LINEAR ALGEBRA AND GEOMETRY F - O

Knowledge: being able to compute the rank of a matrix, with or without a parameter, to study a vector space, to study a linear application, to determine eigenvalues and eigenspaces of an endomorphism, to diagonalize a matrix, to solve problems of linear geometry, to classify conics and quadrics and to study conics bundles in the plane.

Understanding: fundamental definitions and theorems about vector spaces, linear applications and endomorphisms, constructions and theorems about lines and planes in the space.

During the lessons topics and concepts will be proposed in a formal way, together with meaningful examples, applications and exercises. A tutor will carry classroom exercises. The student will be sollicited to carry out exercises autonomously, even during the lessons.

Linear Algebra:

1. General information on sets, operations. Applications between sets, image and counterimage, injectivity, surjectivity, bijective applications. Sets with operations, the main geometric structures: groups, rings, fields.

2. The vectors of ordinary space. Sum of vectors, product of a number and a vector. Scalar product, vector product, mixed product. Components of vectors and operations using components.

3. Vector spaces and their properties. Examples. Subspaces. Intersection, union and sum of subspaces. Linear independence, relative criterion. Generators of a space. Basis of a space, method of successive discards, completion to a basis. Steinitz lemma*, dimension of a vector space. Grassmann formula*. Direct sums.

4. General information on matrices. Rank. Reduced matrices and reduction method. Elementary matrices. Product of matrices. Linear systems, Rouché-Capelli theorem. Solving linear systems with the (Gaussian) reduction method, free unknowns. Inverse of a square matrix. Homogeneous systems and subspace of solutions.

5. Determinants and their properties. Laplace's theorems*. Calculation of the inverse of a square matrix. Binet's theorem*. Cramer's theorem. Kronecker theorem*.

6. Linear maps and their properties. Core and image of a linear application. Injectivity, surjectivity, isomorphisms. The space L(V,W), its isomorphism* with K^{m,n}. Study of linear applications. Change of base, similar matrices.

7. Eigenvalues, eigenvectors and eigenspaces of an endomorphism. Characteristic polynomial. Size of eigenspaces. Independence of the eigenvectors. Simple endomorphisms and diagonalization of matrices.

Geometry:

1. Linear geometry in the plane. Cartesian coordinates and homogeneous coordinates. Lines and their equations. Intersections between lines. Slope coefficient. Distances. Bundles of lines.

2. Linear geometry in space. Cartesian coordinates and homogeneous coordinates. Planes and their equations. The lines, their representation. Improper elements. Angular properties of lines and planes. Distances. Bundles of plans.

3. Coordinate changes in the plane, rotations and translations.

1. S. Giuffrida, A. Ragusa.
Corso di Algebra Lineare  con Esercizi Svolti.
Il Cigno Galileo Galilei, Roma, 1998.

2. G. Paxia. Lezioni di Geometria. 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.

5. E. Sernesi.
Geometria 1.
Bollati Boringhieri, 2000.