Ateneo - UPO

Linear algebra and numerical methods for the solution of some basic mathematical problems. Ordinary differential equations.

+ M. Bramanti, C.D. Pagani, S. Salsa. Analisi matematica 1 con elementi di geometria e algebra lineare, Ed. Zanichelli, 2014.
+ A. Quarteroni , F. Saleri. Introduzione al Calcolo Scientifico. Esercizi e problemi risolti con MATLAB. Springer - Collana Unitext. 2002.
+ Lecture notes uploaded on the DIR page.

The course is based primarily on expository teaching (DE), consisting of lectures aimed at the systematic presentation of the fundamental concepts of linear algebra and the main numerical methods for solving basic mathematical problems. Lectures will be complemented by application examples and guided exercises, with the aim of promoting an operational understanding of mathematical tools and their application to problems of interest in chemistry.

Attendance is strongly recommended, as it encourages active participation, interaction with the professor, and the progressive development of problem analysis, modelling, and problem-solving skills.

At the end of the course, students will be able to:

- Apply the acquired knowledge of the fundamental elements of linear algebra and basic numerical methods to the formulation and solution of simple quantitative problems in the field of chemical sciences.
- Use appropriate technical and scientific language, clearly and rigorously communicating mathematical concepts, procedures, and results related to the topics covered in the course.
- Develop independent judgement and critical thinking skills through the autonomous analysis of problems, the evaluation of the adopted solution strategies, and the formulation of well-supported conclusions regarding the correctness and reliability of the results obtained.

Course contents: Matemtica I.

Lectures integrated with exercises aimed at applying the concepts and methods introduced during the course. Teaching activities will encourage active participation of students through discussion, interaction, and in-depth analysis of the topics covered.

Learning Disabilities or Special Education Needs can request specific services and tools via the Staff Sviluppo e Coordinamento Carriere e Servizi alle Studentesse e agli Studenti, consulting the University webpage:
https://uniupo.it/en/services/services-students-physical-or-learning-disabilities

Students with disabilities, learning disabilities or special education needs, once they have contacted the University Staff, can refer to the tutor in charge of the course to define the examination modalities, concerning academic aspects.

The examination consists of a written test and an oral examination.

The written test consists of solving two exercises related to the topics of linear algebra covered during the course and lasts 30 minutes. Each exercise contributes to the assessment with a maximum score of 16/30, for a total maximum score of 32/30. During the written test, consultation of textbooks and personal paper notes is not permitted.

Students who achieve a score equal to or higher than 16/30 in the written test are admitted to the oral examination. The oral examination consists of a discussion of the written test and an assessment of the understanding of the topics related to the numerical methods part of the course syllabus. The oral examination takes place in the days immediately following the written test; the date and classroom are communicated to students through their institutional email address.

The final grade takes into account the understanding of theoretical concepts, the ability to correctly apply the acquired mathematical methods, and the ability to independently address new problems.

Grading criteria:

18–20/30: essential knowledge of the course topics and basic understanding of the fundamental concepts of linear algebra and numerical methods; ability to apply standard procedures, with possible inaccuracies in the explanation of the steps followed.
21–23/30: fair knowledge of the course contents, correct application of the methods, and adequate ability to interpret the results.
24–26/30: good knowledge of the topics, confident application of the methodologies, ability to establish connections among the different topics of the course, and adequate justification of the procedures.
27–29/30: very good and in-depth knowledge, autonomy in problem analysis, mastery of mathematical language, and ability to critically evaluate the results obtained.
30/30 and honours: excellent mastery of the theoretical and applied contents, autonomy in selecting solution strategies, and full awareness of the limitations and potential of the methods used.

Passing the examination requires a sufficient level of preparation both in understanding the fundamental concepts and in correctly applying the acquired methodologies. Excellence requires, in addition to technical correctness, independent reasoning skills and the ability to establish connections among the topics covered.

Teaching materials:
The teaching material includes the recommended textbooks and the lecture notes provided by the instructor and uploaded to the DIR platform, which represent the main reference for studying the topics covered in the course. During the course, solved exercises and additional supplementary teaching materials will also be provided and made available on the DIR page, with the aim of consolidating the topics and supporting exam preparation. Additional university textbooks on linear algebra, numerical methods, and applied mathematics may also be consulted as supplementary references.

Linear algebra: vectors in the plane and in space, finite-dimensional vector spaces, matrices and linear transformations, kernel and image, determinants, linear systems, computation of eigenvalues and eigenvectors, and matrix diagonalization.

Numerical methods for the solution of basic mathematical problems: nonlinear equations (bisection method and Newton’s method), algebraic linear systems (LU factorization and Gaussian elimination method, QR factorization and Householder method), polynomial interpolation and function approximation through the discrete least squares method, interpolatory quadrature formulas (trapezoidal rule and Simpson’s rule), power method for the numerical computation of eigenvalues and eigenvectors.

Ordinary differential equations: first-order separable differential equations, first-order linear differential equations, second-order linear differential equations with constant coefficients.

At the end of the course, thanks to the educational resources provided by the Degree Programme (lectures, exercises, lecture notes, and supplementary material), students will have acquired knowledge and skills related to the fundamental tools of linear algebra, basic numerical methods, and ordinary differential equations, with particular reference to their application in the modelling and solution of quantitative problems of scientific and chemical interest.

- Knowledge and understanding

Students will acquire knowledge of the fundamental concepts of linear algebra, the main numerical methods for solving basic mathematical problems, and the essential elements of ordinary differential equations.

Satisfactory level: students know the definitions and the main theoretical results related to the topics covered in the course, understand the meaning of the introduced methods, and recognize the main procedures to be applied in solving standard problems.

Advanced level: students demonstrate an in-depth understanding of the mathematical concepts, are able to establish connections among the different topics of the course, interpret the meaning of numerical methods, and evaluate their characteristics, advantages, and limitations in relation to the problem under consideration.

- Applying knowledge and understanding

Students will be able to use the acquired mathematical tools to formulate and solve quantitative problems, applying appropriate analytical and numerical methods and interpreting the results obtained, also in the context of applications in the chemical sciences.

Satisfactory level: students are able to correctly apply the fundamental techniques of linear algebra and numerical methods to solve standard exercises and problems, following established procedures and verifying the consistency of the results obtained.

Advanced level: students are able to independently select the most appropriate mathematical method for the problem at hand, apply it rigorously to problems that may not be immediately reducible to previously encountered examples, and critically evaluate the accuracy and reliability of the solutions obtained, with particular reference to scientific and chemical applications.