# Geometry I

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

**ELENA MARIA GUARDO**

## Expected Learning Outcomes

The aim of the programme is to give students 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.

** **At the conclusion of the course, the students shuold be able to understand the basic notions, to apply their knowledge and understanding. They also should be able to give oral and written presentation of the most important theorems of the contents of the course as well as to work both in collaboration with other people and by themselves, making judgements.

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

## Course Structure

### Teaching Organization

total study 300 hours

206 hours of individual study

70 hours of frontal lecture

24 hours of exercises

Frontal lectures and classroom exercise. There is no standard way of lecturing: some lectures will be written exclusively on blackboards or sometimes the student receive printed notes. The method used depends also on the sort of material that they are covering.

Together with the professor of Algebra and Analysis, periodical meetings will be organized to give students useful suggestions on how to use and apply the acquired knowledge to the other disciplines.

Part of the programme (max 3CFU) could be done by a visiting professor (italian or not).

Information for students with disabilities and / or SLD

To guarantee equal opportunities and in compliance with the laws in force, interested students can ask for a personal interview in order to plan any compensatory and / or dispensatory measures, based on the didactic objectives and specific needs. It is also possible to contact the referent teacher CInAP (Center for Active and Participated Integration - Services for Disabilities and / or SLD) of our Department, prof.ssa Daniele.

## Required Prerequisites

## Attendance of Lessons

## Detailed Course Content

**Linear Algebra**

I) Groups, rings, fields. Z, K[x], C.

II) Matrices over a field. Matrices addition, scalar multiplication, abelian group of matrices, matrix multiplication (or product). Properties. Ring of square matrices. Diagonal, triangular, scalar , symmetric, skew-simmetric matrices and transpose of matrix.

III) Vector spaces and their properties over a filed K. Examples: K[x], K^{n}, K^{m,n.}. Subspaces. Intersection and sum of vector spaces. Direct sum. Linear combinations. Span, Linear Independence and dependence,Finitely generated vector spaces, Base, Dimension. Steinitz’s Lemma *, Grassmann’s formulas*.

IV) Determinants and their properties. Theorems of Binet*,Laplace I*, Laplace II*, Adjunct matrix, Inverse, Rank and Reduction of a matrix. Theorem of Kronecker*. Systems of linear equations. Rouchè-Capelli‘s rule, Cramer’s rule. Solving systems of linear equations.

V) 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. Change of base matrix. Similar matrices.

VI) Eigenvalues, Eigenvectors and Eigenspaces of a matrix. Characteristic polynomial. Dimension of an eigenspace. Relation between Algebraic multiplicity and geometric multiplicity. Linear Independence of the eigenvectors. Diagonalizable linear maps and diagonalization of a matrix.

VII) Real scalar product, hermitian scalar product, Cauchy-Schwarz inequality, Euclidian subspaces and their orthogonal complement. Orthogonal matrix.

VIII) Affine Spaces

**Geometry**

I)Euclidean (geometric) vectors and their properties. Scalar multiplication, dot (or scalar) product, wedge (or cross) product.

II)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 space. 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.

III) 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. Circumferences, Tangents, and pencils of conics.

IV) Quadrics and its associated matrix. Nondegenerate, degenerate and singular quadric surfaces. Cones and cylinders. Classification. Rulings on a quadric, plane sections of a quadric.

## Textbook Information

- Notes on Linear Algebra
- Notes on Geometry
- E. Sernesi, Geometria 1,Bollati- Boringhieri 1989 (Geometry)

## Course Planning

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

1 | Matrices over a field. Matrices addition, scalar multiplication, abelian group of matrices, matrix multiplication (or product). Properties. Ring of square matrices. Diagonal, triangular, scalar , symmetric, skew-simmetric matrices and transpose of matrix. | Testo 1), 3) |

2 | Determinants and their properties. Theorems of Binet*,Laplace I*, Laplace II*, Adjunct matrix, Inverse, Rank and Reduction of a matrix. Theorem of Kronecker*. Systems of linear equations. Rouchè-Capelli‘s rule, Cramer’s rule. Solving systems of linear equations. | testo 1), 3) |

3 | Vector spaces and their properties over a filed K. Examples: K[x], Kn, Km,n.. Subspaces. Intersection and sum of vector spaces. Direct sum. Linear combinations. Span, Linear Independence and dependence,Finitely generated vector spaces, Base, Dimension. Steinitz’s Lemma *, Grassmann’s formulas*.. | testo 1), 3) |

4 | 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. Change of base matrix. Similar matrices. | Testo 1), 3) |

5 | Eigenvalues, Eigenvectors and Eigenspaces of a matrix. Characteristic polynomial. Dimension of an eigenspace. Relation between Algebraic multiplicity and geometric multiplicity. Linear Independence of the eigenvectors. Diagonalizable linear maps and diagonalization of a matrix. | Testo 1), 3) |

6 | Real scalar product, hermitian scalar product, Cauchy-Schwarz inequality, Euclidian subspaces and their orthogonal complement. Orthogonal matrix. Affine spaces | Testo 1), 3) |

7 | Affine spaces of dimension 2 and 3 | 2), 3) Chapter 1, section 7-10 |

8 | Euclidean (geometric) vectors and their properties. Scalar multiplication, dot (or scalar) product, wedge (or cross) product. | testo 2). 3) |

9 | 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 space. 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. | testo 2), 3) |

10 | 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. Circumferences, Tangents, and pencils of conics. | testo 2), 3) |

11 | Quadrics and its associated matrix. Nondegenerate, degenerate and singular quadric surfaces. Cones and cylinders. Classification. Rulings on a quadric, plane sections of a quadric. | testo 2), 3) |

## Learning Assessment

### Learning Assessment Procedures

a) The students are invited to answer some questions during the lectures as self- evaluation.

b) Two Middle-Classes Tests. They are intended passed with a mark greater or equal to 15/30.

The first test will be done during the first pause from lectures.

The second will be done at the end of the course.

The students that pass both tests can do the oral exam without making the general test at the end of the course.

The students that do not pass the one of the test will make the final test at the dn of the course.

To pass the test, the students should solve at least two exercises on Linear Algebra and one in Geometry

Final grades will be assigned taking into account the following criteria:

__Rejected__: Basic knowledges have not been acquired. The student is not able to solve simple exercises.

__18-23__: Basic knowledges have been acquired. The student solves simple exercises and has sufficient communications skills and making judgements.

__24-27__: All the knowledges have been acquired. The student solves all the proposed exercises making few errors and has good communications skills and making judgements.

__28-30 cum laude__: All the knowledges have been completely acquired. The student applies knowledge and has excellent communications skills, learning skills and making judgements.

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 online, should the conditions require it.

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

Lines and planes in the space. Conics: parabole, ellips and iperbole, circumference, quadrics, cones, cilinders, plane sections of quadrics.