# 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

Written examination (preceded by a test) and oral examination.

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.