48
Geometry
REGGIO DI CALABRIA
Overview
Date/time interval
Syllabus
Course Objectives
Knowledge of the basics of linear algebra (matrices, determinants, systems of linear equations, vector spaces, linear applications, eigenvalues and eigenvectors, diagonalization of a matrix, scalar products) and of analytical geometry in dimensions two and three (equations of lines and planes and analytical study of their mutual positions; equations and study of curves and surfaces, with particular reference to conics and quadrics).
Knowledge of the tools and techniques of Linear Algebra for the study of Analytical Geometry. Ability to understand and use appropriate mathematical tools for solving geometric problems of plane and space. Ability to communicate the acquired knowledge through an adequate technical-scientific language.
Course Prerequisites
Knowledge of the basics of mathematics: algebraic equations of first, second and third degree, Ruffini's rule, decomposition of polynomials, algebraic inequalities, analytic geometry of the line
Teaching Methods
Lesson with use of digital whiteboard, carrying out exercises on the blackboard, projection of slides linked to the reference book.
Use of the teams platform to provide students with teaching material and slides of the lessons held.
Carrying out of exercises proposed by the students.
Assessment Methods
The exam consists of a final written test and an oral test which can be accessed if at least a predetermined minimum score has been achieved in the final written test. Passing any written tests in itinere exempts the student from the final written test or part of it.
Passing the written test gives the right to take the oral exam only in the appeal in which the written exam was passed or in the appeals of the same session.
The possible topics on which the written exam will focus are:
1. Resolution of linear systems (6 pt)
2. Operations between matrices, inverse matrix calculation, similar matrices (6 pts)
3.Linear applications (injectivity, surjectivity, image and core, diagonalization, base change) (10 pt)
5. Equation of the line in plane and space (8 pt)
The written test evaluates the critical skills achieved by the student in framing the topics covered by the course and the methodological rigor of the resolutions proposed in response to the questions formulated. This test shall last a maximum of two hours. The oral exam consists of an interview on the topics of the written test and on the theoretical topics that are part of the course program.
The student's ability to communicate the notions acquired through an adequate scientific language and the ability to expose is evaluated.
The final grade will be assigned according to the following evaluation criterion:
30 - 30 laude: excellent knowledge of the topics, excellent language properties, complete and original interpretative skills, strong ability to independently apply knowledge to solve the proposed problems;
26 - 29: complete knowledge of the topics, good language properties, complete and effective interpretative skills, able to independently apply knowledge to solve the proposed problems;
24 - 25: knowledge of topics with a good degree of learning, good language properties, correct and safe interpretative skills, ability to correctly apply most of the knowledge to solve the proposed problems;
21 - 23: adequate knowledge of the topics, but lack of mastery of the same, satisfactory language properties, correct interpretative ability, limited ability to apply knowledge autonomously to solve the proposed problems;
18 - 20: basic knowledge of the main topics and technical language, sufficient interpretative skills, ability to apply the acquired knowledge;
Insufficient: does not have an acceptable knowledge of the topics covered during the course.
Texts
Francesco Bottacin, Linear Algebra and Geometry, Esculapio, Bologna, 2023
Francesco Bottacin, Exercises in Linear Algebra and Geometry. Esculapio, Bologna, 2023
Contents
Vector spaces (1CFU)
Definition of k-field and examples of numerical sets. Definition of k-vector space. Examples. Subspaces. Operations with subspaces: sum, intersection, union and direct sum. Intersection of two subspaces (with proof). Criterion for the direct sum of two subspaces. Linear combination of a set of vectors of a vector space. Linearly independent vectors. Criterion for the linear independence of vectors. Finite dimensional vector spaces. Generators and bases of a vector space. Completion method and waste method for determining a basis. Canonical bases. Components of a vector and base changes. Theorem on the dimension of a subspace. Grassmann formule
Linear systems and matrices (2CFU)
Systems of linear equations. Homogeneous linear systems. Matrices. Diagonal, symmetric and antisymmetric matrices. Transposed matrix. Triangular matrices. Reduced matrix by rows. Row and column reduction of a matrix. Equivalent linear systems. Reduced linear systems. Solving systems of linear equations. Gauss-Jordan method. Product of matrices. Invertible matrices. Rank of a matrix. Rouchè-Capelli theorem (with proof). Determinant of a matrix. Sarrus rule. Laplace theorems. Calculation of determinants and properties. Determinants and invertible matrices. Added matrix. Inverse of a matrix with the adjoint matrix method. Cramer's rule. Less than a matrix. Kronecher theorem. Parametric linear systems.
Linear applications and Euclidean vector spaces (1CFU)
Definition and examples of linear application. Core and Image of a linear application. Linear and matrix applications. Injective, surjective and bijective linear maps. Isomorphisms. Injectivity criterion (with proof). Theorem on image generators of a linear application. Dimension theorem. Composition between two linear applications and associated matrix. Trace of a matrix. Similar matrices. Eigenvalues and eigenvectors. Algebraic and geometric multiplicity of an eigenvalue. Simple endomorphisms and theorem on simple endomorphism. Theorem on the dimension of eigenspaces. Orthogonal matrices. Orthogonal bases. Orthogonal matrices.
More information
Office hours for students both in the studio and in the classroom. Possibility also of support for carrying out exercises or any explanations also via email or online connection.