LINEAR ALGEBRA AND GEOMETRY F - O

4. Communication skills: the frequence of the lessons and the suggested books will help the student to familiarize with the rigour of mathematical language and to learn the specific language of linear algebra and geometry. Through the continuous interaction with the teacher, the student will learn to communicate with rigour and clarity his/her acquired knowledge, both in an oral and in a written way. At the end of the course, the student will have learnt that the mathematical language is useful in order to be able to communicate in a clear way in a scientific setting.

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:

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. Elementary matrices. 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. Pencils of planes.
3. 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.
4. Quadrics in the space and associated matrices. Irreducible quadrics. 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 proofs of the theorem signed with * can be ometted.

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.

More didactic material is available at https://studium.unict.it/ and https://www.dmi.unict.it/bonacini/didattica/

The examination is written and oral. The written examination, which usually lasts 3 hours, is compulsory to take the oral examination.

Exminations may also be carried out on line, should the conditions require it. In such a case, the length of the written examination might change.

Linear Algebra exercises:

1. study of a linear application with parameter, determining kernel and image.

2. study of the diagonalization of an endomorphism with parameters, determining, if possible, a bases of eigenvectors.

3. inverse image of a vector, resolution of a linear system with parameter, inverse image of a vector space, image of a vector space.

4. exercises on vector spaces and their dimension, direct sum, operations on linear applications, induced linear applications, restrictions and extensions.

Geometry exercise:

1. linear geometry exercises in the 3-dimensional space: parallelism and orthogonality, distances, orthogonal projections, angles.

2. pencil of conics, complete study of a conic, conics under conditions.

3. classification of quadrics with parameter, quadrics under conditions, intersection of quadrics with planes.