# GEOMETRIA 1Module ESERCITAZIONI

**Academic Year 2023/2024**- Teacher:

**Antonio CAUSA**

## Expected Learning Outcomes

The aim of the programme is to give some preliminaries and tools for a basic introduction to Linear

Algebra and Analytical Geometry. In this course we look at properties of matrices, systems of linear equations

and vector spaces useful to find real eigenvalues and eigenvectors of applications.

We will learn about classification of plane conics and quadric surfaces, using their invariants and polar coordinates.

We will also solve some problems similar to the ones assigned at the final exam.

## Course Structure

Frontal lectures and classroom exercise. The teaching approach is a traditional one. The program offers personal feedback and attention from tutors in order to help students in their studies.

*Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the programme planned and outlined in the syllabus.*

*Learning assessment may also be carried out on line, should the conditions require it.*

## Required Prerequisites

## Attendance of Lessons

## Detailed Course Content

**Linear Algebra**

- Matrix operation: addition, scalar multiplication, matrix multiplication.
- Vector spaces and their properties over R. Examples: R[x], R
^{n}, R^{m,n.}. Subspaces. Intersection and sum of vector spaces. Direct sum. Linear combinations. Span, Linear Independence and dependence, Bases of a vector space, Dimension of a vector space. - Determinants and their properties. Applications of Laplace theorem, Inverse of a matrix, Rank and reduction of a matrix. Solving systems of linear equations. Application of Rouchè-theorem.
- Linear maps between vector spaces and their properties. Kernel and image of a linear map. Injective, surjective maps and isomorphisms. Study of linear maps. Matrices associated to linear maps.
- Eigenvalues, Eigenvectors and Eigenspaces of a matrix. Characteristic polynomial. Diagonalization of a matrix.

**Geometry**

- Euclidean (geometric) vectors and their properties. Scalar multiplication, dot (or scalar) product, wedge (or cross) product.
- Cartesian coordinates. Points, lines , Homogeneous coordinates, Points at infinity (Improper Points), Parallel and orthogonal Lines.
- Slope of a line. Distances from a point to a line. Pencil of lines.
- Planes in the euclidean space, palne through three points.
- Coplanar and Skew lines. Pencil of Planes. Angles between lines and planes.
- Distance from a point to a plane and from a point to a line in the space.
- Conics and their associated matrices. Orthogonal Invariants. Canonical reduction of a conic.
- Irreducible and degenerate conics. Rank of its associated matrix. Discriminant of a conic. Parabolas, Ellipses, Hyperbolas: equations, focus, eccentricity, directrix, semi-maior axis, center.
- Circles. Circle through three points, tangent line to a circle.
- Tangents, and pencils of conics: construction, inspection.
- Quadrics and its associated matrix. Nondegenerate, degenerate and singular quadric surfaces.
- Cones and cylinders. Classification of quadrics, pencils of quadrics.

## Textbook Information

1) S. Giuffrida, A.Ragusa, Corso di Algebra Lineare, Ed. Il Cigno G.Galilei, Roma 1998 (Linear Algebra).

2) G. Paxia, Lezioni di Geometria, Spazio Libri, Catania, 2005 (Geometry) available at www.giuseppepaxia.com

## Course Planning

Subjects | Text References | |
---|---|---|

1 | Linear spaces and basic properties. R^n, R^m,n, R[X]. Intersection and sum of subspaces. Linear independence, bases and dimension of a vector space. | Testo 1 |

2 | Matrix algebra, dot product. | Testo 1 |

3 | Determinant of a matrix. Inverse of a matrix. Rango di una matrice. reduction of a matrix in echelon form. Systems of linear equations. | Testo 1 |

4 | Linear transformations. Application fo rank-nullity theorem. Linear transformation depending on a parameter. | Testo 1 |

5 | Eigenvalues and eigenvectors. Characteristic polynomial. Diagonalization of matrices. | Testo 1 |

6 | Free vectors in euclidean space. Sum. dot product, cross product. Coordinates of a vector referred to a base. | Testo 2 |

7 | Projective coordinates. Lines in plane. parallelism and orthogonality. Pencils of lines. Lines and plenes in space. Distance point-plane. | Testo 2 |

8 | Conics classification. Tangent lines to a conic. Pencils of conics. | Testo 2 |

9 | Quadrics classification. Pencils of quadrics. Double points of a quadric. | Testo 2 |

## Learning Assessment

### Examples of frequently asked questions and / or exercises

Linear Algebra

- Definition of vector space. Examples of vector space. Linearly independent vectors. Calculation of the components of a vector given a basis.
- Calculation of the basis of a subspace given the Cartesian equations. Calculation of the Cartesian equations of a subspace given a basis.
- Basis of a vector space. Dimension of a vector space.
- Calculation of the rank of a matrix, also dependent on parameters. Basic change matrix.
- Applications of the Grassmann Formula. Verification of the linearity of the applications. Calculation of kernel and image of a linear application. Matrix associated with a linear application.
- Applications of rank-nullity theorem.
- Systems of linear equations, determining solutions with row reduction and/or Carmer's theorem and Rouché's theorem.
- Calculation of eigenvalues and eigenvectors of an endomorphism. Similarity transformations and diagonalizable matrices.
- Scalar products, calculation of an orthogonal basis with the Gram-Schmidt method.

Geometry

- Equations of a line in the plane and in space. Interpretation of the direct numbers of a line.
- Exercises about planes in euclidean space.
- Exercises on spheres.
- Affine classification of conics. Lines tangent to a conic.
- Affine classification of quadrics in space. Planes tangent to a quadric.
- Nature of the points of an irreducible quadric. Double points of a quadric.