Geometry 2
A.Y. 2024/2025
Learning objectives
The course aims at completing the linear algebra background and at introducing to n-dimensional geometry in affine-euclidean spaces and projective spaces. Quadric hypersurfaces will be discussed in these frameworks.
Expected learning outcomes
Knowledge of linear algebra tools and ability to apply them. Ability to deal with geometric problems in the most appropriate context.
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
1. Endomorphisms of vector spaces and their canonical forms
Eigenvalues and eigenvectors of an endomorphism. Diagonalizable endomorphisms. Minimum polynomial. Cayley Hamilton theorem. Jordan's canonical form ..
2. Euclidean vector spaces
Internal products in real and complex vector spaces. Orthonormal bases; Gram-Schmidt procedure. Isometries and orthogonal group. Symmetric endomorphisms. Real spectral theorem. The complex case.
3. Bilinear and quadratic forms
Multi-linear forms. Bilinear forms; congruent matrices. Canonical reduction of a quadratic form. Real quadratic forms. Sylvester's theorem. Complex quadratic forms.
4. Geometry in n-dimensional spaces on an arbitrary field
Euclidean affine spaces. Orthonormal coordinate systems. Linear subspaces and their representations. Distances, angles. Coordinate changes and transformations. Projective spaces. Projective linear subspaces and their representations. Grassmann's formula. Fundamental theorem of projective geometry. The affine complementary space of a hyperplane. Projectivities and affinities.
5. Quadrics and conics
Hyperquadrics from the real / complex projective point of view: singular points; reducibility, classification. Hyperquadrics in the affine space. Hyperquadrics in Euclidean space.
Eigenvalues and eigenvectors of an endomorphism. Diagonalizable endomorphisms. Minimum polynomial. Cayley Hamilton theorem. Jordan's canonical form ..
2. Euclidean vector spaces
Internal products in real and complex vector spaces. Orthonormal bases; Gram-Schmidt procedure. Isometries and orthogonal group. Symmetric endomorphisms. Real spectral theorem. The complex case.
3. Bilinear and quadratic forms
Multi-linear forms. Bilinear forms; congruent matrices. Canonical reduction of a quadratic form. Real quadratic forms. Sylvester's theorem. Complex quadratic forms.
4. Geometry in n-dimensional spaces on an arbitrary field
Euclidean affine spaces. Orthonormal coordinate systems. Linear subspaces and their representations. Distances, angles. Coordinate changes and transformations. Projective spaces. Projective linear subspaces and their representations. Grassmann's formula. Fundamental theorem of projective geometry. The affine complementary space of a hyperplane. Projectivities and affinities.
5. Quadrics and conics
Hyperquadrics from the real / complex projective point of view: singular points; reducibility, classification. Hyperquadrics in the affine space. Hyperquadrics in Euclidean space.
Prerequisites for admission
Gli argomenti di matematica presentati nel corso di Elementi di matematica di base e negli insegnamenti del primo semestre.
Teaching methods
Traditional: lessons and exercise classes.
Tutoring: 2 hours a week for 12-13 weeks.
Tutoring: 2 hours a week for 12-13 weeks.
Teaching Resources
C. Ciliberto. Algebra Lineare. Bollati Boringhieri, Torino, 1994.
E. Sernesi, Geometria 1, Bollati Boringhieri, Torino, 1989.
Further material will be available on the Ariel teaching site.
E. Sernesi, Geometria 1, Bollati Boringhieri, Torino, 1989.
Further material will be available on the Ariel teaching site.
Assessment methods and Criteria
The exam consists of a written test followed by an oral exam (if the written test is passed).
The written test requires the solution of exercises with contents and difficulties similar to those faced in the exercise classes, and is aimed at ascertaining the acquired skills to solve problems through the techniques developed during the course. Part of the written test can be passed through an intermediate test which takes place about halfway through the course.
The oral exam consists of an interview on the topics of the program, mainly aimed at ascertaining the knowledge of the theoretical topics addressed in the course.
The complete final examination is passed if all two parts (written, oral) are successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
The written test requires the solution of exercises with contents and difficulties similar to those faced in the exercise classes, and is aimed at ascertaining the acquired skills to solve problems through the techniques developed during the course. Part of the written test can be passed through an intermediate test which takes place about halfway through the course.
The oral exam consists of an interview on the topics of the program, mainly aimed at ascertaining the knowledge of the theoretical topics addressed in the course.
The complete final examination is passed if all two parts (written, oral) are successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
MAT/03 - GEOMETRY - University credits: 9
Practicals: 48 hours
Lessons: 45 hours
Lessons: 45 hours
Shifts:
Professor:
Turrini Cristina
Turno 1
Professor:
Tortora Alfonsoturno 2
Professor:
Lombardi LuigiProfessor(s)
Reception:
Wednesday 15:30-16:30 or by appointment
Math Department in via C. Saldini 50. Office: 1.109
Reception:
by appointment (by e-mail)
Math. Dept. - via C. Saldini 50 - Milano