# Geometry

**Academic Year 2024/2025**- Teacher:

**Pietro URSINO**

## Expected Learning Outcomes

The course introduces the student to the language, the precision and the accuracy necesary for the study

of basic concepts of Linear Algebra and Analytic Geometry: among these, vector space theory, matrix

calculus, resolutions of linear systems, linear applications, computation of eigenvalues and eigenvectors,

diagonalizations of matrices, lines and planes in the 3-dimensional space, conics in the plane.

The student is required to apply these notions and methods to the resolution of concrete problems of

linear algebra and analytic geometry that concern the study of simple geometric objects in 2 and 3-

dimensional spaces.

The student will face various theoretical aspects of the topics covered, improving logic skills in order to

use with precision and accuracy some significant mathematics proof methods. Such proofs are presented

in order to catch every detail necessary to reach the target.

Studying Linear Algebra and Geometry and testing their skills through exercises, the student will will

learn to communicate with clarity and rigour both, verbally and in writing. The student will learn that

using a correct terminology is one of the most important tools in order to communicate correctly in

scientific language, not only in mathematics.

Students will be able to use acquired notions, concepts and methods in their further studies and will be

encouraged to deepen specific aspects.

## Course Structure

During the lessons topics and concepts will be proposed in a formal way, together with meaningful

examples, applications and exercises. The student will be sollicited to carry out exercises autonomously,

even during the lessons.

## Required Prerequisites

## Attendance of Lessons

## Detailed Course Content

Linear Algebra:

1. Generalities on set theory and operations. Maps between sets, image and inverse image, injective

and surjective maps, bijective maps. Sets with operation, gropus, rings, fields.

2. Vectors in the ordinary space. Sum of vectors, product of a number and a vector. Scalar product. Components of vectors and operations with components.

4. Vector spaces and properties. Examples. Subspaces. Intersection, union and sum of subspaces.

Linear independence. Generators. Base of a vector space, completion of a base. Steinitz Lemma*,

dimension of a vector space. Grassmann formula*. Direct sum.

5. Generalities on matrices. Rank. Reduced matrix and reduction of a matrix. Product of matrices.

Linear systems. Rouchè-Capelli theorem. Solutions of linear systems. Homogeneous systems and

space of solutions.

6. Determinants and properties. Laplace theorems*. Inverse of a square matrix. Binet theorem*.

Cramer thoerem. Kronecker theorem*.

7. Linear maps and properties. Kernel and image. Injective and surjcetive maps. Isomorphisms. L(V,W)

and isomomorphism with k^{m,n}. Study of a linear map. Base change.

8. Eigenvalues, eigenvectors and eigenspaces of an endomorphism. Characteristic polynomial.

Dimension of eigenspaces. Independence of eigenvectors. Simple endomorphisms and

diagonalization of matrices.

Geometry

1. Linear geometry on the plane. Cartesian coordinates and homogeneous coordinates. Lines and

their equations. Intersection of lines. Angular coefficient. Distances. Pencils of lines.

2. Linear geometry in the space. Cartesian coordinates and homogeneous coordinates. Planes and

their equation. Lines and their representation. Ideal elements. Angular properties of lines and

planes. Distances. éencils of planes.

3. Change of coordinates in the plane, rotations and translations. Conics and associated matrices,

ortogonal invariants. Reduced equations, reduction of a conic in canonic form. Classification of

irreducible concis. Study of equations in canonic form. Circle. Pencils of conics.

The prooves of the theorem signed with * can be ometted.

## Textbook Information

1. S. Giuffrida, A. Ragusa: Corso di Algebra Lineare. Il Cigno Galileo Galilei, Roma, 1998.P. Bonacini, M. G. Cinquegrani, L. Marino. Algebra lineare: esercizi svolti. Cavallotto Edizioni,

Catania, 2012.

2. G.Paxia: Lezioni di Geometria. Spazio Libri, Catania, 2000.

3. .P. Bonacini, M. G. Cinquegrani, L. Marino. Algebra lineare: esercizi svolti. Cavallotto Edizioni,

Catania, 2012.

4. P. Bonacini, M. G. Cinquegrani, L. Marino. Geometria analitica: esercizi svolti. Cavallotto Edizioni,

Catania, 2012.

## Course Planning

Subjects | Text References | |
---|---|---|

1 | Introduction to set theory. Introduction to vector fields and spaces. Determinant of a matrix. Calculating the rank and reduction of a matrix. Solving linear systems. Time required: 8 hours | Libro di teoria: capitoli 1,3 Libro di esercizi: capitolo 1 |

2 | Matrix Operations. Time Required: 2 hours | Libro di teoria: capitolo 3 Libro di esercizi: capitolo 1 |

3 | Vector spaces. Generators and free sets. Subspaces. Basis and components with respect to a basis. Dimension of a vector space. Time required: 8 hours | Libro di teoria: capitolo 2 Libro di esercizi: capitolo 2 |

4 | Sum and intersection of vector spaces. Extraction of a basis from a system of generators and completion of a free set. Time required: 2 hours | Libro di teoria: capitolo 2 Libro di esercizi: capitolo 2 |

5 | Linear applications and their assignment. Study of a linear application. Calculation of images and counterimages. Time required: 9 hours | Libro di teoria: capitolo 4 Libro di esercizi: capitoli 3,4 |

6 | Base change matrices and similar matrices. Operations with linear applications. Time required: 2 hours | Libro di teoria: capitolo 4 Libro di esercizi: capitolo ,5 |

7 | Eigenvalues, eigenvectors and eigenspaces. Characteristic polynomial. Algebraic and geometric multiplicity of an eigenvalue. Simple endomorphisms. Diagonalization of a matrix. Time required: 8 hours | Libro di teoria: capitolo 5 Libro di esercizi: capitolo 6 |

8 | Generalities on vector calculus. Cartesian coordinates and homogeneous coordinates. Assignment of a line and a plane and their equations. Ideal points. Intersections. Parallelism and orthogonality. Bundles of lines and planes. Distances. Time required: 9 hours | Libro di teoria: capitoli 1, 2, 3 Libro di esercizi: capitolo 1 |

9 | Conics and associated matrices. Coordinate changers in the plane, orthogonal invariants and reduced equations of a conic. Classification of conics. Circles. Tangent lines. Conic bundles. Time required: 8 hours | Libro di teoria: capitolo 4 Libro di esercizi: capitolo 2 |

10 | Complete study of conics. Conics under condition. Time required: 4 hours. | Libro di teoria: capitolo 4 Libro di esercizi: capitolo 2 |