Ateneo - UPO

The course covers the main topics of linear algebra and some fundamental concepts of analytic geometry in the plane and in space.
Specifically, the key topics include: vector spaces and bases; vector operations, matrices and matrix operations, matrix rank, the determinant of a square matrix, invertibility of square matrices and calculation of the inverse matrix, linear systems and the Rouché-Capelli theorem, the dot product in n-dimensional Euclidean space, the Hermitian product for complex vector spaces, the cross product in three-dimensional Euclidean space, linear maps between abstract vector spaces, eigenvalues ​​and eigenvectors, diagonalization of symmetric operators.

Recommended text:

Bramanti, Pagani, Salsa, "Mathematical Analysis 1 with elements of geometry and linear algebra", Ed. Zanichelli, 2014.

The text indicated is the same that will be adopted for the course of Mathematics I.

The course aims to enable students to acquire the fundamental knowledge of linear algebra and analytic geometry required in fields that utilize advanced mathematical tools. It covers the basics of vector and matrix calculus, with applications to analytic geometry in the plane and in space.

Knowledge of the concepts and content of the Calculus I course is required, specifically the fundamentals of set theory and functions. In particular, students intending to take this course and pass the corresponding exam must have achieved certain learning outcomes from the Calculus I course, as outlined below:
- ability to apply knowledge and understanding: students must have developed a mastery of the concepts learned, enabling them to independently interlink these concepts and apply them jointly to solve simple theoretical problems;
- ability to communicate what has been learned: students must be able to clearly explain both the concepts themselves and their application, whether in general terms or within the specific context of a problem.

The course is structured around lectures comprising both theoretical content and exercises. Each topic is introduced via a general discussion designed to make the material as accessible as possible. Next, the fundamental concepts of the topic are presented, followed by examples to clarify their meaning. The third stage covers the statements of key theorems and their proofs. The final part is dedicated to exercises. Active participation is encouraged through questions that also aim to gauge the level of difficulty students experience in following the lectures. The university platform (www.dir.uniupo.it) is used to provide supplementary teaching materials, such as lecture notes, various exercises, and past written exam papers.

In addition to the suggested books, further material for the preparation of the exam will be provided during the development of the course. It is made up of the slides used during the course of the last year and of the written exams of the previous years.

Students with physical disabilities, 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/it/servizi/servizi-studentesse-e-studenti-condizione-di-disabilit%C3%A0-e-dsa
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 exam consists of a written test and an oral exam. The written test usually comprises three exercises covering various course topics. The main topics that may appear in these exercises are as follows:
a) analyzing the linear independence of given vectors;
b) determining whether a given set of vectors constitutes a spanning set or a basis for a specific vector space;
c) exercises involving scalar, Hermitian, vector and mixed products;
d) matrix operations, analyzing matrix invertibility, and calculating the inverse matrix;
e) determining the rank of a matrix;
f) analyzing the existence and uniqueness of solutions to linear systems using the Rouché-Capelli theorem;
g) calculating solutions to linear systems using various methods, including Cramer's rule;
h) exercises involving lines and planes in three-dimensional Euclidean space;
i) determining eigenvalues ​​and eigenvectors for linear operators and/or their associated matrices;
l) analyzing the diagonalizability of linear operators.

The score assigned to each exercise varies depending on the type and difficulty of the exercise.
This score is indicated next to the text of each exercise.

Admission to the oral exam is contingent upon achieving a minimum score on the written test.
The oral exam involves an assessment of the following points:
a) understanding of any errors made in the written test;
b) discussion of a theoretical topic of the student's choice;
c) an additional question on a topic chosen by the instructor.
If the answers and the language used are satisfactory, it is possible to achieve a final grade higher than the one obtained in the written test. However, the final score may not exceed the written test score by more than 7 points.
It is also possible for the final grade to be lower than the written test score; Please note that if the oral examination proves unsatisfactory, the overall exam result may be deemed negative.
Achieving an excellent grade requires a strong grasp of the theoretical topics combined with excellent proficiency in calculations.

The final grade for the Calculus II exam will be determined based on an overall assessment of the results achieved in the individual modules.

The course aims to provide the fundamentals of vector and matrix calculus. The main topics covered in the course are as follows:
1. geometric vectors in the plane and in three-dimensional space, and related operations;
2. introduction to the concept of abstract vector spaces over real and complex fields;
3. spanning sets, linearly independent vectors, and bases of a vector space;
4. matrices and matrix operations;
5. rank of a matrix and determinant of a square matrix;
6. dot product in n-dimensional Euclidean space, Hermitian product in complex space C^n;
7. cross product and scalar triple product for vectors in three-dimensional Euclidean space;
8. invertibility of square matrices and calculation of the inverse matrix;
9. existence and uniqueness of solutions to linear systems, Rouché-Capelli theorem, Cramer's rule;
10. lines and planes in three-dimensional Euclidean space;
11. linear operators, eigenvalues, and eigenvectors;
12. diagonalizability of linear operators;
13. diagonalizability of symmetric linear operators over the real field and Hermitian operators over the complex field.

To achieve the knowledge and skills corresponding to the minimum passing level, students must demonstrate:

(Knowledge)
- familiarity with the fundamental concepts of vectors, matrices, and their associated operations;
- familiarity with the fundamental concepts regarding the study of linear systems;
- familiarity with the fundamental concepts of the analytic geometry of lines and planes in three-dimensional Euclidean space;
- familiarity with the fundamental concepts of linear maps between vector spaces;
- familiarity with the fundamental concepts regarding eigenvalues, eigenvectors, and the diagonalizability of linear operators.

(Competencies)
- the ability to apply the fundamental concepts of vector and matrix calculus;
- the ability to identify, within a specific application area, the appropriate mathematical concept required for the context at hand.

(Transversal skills)
- the ability to use correct mathematical terminology within an applied project;
- the ability to grasp the deeper meaning of material from sources such as textbooks and articles, even when advanced mathematical language is employed.

Achievement of an advanced level: students must demonstrate a high degree of mastery of the subject matter, both in terms of their ability to solve problems and their knowledge of the key theoretical results of vector and matrix calculus.