Geometry

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.

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.

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.

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.