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
(In case of multiple editions, please check the period, as it may vary)
1. Endomorphisms of vector spaces and their canonical forms Review of 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, volumes. 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. The concept of geometry according to F. Klein. 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. 6. Duality Dual of a vector space. Dual basis. Transposed homomorphism. Anihilators subspaces and their properties. Principle of duality in projective geometry.
Prerequisites for admission
The topics presented in the course Elementi di matematica di base and in the first semester courses.
Traditional: lessons and exercise classes. Tutoring: 2 hours a week for 12-13 weeks.
C. Ciliberto. Algebra Lineare. Bollati Boringhieri, Torino, 1994. E. Sernesi, Geometria 1, Bollati Boringhieri, Torino, 1989.
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. The first 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.