LINEAR ALGEBRA AND GEOMETRY Cp - I

Knowledge and understanding: fundamental definitions and theorems about vector spaces, linear
applications and endomorphisms, constructions and theorems about lines and
planes in the space and conics in the plane, definitions and theorems about the
classifications of quadrics.

Applying knowledge and understanding: 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.

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. Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the programme planned and outlined in the syllabus. Learning assessment may also be carried out on line, should the conditions require it.

Linear Algebra:

 

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

inverse image, injective and surjective maps, bijective maps. Sets with

operation, groups, rings, fields.

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.

Complex numbers, operations and properties. Algebraic and trigonometric

form of complex numbers. De Moivre formula. nth root of complex numbers.

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.

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

Elementary matrices. Product of matrices. Linear systems. Rouchè-Capelli

theorem. Solutions of linear systems. Homogeneous systems and space of

solutions.

Determinants and properties. Laplace theorems*. Inverse of a square

matrix. Binet theorem*. Cramer thoerem. Kronecker theorem*.

Linear maps and properties. Kernel and image. Injective and surjective

maps. Isomorphisms. L(V,W) and isomorphism with k^{m,n}. Study of a linear

map. Base change.

Eigenvalues, eigenvectors and eigenspaces of an endomorphism.

Characteristic polynomial. Dimension of eigenspaces. Independence of

eigenvectors. Simple endomorphisms and diagonalization of matrices.

 

Geometry

 

Linear geometry on the plane. Cartesian coordinates and homogeneous

coordinates. Lines and their equations. Intersection of lines. Angular

coefficient. Distances. Pencils of lines.

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. Pencils of

planes.

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

associated matrices, orthogonal invariants. Reduced equations, reduction of a

conic in canonic form. Classification of irreducible conics. Study of equations

in canonic form. Circle. Tangent lines. Pencils of conics.

Quadrics in the space and associated matrices. Irreducible quadrics.

Vertices and degenerate 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 proofs of the theorem signed with * can be ometted.

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

2. 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.

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