- Docente: Monica Idà
- Credits: 7
- SSD: MAT/03
- Language: Italian
- Moduli: Monica Idà (Modulo 1) Patrizio Frosini (Modulo 2)
- Teaching Mode: Traditional lectures (Modulo 1) Traditional lectures (Modulo 2)
- Campus: Bologna
- Corso: First cycle degree programme (L) in Mathematics (cod. 8010)
Course contents
Canonical forms for endomorphisms: recalls on diagonalizability, the Jordan form of an endomorphisms.
Bilinear forms on a vector space over a field K. The matrix of a bilinear form with respect to a given basis. Two matrices are congruent if and only if they represent the same bilinear form with respect to different basis. Rank of a bilinear form. The quadratic form associated to a symmetric bilinear form. The existence of a diagonalizing basis for a symmetric bilinear form over a K-vector space. Canonical form for a real quadratic form and for a complex quadratic form. Signature of a real quadratic form.
Scalar products and finitely dimensional euclidean vector spaces. Schwarz inequality. Existence of orthonormal basis. Orthogonal projection of a vector on a subspace and the Pythagorean identity. Orthogonal matrices. Symmetric endomorphisms and the Spectral Theorem.
Affine spaces. Affine subspaces, affine frames. Cartesian and parametric equations of an affine subspace. Parallel, skew and incident subspaces. The affine group and its subgroups. Equations of an affine transformation.
Euclidean spaces. Cartesian coordinates. Distance between two points. The convex angle between two oriented lines. Direct and inverse isometries, rotations. The isometries of the plane: Chasles Theorem.
Readings/Bibliography
E.Sernesi: Geometria 1 (Bollati Boringhieri)
The lecture notes, together with additional exercise sheets and previous written examinations, will be uploaded at the end of the semester on
http://www.dm.unibo.it/~ida/
Teaching methods
Lectures and exercise sessions
Assessment methods
Written and oral examination. The written assignment consists in some exercises through which the student should prove to be able to use the instruments acquired during the lessons. If the written exam is sufficient (i.e. >=14/30), the student is admitted to the oral examinations, where he/she discusses the written assignment and shows that he/she knows the subjects treated in the course and is able to reason about arguments related to the course.
Teaching tools
Additional excercise sheets and previous written examinations can be found at http://www.dm.unibo.it/~ida/annoincorso.html
Office hours
See the website of Monica Idà
See the website of Patrizio Frosini