# ELEMENTS OF MATHEMATICS FOR ENGINEERING

OVERVIEW

The course provides basic knowledge in linear algebra and geometry of euclidean plane and space. The concept of a vector space is introduced first by means of examples, and then formalized. The course ends by introducing the notion of linear map and studying the problem of diagonalization.

## AIMS AND CONTENT

LEARNING OUTCOMES

The course provides the basic knowledge of linear algebra and geometry needed for applications, with special emphasis on vector calculus and linear transformations

AIMS AND LEARNING OUTCOMES

The course focuses on the concept of vector space. To make it easier to understand, some arguments are first introduced (complex numbers, polynomials, matrices, linear systems) which, besides having a specific interest, serve to illustrate the "abstract" theory that is developed later. This theory is then applied to the study of geometry. In the final part of the course the fundamental aspects of the diagonalization theory and linear applications are outlined.

Teaching methods

The course includes lectures at the blackboard in which the topics of the program are presented. Examples and exercises designed to clarify and illustrate the concepts of the theory are also carried out.

SYLLABUS/CONTENT

- Preliminaries. Sets. Operations between sets. Cartesian product of sets. Applications, injectivity and surjectivity.
- Complex numbers. Trigonometric and algebraic representation of a complex number. Euler formulas and exponential form of a complex number. N-th roots of a complex number. The fundamental theorem of algebra. Decomposition of a real polynomial.
- Linear systems, Gauss algorithm, Gauss Jordan algorithm, Rouche'-Capelli theorem.
- Matrices, operations with matrices, Reduced matrices, Rank, Elementary matrices. Linear systems, Determinant, Inverse.
- Vector spaces. Subspaces, Linear independence. bases and dimension.
- Elements of the theory of vectors. The vector space of geometric vectors in space. Scalar and vector product of two vectors. Mixed product. Calculating the coordinates of a vector with respect to an arbitrary base. Orthogonal reference frames and vectors.
- Analytic geometry of euclidean plane and space. Cartesian equation of a plane. Analytic representations of a straight line in space. Parallelism and orthogonality between planes, between straight lines, between straight lines and planes. Pencil of planes. Angle between two straight lines, two planes, a line and a plane.
- Linear transformations, associated matrices and change of coordinates.
- Diagonalization. (outline) Eigenvalues, eigenvectors and eigenspaces. Characteristic polynomial. Diagonalizability of a square matrix. Scalar product. Diagonalization of real symmetric matrices.

RECOMMENDED READING/BIBLIOGRAPHY

1) Handouts of the lecturer on AulaWeb.

2) Caligaris Oliva Ferrando Elementi di algebra lineare e geometria analitica available at http://web.inge.unige.it/DidRes/Analisi/AMindex.html

3) E. Carlini, M.V. Catalisano, F. Odetti, A. Oneto, M. E. Serpico, Geometria per Ingegneria, Esculapio

4) Schlesinger Algebra lineare e geometria Zanichelli

5) Fioresi R., Morigi M. Introduzione all’algebra lineare Casa Editrice Ambrosiana

6) Catalisano Perelli, Dispense (disponibili online)

Organization

## TEACHERS AND EXAM BOARD

**Ricevimento:** Appointment on student's request (send an email to carm@sv.inge.unige.it).

Exam Board

CLAUDIO CARMELI (President)

MAURIZIO SCHENONE

OTTAVIO CALIGARIS

## LESSONS

Teaching methods

The course includes lectures at the blackboard in which the topics of the program are presented. Examples and exercises designed to clarify and illustrate the concepts of the theory are also carried out.

## EXAMS

Exam description

The exam consists of a written and an oral exam. Once the written exam is passed, the student can access the oral exam.

The written test lasts three hours and consists of ten problems on the topics of the course.

The written exam can be replaced by two partial tests (one towards the middle of the course and one at the end). The student must pass both partial tests to access the oral exam. If the student has passed one of the two partial tests, he has another opportunity to pas the other during the first written test of the year. Partial tests last two hours and consist of 6 problems.

To participate in any type of test, students must register at least two days before the date of the exam on the website

https://servizionline.unige.it/studenti/esami/prenotazione

Assessment methods

The written examination tests the ability in solving problems in linear algebra and geometry. The oral part certifies that the student has overcome the gaps revealed by the written test and that he has understood the results of the theory. The quality of the presentation, the correct use of the specialized lexicon and the critical reasoning skills contribute to the final evaluation.

## FURTHER INFORMATION

Pre-requisites :

Good knowledge of mathematics at the high school level. In particular: trigonometry. Some acquaintance with cartesian geometry, although not strictly necessary, is recommended.