Subject: LINEAR ALGEBRA AND GEOMETRY (A.A. 2022/2023)
Unit Algebra e Geometria
Mathematics, Information Technology and Statistics (lesson)
OFA - Obblighi formativi aggiuntivi
The aim of the course is to provide some basic concepts of linear algebra and analytic geometry.
The prerequisites are the knowledge of High School Mathematics, with focus on the following topics. Set operations. The sets of natural numbers, integer numbers, rational numbers, real numbers and their main properties. Polynomial algebra. Algebraic equations and inequalities. Powers, n-th roots and logarithms. Trigonometric functions.
The course is divided into six Chapters.
Chapter I (1 CFU): the definition of a vector space and the first elements of Matrix Theory
Chapter II (1 CFU): vector subspaces, rank and the Gauss-Jordan Algorithm
Chapter III (1 CFU): bases and linear transformations
Chapter IV (1 CFU): determinants and systems of linear equations. The theory of determinants: definitions, properties, the rule of Sarrus and triangulation methods, The theorem of Laplace; Invertible matrices; Cramer’s formula; homogeneous systems of linear equations.
Chapter V (1 CFU): euclidean vector linear spaces. The theory of eigenvalues and eigenvectors. Orthogonality. A diagonalization theorem.
Chapter VI (1 CFU): affine and euclidean spaces. Definitions and main properties
The course consists in formal lectures in the classroom, combining theoretical sessions and practical sessions. The latter ones aim at applying the various techniques to the resolution of exercises/problems of different types.
The final exam occurs in two steps: a written test and an oral interview. The written test (duration: 2 hours) requires the resolution of some exercises of standard type, very similar to those presented during the lectures. Admission to the oral interview is achieved with a score of 18/30 or more in the written test. Oral interviews of the candidates passing the written test take place immediately after the assessment of the written test. The interview consists of a discussion of the main concepts presented in the lectures, definitions, main results, some particularly meaningful proofs and, above all, a logical connection among the various concepts. The final score is assigned at the end of the oral interview. Students passing the written test but not the oral examination must begin the entire process again (starting from the written test) at a a later round of exams.
At the end of the course students should be able to solve some problems of standard type using the techniques they learned. They should also be able to define the fundamental theoretical concepts and to explain the logical links connecting them
S. LANG, Algebra Lineare, Ed. Bollati Boringhieri
M.R.CASALI, C. GAGLIARDI, L. GRASSELLI, Geometria, Societa` Editrice Esculapio, Bologna, 2010. ISBN: 978-88-7488-378-3
E. SERNESI, Geometria 1 Seconda Edizione, Bollati Boringhieri, Torino, 2000. ISBN: 978-88-339-5447-9