81856 - GEOMETRIA 1B

Course Unit Page

Academic Year 2018/2019

Learning outcomes

At the end of the course students have deepened their knowledge of the fundamental concepts of linear algebra seen in Geometry 1A; they are able to diagonalize an endomorphism and know the canonical Jordan form. They have also studied bilinear and quadratic forms and their application to some geometrical problems (Euclidean spaces, classification of conics and quadrics).

Course contents

Endomorphisms. Invariant subspaces. Decomposition of a space into the direct sum of invariant subspaces and its matrix formulation. An example: the Chinese remainder theorem and the associated decomposition. Characteristic polynomial of an endomorphism. Eigenvalues, eigenvectors, diagonalizable endomorphisms. Geometric and algebraic multiplicity of an eigenvalue. Necessary  and sufficient conditions for diagonalisation. Matrix polynomials. The minimum polynomial of an endomorphism. The Cayley-Hamilton theorem. Generalized autospaces. Nilpotent endomorphisms. Jordan's form of a nilpotent endomorphism. Example of calculation. Jordan form of an endomorphism on an algebraically closed field. Some elements on  the rational canonical form. Bilinear forms. Symmetric and antisymmetric forms. Radical of a bilinear form. Non-degenerate forms. Orthogonal space to a subspace and its dimension. Matrix associated to a bilinear form with respect to the choice of a basis. Dependence of the matrix on the chosen basis. Symmetric bilinear forms: Existence of orthogonal bases. Isotropic subspaces. Bilinear forms over the real field and over the complex field. Sylvester theorem. Scalar products over the real field. Orthogonal bases. Gram-Schmidt orthogonalization process. Its formulation in terms of factoring of invertible matrices. Transposed of a linear application. Symmetric endomorphisms. Spectral theorem. Applications to analytic geometry: Affine transformations of the plane and space, isometries, orthogonal projections. Conics and quadrics in the plane and in the space, affine classification.


M. Manetti. Algebra Lineare per Matematici.


S. Lang. Algebra Lineare. Bollati Boringhieri

Teaching methods

Lectures at the blackboard.

Assessment methods

Exam consisting of a written test and an oral test.

Students are admitted to the oral exam only if the written exam is sufficient. Both tests must be held in the same exam session.

The written exam consists of a section with multiple-choice questions and 3 exercises to be solved. The exercises will be corrected only if all the answers to the multiple choice questions are correct.

The answers to the multiple-choice questions and the solution of the exercises will be available on the teachers' web page.

Office hours

See the website of Luca Migliorini

See the website of Nicoletta Cantarini