin 3) to propose the most suitable approach to argue about the posed questions. The oral examination also aims to verify the student's ability to answer with proper language to the questions proposed by the Commission, to support a dialectical relationship during the discussion and to show logical deductive skills and synthetic exposition (descriptor Dublin 4). The final evaluation will be made by the Commission taking into account also of the evaluation of the written test. For information on support services for special needs students, please visit the page https://www.unipg.it/en/international-students/general-information/facilities-for-special-needs-students . In any case, the teacher is available to personally evaluate, in specific cases, any compensatory measures and / or personalized paths in the case of students with special needs.
Extended program	Set theory, number sets, equations, inequalities. Functions: main definitions, injectivity, surjectivity, one-to-one functions, inverse functions, composition, graphs and main functions (power functions, exponential, logarithmic, trigonometric functions). Concept of limit: calculation and properties. Infinite and infinitesimal. Continuity and main results about continuous functions. Derivatives: geometric meaning, calculation and main results. Fundamental theorems on differentiable functions. Convexity. Study of the graph of a function of one real variable. Riemann integration: definition, geometric meaning, calculation rules and main results.
Obiettivi Agenda 2030 per lo sviluppo sostenibile