B005475 - GEOMETRY I

Italian

The space R^n.
Vector spaces.
Matrices.
Linear Applications.
Linear Applications and matrices.
Determinants.
Scalar products and orthogonality.
Matrices and bilinear applications.
Polynomials and matrices.
Triangulation for matrices and linear applications.
Spectral theorem.
Jordan canonical form.
Projective geometry.
Affine geometry.
Euclidean geometry.
Conics and quadrics.

Serge Lang, ALGEBRA LINEARE, Bollati Boringhieri

Marco Abate, GEOMETRIA, McGraw-Hill

W. Keith Nicholson, ALGEBRA LINEARE - DALLE APPLICAZIONI ALLA TEORIA, McGraw-Hill

Gareth Williams, LINEAR ALGEBRA WITH APPLICATIONS, Jones and Bartlett Mathematics

Nigel Hitchin, PROJECTIVE GEOMETRY, https://people.maths.ox.ac.uk/hitchin/hitchinnotes/hitchinnotes.html

The course aims to provide the students with fundamental knowledge and understanding in Linear Algebra, Analytic Geometry and Projective Geometry. One of the aims is to let the students develop basic technical skills and critical thinking, needed when modelling and solving mathematical problems in different settings. Special attention is paid to help the students develop communication skills necessary for teamwork. The course covers topics and provides learning skills that are needed, or strongly suggested, to pursue a degree in mathematics or in any scientific subject.

Basic notions of algebra and geometry taught in high school.

Lectures: presentation of the theory described in the course program, with teacher-student direct interaction, to ensure a full understanding of the subject.

Training sessions: training of the students to modelling and solving a wide selection of problems in Linear Algebra, Analytic Geometry and Projective Geometry. The training sessions are conducted so to:
-- help the students develop communication skills and apply theoretical knowledge;
-- encourage independent judgement in the students.

Moodle learning platform: online teacher-student interaction; posting of additional notes, weekly exercise sheets, and copies of past tests.

Remark: the suggested reading includes supplementary material that may be useful for further personal studies in mathematics or any scientific subject.

Intermediate and final written examination: a selection of exercises is proposed. The tests are designed to assess the ability of the students to apply their skills to problem modelling and solving. In the evaluation, special attention is paid to the correctness of the solution procedure, as well as to the originality and effectiveness of the methods adopted.

Oral examination: a number of questions are posed. The oral examination is designed to evaluate the degree of understanding of the theory presented in the course. In the assessment, special attention is paid to communication skills, critical thinking and appropriate use of mathematical language.

The space R^n. Points of the n-space; vectors; canonical scalar product; norm of a vector; orthogonality and parallelism; distance; Euclidean geometry of the n-space; lines, planes, hyperplanes.

Vector spaces. Subsets of linearly dependent, or linearly independent, vectors; bases of a vector space; dimension of a vector space; sums of vector spaces; direct sums of vector spaces.


Matrices. The vector space of matrices; sum and product of matrices; linear equations; systems of linear equations and matrices; the Gauss algorithm to solve linear systems.

Linear maps. Kernel and image of a linear map; dimension of the kernel and the image; composition of linear maps.

Linear maps and matrices. Linear map associated to a matrix; matrix associated to a linear map; composition of linear maps and matrices.

Determinants. Determinants of order 2; Properties of determinants; The Cramer rule; existence of determinants; uniqueness of the determinant; determinant of the transpose of a matrix; determinant of the product of two matrices; inverse of a matrix; determinant of a linear map.

Scalar products and orthogonality. Scalar products; positive definite scalar products; orthogonal bases in the general case; dual space; rank of a matrix and systems of linear equations.

Matrices and bilinear maps. Bilinear forms; quadratic forms; symmetric operators; Hermitian operators; Sylvester's theorem.

Polynomials and matrices. Polynomials of matrices and linear maps; eigenvectors and eigenvalues; characteristic polynomial.

Triangulation of matrices and linear maps. Fan; fan basis; existence of triangulation; Hamilton-Cayley's theorem; diagonalization of unitary matrices.

The spectral theorem. Eigenvectors of linear symmetric maps; The spectral theorem; the complex case.

The Jordan normal form. Generalized eigenspaces of a linear map; Jordan normal form for a nilpotent map; Jordan basis and Jordan form for a linear map.

Projective geometry. Projective spaces; projective subspaces; the group of projective transformations; points in general position; Desargues' and Pappus' theorems; duality.

Quadrics. Quadratic forms; conics and quadrics; polarity with respect to a conic; projective classification of conics; pencils of conics.

Affine geometry. The affine plane and the affine space; the group of affine transformations; affine classification of conics.

Euclidean geometry. The group of isometries; euclidean classification of conics.

Exercises and examples.