# GEOMETRY

## Learning outcomes of the course unit

Knowledge and understanding:

the theory of vector spaces.

Applying knowledge and understanding:

a) solve systems of linear equations;

b) diagonalize (symmetric) matrices;

c) solve easy problems of analytic geometry;

d) recognize the type of a conic and write its canonical form.

Making judgements:

evaluate the correctness of a simple proof.

Communication and learning skills:

properly express themselves with mathematical language.

## Prerequisites

Precourse. This exam is preparatory to "Analisi matematica 2".

## Course contents summary

1. Real and complex vector spaces.

2. Determinants and rank of a matrix.

3. Linear systems.

4. Linear applications.

5. Endomorphisms of a vector space.

6. Scalar products.

7. Affine geometry of space.

8. Elements of analytic geometry of the three-dimesional space.

9. Complements of algebra/geometry.

## Course contents

0. Preliminaries: equivalence relations and partitions; algebraic structures

(groups and fields).

1. Real and complex vector spaces. Linear subspaces: sum and intersection. Linear combinations of vectors: linear dipendence/indipendence. Generators, bases and dimension of a vector spaces. Grassmann formula for subspaces.

2. Determinants: Laplace expansion and basic properties. Binet theorem. Row and column elementary operations on matrices. Computation of the inverse matrix. Rank of a matrix.

3. Linear systems: Gauss-Jordan method and Rouché Capelli theorem.

4. Linear maps. Definition of kernel and image; fundamental theorem on linear maps. Matrix representation of a linear map and change of bases. Isomorphisms and inverse matrix.

5. Endomorphisms of a vector space: eigenvalues, eigenvector and eigenspaces. Characteristic polynomial. Algebraic and geometric multiplicity. Diagonalizable endomorphisms.

6. Scalar products. Orthogonal complement of a linear subspace. Gram-Schmidt orthogonalization process. Representation of isometries by orthogonal matrices. The orthogonal group. Diagonalization of symmetric matrices: spectral theorem. Positivity criterion for scalar product: Sylvester theorem.

Outline of the complex case.

7. Affine geometry.

Parallelism and mutual position of affine subspaces.

8. Two and three dimensional analytic geometry. Parametric and Cartesian equations of a line. Mutual position between two lines in the space; skew lines. Equation of a plane. Canonical scalar product and distance. Vector product and its fundamental properties. Distance of a point from a line and from a plane.

9. Complements of algebra/geometry.

## Recommended readings

F. Capocasa, C.Medori: "Corso di Geometria", ed. S.Croce (Parma, 2013).

## Teaching methods

Lessons.

## Assessment methods and criteria

Course grades will be based on a final exam which consists of a preliminary multiple-choice test, a written exam and an oral interview. There will be the possibility of intermediate written exams to avoid the final written exam. The written exam, through tests and exercises, should establish that students have learned the course materials to a sufficient level. In the colloquium, students should be able to repeat definitions, theorems and proofs given in the lectures using a proper mathematical language and formalism.

The examamination has to ensure the intellectual maturity of the candidate and his organic preparation on the arguments of the course.

## Other informations

Lecture attendance is highly recommended.