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 aims to train the use of linear algebra and geometry for applications, with special attention to vector calculus and linear transformations. The course objective is also to develop the ability to understand and express themselves with precision.

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

Exam Board

CLAUDIO CARMELI (President)

OTTAVIO CALIGARIS

RANIERI ROLANDI

MAURIZIO SCHENONE

LESSONS

Class schedule

ELEMENTS OF MATHEMATICS FOR ENGINEERING

EXAMS

EXAM DESCRIPTION

The exam consists of a written and an oral examination. The student can access the oral exam after he passes the written part.

ASSESSMENT METHODS

The written examination tests the ability in solving problems in linear algebra and geometry. The oral part certifies that the student has filled the gaps revealed by  the written test and that he is able to express  properly the results of the theory.

Exam schedule

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.