# GEOMETRY 1B

## Learning outcomes of the course unit

Knowledge and understanding: The course aims to provide the basic knowledge of the spectral theory of the operators on a real or complex Euclidean space, of the theory of bilinear and sesquilinear forms, of the scalar and hermitian products, and of the forms on a Euclidean space. The student should be able to independently read and understand results of Analytical Geometry and Linear Algebra also by consulting scientific monographs.

Skills: The student should be able to solve exercises of Analytical Geometry and Linear Algebra also non-elementary.

Making judgments: The student should be able to construct and develop logical arguments with a clear identification of assumptions and conclusions; it should also be able to recognize correct demonstrations and identify fallacious reasoning.

## Prerequisites

Geometria 1a

## Course contents summary

Dual space and the transpose of a linear transformation. Inner product spaces. Linear isometries and unitaty operators.

Linear functionals and adjoints. Spectral theory of operators on an inner product space: self-adjoint and normal operators. Bilinear and sesquilinear forms. Scalar and Hermitian products. Forms on inner product spaces. Quadrics. Affine e projective geometry (outline).

## Course contents

Dual of a vector space and the dual basis. The double dual of a vector space. Annihilator of a subset of a vector space. The transpose of a linear transformation.

Jordan's canonical form. Hamilton-Cayley theorem.

Real or complex inner product spaces: orthogonal complement of a linear subspace. Linear isometries and unitary operators: unitary and orthogonal matrices. Representation of a linear functional and adjoint of a linear operator. Self-adjoint operators: symmetric and hermitian operators. Spectral theorem for self-adjoint operators and for normal operators. Bilinear and sesquilinear forms: basis change, congruence, polar form. Scalar and hermitian products: existence of orthonormal bases. The Sylvester theorem: congruence criterion for symmetric matrices. Forms on inner product spaces: reduction to principal axes, simultaneous reduction of a pair of quadratic forms to a sum of squares.

Quadrics. Affine e projective geometry (outline).

## Recommended readings

Serge Lang. Algebra lineare. (Terza edizione) Bollati Boringhieri, 1970.

Paolo De Bartolomeis. Algebra lineare. La Nuova Italia, 1993.

Other references:

Morton L. Curtis. Abstract linear algebra. Springer, 1990.

Seymour Lipschutz, Marc Lipson. Algebra lineare. McGraw-Hill, 1992.

Ciro Ciliberto. Algebra lineare. Bollati Boringhieri, 1994.

Marco Abate. Geometria. McGraw-Hill, 1996.

Mauro Nacinovich. Elementi di geometria analitica. Liguori Editore, 1996.

## Teaching methods

The theoretical topics of the course are presented during class lectures and illustrated with significant examples, applications and several exercises. Homework assignments are proposed during the course, which are then discussed in recitation sessions during class time.

## Assessment methods and criteria

The final exam consists of a written part, where students are required to answer to a test and to some questions, and of an oral part about the theoretical topics and the applications discussed during the course. Access to the oral part is not recommended if the written part is insufficient.

The final mark turns out to be a weighted average between the evalution of the written and oral parts.

## Other informations

Part of the educational material may be present on Elly.