AIMS AND CONTENT

AIMS AND LEARNING OUTCOMES

The purpose of this course is to consolidate the techniques already learned in the previous module of the same course. In particular, the aims are the followings:

1. Diagonalizability and diagonalization of linear applications: definitions and properties of diagonalizable endomorphisms; eigenvalues, eigenvectors, eigenspaces; characteristic polynomial of an endomorphism, properties and relation to eigenvalues; diagonalizability criterion.
2. Triangulability and triangulation of endomorphisms: definition and properties of triangulable endomorphisms; triangulability criterion; characteristic spaces, Lemma of cores and Cayley-Hamilton Theorem; nilpotent endomorphisms and their triangulation by completion of the bases and by Jordan decomposition; triangulation of endomorphisms in general.
3. Bilinear applications: dual vector space, dual basis, transposed linear application; bilinear forms and applications and properties, matrix of a bilinear form and properties (symmetric bilinear forms, nondegenerate bilinear forms), congruence of matrices. Defining character of a symmetric bilinear form on a real vector space, Euclidean vector spaces, norms and angles, orthogonality, orthogonal projections; orthonormal bases Gram-Schmidt theorem, orthogonal complement of a subspace.
4. Endomorphisms between Euclidean vector spaces: isometries and their characterizations, orthogonal and special orthogonal matrices, properties, description of plane isometries; self-adjoint endomorphisms and their characterizations, symmetrical matrices, real spectral theorem, diagonalization of self-adjoint endomorphisms by means of orthonormal bases. Signature of a real symmetric matrix, Sylvester's inertia theorem.
5. Conics and quadrics: definition of a quadric, matrix associated with a quadric and quadratic form matrix. Classification of quadrics by affine transformations, geometric properties of conics and quadrics, bundles of conics.
6. Affine and projective spaces: definition of affine space over a field, and Euclidean affine space, and their properties. Lines, planes, hyperplanes in an affine space, Euclid's five postulates; affine transformations, coordinate systems and coordinate changes. Projective spaces: motivations, models of projective spaces. Straight lines, planes, hyperplanes in a projective space, Euclid's fifth postulate does not apply in projective geometry. Homogeneous coordinates, equations of lines and planes, affine maps, points at infinity. Projectivity: definition, properties, characterization of fixed points, fixed lines, fixed point lines using linear algebra.

At the end of the course the students will be able to:

1. Determine whether an endomorphism is diagonalizable and diagonalize it.
2. Determine whether an endomorphism is triangulable and determine its canonical Jordan form.
3. Work with bilinear forms.
4. Classify projective and affine conics and quadrics.

TEACHING METHODS

Standard Frontal Lesson 

SYLLABUS/CONTENT

1. Diagonalization and triangulation
2. Euclidean vector spaces
3. Isometries and self-adjoint endomorphisms
4. Conics and quadrics
5. Affine and projective spaces

RECOMMENDED READING/BIBLIOGRAPHY

• M.E. Rossi, Algebra lineare, Dispense disponibili nella pagina del corso
• F. Odetti - M. Raimondo, Elementi di Algebra Lineare e Geometria Analitica, ECIG Universitas.
• Marco Abate, Algebra Lineare , ed. McGraw-Hill
• E. Sernesi, Geometria vol 1, ed Bollati-Boringhieri.
• A.Bernardi, A.Gimigliano, Algebra Lineare e Geometria Analitica, Citta'Studi Edizioni. 

TEACHERS AND EXAM BOARD

Exam Board

ARVID PEREGO (President)

ELEONORA ANNA ROMANO

MATTEO PENEGINI (President Substitute)

MARIA EVELINA ROSSI (President Substitute)

SIMONE DI MARINO (Substitute)

LESSONS

EXAMS

EXAM DESCRIPTION

The examination consists of a written and an oral part.

Students with a certified DSA, disability or other special educational needs are advised to contact the lecturer at the beginning of the course in order to agree on teaching and examination methods that, while respecting the teaching objectives, take into account individual learning methods and provide suitable compensatory tools.

ASSESSMENT METHODS

In the written test, the student is asked to solve exercises covering the entire course syllabus.

In the oral test, students in Mathematics and SMID are required to know and be able to present the definitions, statements of theorms and their proofs seen throughout the course.

Exam schedule