GEOMETRIA M - Z

Basic knowledge of linear algebra and geometry

The course is structured in participatory lectures and cooperatives that provide for ongoing checks. It will aim to ensure consistency between the training objectives and the methods used, combining the frontal methodology with the dialogue one but including the possibility of applying knowledge through exercises .

Geometric vectors in the plane and space- Vector spaces, subspaces and operations with them- Matrices, determinants and linear systems and their properties- Linear fuctions and their properties – Eigenvalues and eigenvectors of an endomorphism- Similarity transformations between matrices- Diagonalization of an endomorphism- Orthogonal projections - Matrix of a projection and Decomposition Theorem orthogonal projections -Bilinear forms, real scalar products and hermitian products- Self-adjoint endomorphisms (or hermitian if the scalar field is C, symmetric if the scalar field is R) – Spectral theorem and its corollaries- Decomposition Spectral Theorem Hermitian matrix - Real quadratic forms- Congruence transformations between matrices associated with a quadratic form- Nullity and signature of a quadratic form- Invariants by congruence- Examples.

Linear geometry in the plane and space - Conics, orthogonal invariants- Matrix associated with a Conic- Affine classification of conics non-degenerate- Degenerate and not- degenerate quadrics, orthogonal invariants - Matrix associated with a Quadric- Polarity in the plane and space- Cones and cylinders- Affine classification of not- degenerate quadrics- Classification of quadrics through Signature- Tangent lines and tangent planes for a quadric- Plane sections of irreducible quadrics. Examples.

Lezioni di Algebra lineare e Geometria -- Autori: Greco-Valabrega

Elementi di Algebra Lineare e Geometria -- Autore: Silvana Abeasis

Complementi di Algebra Lineare e Geometria -- Autore: Silvana Abeasis