# LINEAR ALGEBRA AND GEOMETRY J - Pr

MAT/03 - 9 CFU - 2° Semester

PAOLA BONACINI

## Learning Objectives

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 and quadrics in the 3-dimensional space.

The student at the end of the course will be able to: compute the rank of a matrix, solve linear systems, determine the dimension of a vector space and compute a base, study linear applications between vector spaces, compute eigenvectors and eigenvalues of endomorphisms, diagonalize matrices, solve problems of linear geometry with points, lines and planes in the 3-dimensional space, classify and study conics, study conic bundles, classify quadrics in the 3-dimensional space.

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. Moreover, for every topic covered the students are proposed various exercises, to do in the room during the lesson or at home.

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.

## 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.

## 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, vector product. Components of vectors and operations with components.
3. Complex numbers, operations and properties. Algebraic and trigonometric form of complex numbers. De Moivre formula. nth root of complex numbers.
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. Tangent lines. Pencils of conics.
4. Quadrics in the space and associated matrices. Irreducible concis. Vertices and dengerate quadrics. Cones and cylinders. Reduced equations, reduction in canonic form. Classification of non degenerate quadrics. Sections of quadrics with lines and planes. Lines and tangent planes.

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

## Textbook Information

1. P. Bonacini, M. G. Cinquegrani, L. Marino. Algebra lineare: esercizi svolti. Cavallotto Edizioni, Catania, 2012.
2. P. Bonacini, M. G. Cinquegrani, L. Marino. Geometria analitica: esercizi svolti. Cavallotto Edizioni, Catania, 2012.
3. S. Giuffrida, A. Ragusa: Corso di Algebra Lineare. Il Cigno Galileo Galilei, Roma, 1998.
4. Lezioni di Geometria. Spazio Libri, Catania, 2000.

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