## 20410340 - GE210 - Geometry and linear algebra 2

Acquire a good knowledge of the theory of bilinear forms and their geometric applications. An important application will be the study of Euclidean geometry, mainly in the plane and in the space, and the Euclidean classification of the conics and of the quadratic surfaces.

Curriculum

teacher profile | teaching materials

Programme

Euclidean geometry
Bilinear forms and quadratic forms. Diagonalization of quadratic forms. Scalar products.
The vector product operation. Euclidean spaces. Unitary operators and isometries. Isometries of plane and three-dimensional spaces. Diagonalization of symmetric operators. The complex case.

Projective geometry
Projective spaces. Affine geometry and projective geometry. Duality. Homogeneous coordinate changes
and projectivities.

Plane algebraic curves
Generality. Real algebraic curves. Classification of projective conics. Classification of affine conics
and of euclidean conics.

Core Documentation

E. Sernesi: Geometria I, Bollati Boringhieri (1989)

Type of delivery of the course

lessons in class, exercises classes, group work assisited by a tutor, midterm and final evaluation

Type of evaluation

Verification of learning will be made either by two partial written exams (each of two hours and a half) to be done by the end of the course or, in alternative, by a three hours written exam after the end of the course, and, in any case, an oral examination. The written exam consists of three exercises (divided in points), with the goal to verify the level of understanding of the concepts and the capacity of applying them in practice. The oral exam consists in a verification of the knowledge of the most important theorems and results of the course. All written exams (and partial exams) of the previous years are available in the webpage: http://ricerca.matfis.uniroma3.it//users/lopez/didattica-passata.html

teacher profile | teaching materials

Programme

Euclidean geometry
Bilinear forms and quadratic forms. Diagonalization of quadratic forms. Scalar products.
The vector product operation. Euclidean spaces. Unitary operators and isometries. Isometries of plane and three-dimensional spaces. Diagonalization of symmetric operators. The complex case.

Projective geometry
Projective spaces. Affine geometry and projective geometry. Duality. Homogeneous coordinate changes
and projectivities.

Plane algebraic curves
Generality. Real algebraic curves. Classification of projective conics. Classification of affine conics
and of euclidean conics.

Core Documentation

E. Sernesi: Geometria I, Bollati Boringhieri (1989)

Type of delivery of the course

lessons in class, exercises classes, group work assisited by a tutor, midterm and final evaluation

Type of evaluation

Verification of learning will be made either by two partial written exams (each of two hours and a half) to be done by the end of the course or, in alternative, by a three hours written exam after the end of the course, and, in any case, an oral examination. The written exam consists of three exercises (divided in points), with the goal to verify the level of understanding of the concepts and the capacity of applying them in practice. The oral exam consists in a verification of the knowledge of the most important theorems and results of the course. All written exams (and partial exams) of the previous years are available in the webpage: http://ricerca.matfis.uniroma3.it//users/lopez/didattica-passata.html