### Sciences

## Subject: GEOMETRY (A.A. 2020/2021)

### degree course in PHYSICS

Course year | 1 |
---|---|

CFU | 6 |

Teaching units |
Unit Geometria
A12 (lesson)
- TAF: Supplementary compulsory subjects SSD: MAT/03 CFU: 6
Fulvia SPAGGIARI, Lucia MAZZALI |

Exam type | oral |

Evaluation | final vote |

Teaching language | Italiano |

### Teachers

*Fulvia SPAGGIARI*

*Lucia MAZZALI*

### Overview

To introduce the concepts and the basic structures of linear algebra and of Euclidean geometry of dimension two and three, in relation with their use in other courses.

### Admission requirements

Is required to pass a test on basic skills in mathematics

### Course contents

Fundamental algebraic structures: groups and fields.

Vector spaces and subspaces. Fundamental models. Intersection of subspaces. Systems of generators. Linear dependence and independence of vectors. Bases and dimension of a vectorial space.

Matrices. Matrix operations. Special matrices. Determinant of a matrix. Properties of the determinant and methods for computing the determinant. The inverse of a matrix.

Linear transformations and their properties. The matrix of a linear transformation. Rank of a matrix. Algorithms for computing the rank of a matrix.

Systems of linear equations. Cramer linear systems. Algorithms for the resolution of linear systems. Similar matrices. Diagonalizable matrices. Eigenvectors, eigenvalues and eigenspaces.

Inner product. The length of a vector. Angle between two vectors. Orthogonal vectors. Orthonormal bases. Gram-Schmidt method. Orthogonal complement of a vectorial Euclidean subspace. Outer product.

The real Euclidean plane. Parametric and cartesian representations of a line of the Euclidean plane. Reciprocal position of lines. Parallelism and orthogonality of lines. Euclidean distance. Angles between lines. Areas. Plane transformations. Polar coordinates.

The real three-dimensional Euclidean space. Parametric and cartesian representations of Lines and planes. Reciprocal positions of lines, planes, line and plane. Parallelism and orthogonality of lines, planes, between line and plane. Distances. Angles. Volumes. Space transformations.

Conic theory. Projective extension of a Euclidean space; points at infinity. Homogeneous and non-homogeneous equations of a conic. Tangent and polar of a conic. Classification of conics of the Euclidean plane. Centre and axes. Canonical equations. Foci and directrices of non-degenerate conic.

### Teaching methods

The teaching is based on lectures supplemented by exercise activities in the classroom. A teaching support after hours of lessons, based on exercises and mainly aimed at preparing for the written test, is proposed. All above will be given on line with asincrone registration according to the evolution of the COVID19 pandemic situation.

### Assessment methods

The exam consists of a written test and an oral test. The written exam is a multiple choice test of numerical exercises and theory questions. There are two written tests during the course, the passing of which allows students to have direct access to the oral test. The interview focuses on theoretical arguments of the program and is designed to test the level of knowledge and understanding of the topics of the program, the ability to use mathematical language and to apply correctly the hypothetical deductive method. Such examinations will be in presence or will be on line according to the evolution of COVID19 pandemic problem.

### Learning outcomes

Through lectures and individual study, knowledge and understanding of: vector spaces, matrices and their properties, linear applications and their properties, algorithms for solving linear systems, conditions of diagonalizability of a matrix, scalar and vector product, Euclidean spaces and the notions of parallelism, orthogonality and Euclidean distance between subspaces (with particular reference to the Euclidean plane and the Euclidean space of dimension three), the conics of the Euclidean.

Through classroom exercises, the support activities and individual work, ability to:

determine basis and dimension of a vector space

determine kernel and image of a linear transformation,

calculate the determinant and the rank of a matrix

discuss and solve linear systems

calculate the eigenvalues and discuss diagonalizability of a matrix

construct orthonormal bases in a Euclidean vector space

determine the orthogonal complement of a Euclidean subspace

represent by cartesian and parametric equations Euclidean subspaces

determine the relative position between Euclidean subspaces and calculate the distance

calculate areas and volumes in the Euclidean space

solve simple problems of Euclidean geometry

classify conics in the Euclidean plane and determine the basic elements.

Attitude towards a methodological approach that leads to verify the claims through rigorous arguments and methods presented.

Ability to self-assessment of their skills and abilities.

Ability to deal with accurately and coherent a dialectic, arguing with precision

Acquisition of mathematical knowledge as its assets, that can use at any time of the cultural path

Attitude towards a methodological approach that leads to an improvement in the method of study and in the learning ability

### Readings

Testo di riferimento:

A. Cavicchioli - F. Spaggiari, PRIMO MODULO DI GEOMETRIA, Pitagora ed., Bologna, 2002.

A. Cavicchioli - F. Spaggiari, SECONDO MODULO DI GEOMETRIA, Pitagora ed., Bologna, 2004.

Altri testi:

R.Betti, Lezioni di Geometria (Volumi I-II), Masson ed., 1995;

M.Rosati, Lezioni di Geometria - nuova edizione, ed. Libreria Cortina, 1997;

C.Bignardi - B.Ruini - F.Spaggiari, Esercizi di algebra lineare, Pitagora ed., 1996;

B. Ruini - F. Spaggiari, Esercizi di geometria, Pitagora ed., 2002;

L. Gualandri, Esercizi di algebra lineare e geometria, Esculapio ed., 1995.

D. C. Lay, Linear Algebra and Its Applications, Addison-Wesley Publishing Company, 1994