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