OVERVIEW

The course provides basic knowledge of linear algebra and analytical geometry. In particular, matrix calculus, the resolution of linear systems, the concepts of vector space and linear application, the resolution of analytical geometry problems in space are addressed.

AIMS AND CONTENT

LEARNING OUTCOMES

The course provides the basic notions and tools of linear algebra and analytic geometry in plane and space, with particular reference to vector calculus and linear applications.

AIMS AND LEARNING OUTCOMES

Attendance and active participation in lessons and individual study will allow the student to acquire the basic notions of linear algebra. Initially, some topics are introduced (complex numbers, polynomials, matrices, linear systems) which, in addition to having their own relevance, facilitate the understanding of the concepts illustrated subsequently. The theory is then applied to the study of geometry in space. Finally, the fundamental aspects of the theory of diagonalization and linear applications are illustrated.

At the end of the course the student will be able to:

- apply the concepts of linear algebra in the study of the topics;
- logically structure a mathematical argument;
- understand the concepts underlying algorithmic problem-solving procedures;
- solve problems and exercises relating to the topics covered in the course; in particular, in relation to specific problems he will be able to understand which techniques can be used, and which already known results applied, to resolve the issue addressed;
- explain the procedures adopted to solve the exercises.

TEACHING METHODS

The course consists of lectures held by the teacher in which the theory will be exposed, which will be applied to examples and exercises.

In their personal work, the student must acquire the knowledge and concepts illustrated during the lessons and be able to solve exercises.

Attendance is not mandatory but recommended, in order to make learning more effective.

Students who have valid certifications for Specific Learning Disorders (DSA), for disabilities or other educational needs are invited to contact the professor and the contact person for disabilities of the Polytechnic School, Prof. Federico Scarpa (federico.scarpa@unige.it), at the beginning of the course to agree on any teaching methods that, in compliance with the objectives of the course, take into account their individual learning approach.

SYLLABUS/CONTENT

• Recall of set theory. Operations between sets. Cartesian product of sets. Applications and their properties.
• Algebraic structures.
• Complex numbers. Algebraic and trigonometric representation of a complex number. Euler's formulas and exponential form of a complex number. Roots of a complex number. Fundamental theorem of algebra. Decomposition of a real and complex polynomial into factors of minimum degree.
• Linear systems. Gauss algorithm. Associated matrices and characteristic of a matrix. Rouché-Capelli theorem.
• Matrices. Product rows by columns. Reduced matrices. Elementary matrices.  Determinants. Inverse matrix. 
• Vector spaces.  Vector subspaces. Linear independence. Span. Bases and dimension. 
• Elements of vector theory. The vector space of geometric vectors in space. Scalar and vector product of two vectors. Mixed product. Double vector product. Orthogonal Cartesian reference and vectors.
• Analytical geometry of space. Cartesian equation of a plane. Analytical representations of a straight line in space. Parallelism and orthogonality between planes, between lines, between line and plane. Skewed lines and coplanar lines. Bundles of plans. Angle of two lines, of two planes, of a line and a plane. Scalar product and orthonormal bases. 
• Linear transformations, associated matrix. Coordinate changes.
• Diagonalizability of an endomorphism. Eigenvalues, eigenvectors and eigenspaces. Characteristic polynomial. Diagonalization of matrices.

RECOMMENDED READING/BIBLIOGRAPHY

The handouts used and the lessons' blackboards, as well as other teaching materials, will be available on Aulaweb. This material and the notes taken by the student during the lessons are sufficient to prepare the exam.

LESSONS

LESSONS START

https://corsi.unige.it/en/corsi/11941/studenti-orario

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

The exam includes a written test that focuses on the topics covered in the course. To take the written test you must register for the official exam sessions available on the website https://servizionline.unige.it/studenti/esami/prenotazione.

Three partial written tests are scheduled during the course. Each intermediate test is passed if the student has achieved a score of at least 15. The student is exempt from taking the written test if he/she has passed all the intermediate tests with an average of scores equal to at least 18. In this case, the grade awarded will be equal to the average of the scores of the intermediate verification tests.

ASSESSMENT METHODS

In the written test, the questions aim to verify both the operational skills acquired by the student, through the resolution of exercises, and the learning of the theoretical concepts covered in the course.

The use of notes, texts or electronic devices is not permitted.

The correct use of specialist vocabulary and critical reasoning ability contribute to the final assessment.

FURTHER INFORMATION

Ask the professor for other information not included in the teaching schedule.