# LINEAR ALGEBRA AND GEOMETRY P - Z

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

**MONICA LA BARBIERA**

## Expected Learning Outcomes

**1. Knowledge and understanding: **definitions and theorems about vector spaces, linear applications and endomorphisms, fundamental constructions and theorems about lines and planes in the 3-dimensional space and conics in the plane, definitions and theorems about quadrics.

**2. Applying knowledge and understanding:** being able to compute the rank of a matrix, to study a vector space, to study a linear application, to determine eigenvalues and eigenvectors of endomorphisms, to diagonalize a matrix, being able to solve linear geometry problems about points, lines and planes in the 3-dimensional space, to classify conics and quadrics and to study pencils of conics in the plane.

**3. Making judgements:**the student will be able to autonomally deepen his/her knowledge and do exercises about the topics studied and at the end of the course he /she will be able to autonomally determine solutions about the main topics, choosing the best strategy. Moreover, the interaction between students and the teacher will be encouraged in such a way that the student might be able to critically monitor his/her understanding process.

**4. Communication skills:** the frequence of the lessons and the suggested books will help the student to familiarize with the rigour of mathematical language and to learn the specific language of linear algebra and geometry. Through the continuous interaction with the teacher, the student will learn to communicate with rigour and clarity his/her acquired knowledge, both in an oral and in a written way. At the end of the course, the student will have learnt that the mathematical language is useful in order to be able to communicate in a clear way in a scientific setting.

**5. Learning skills**

**:**the aim of the course is to provide the student a way of studying, the "forma mentis" and the logical rigour that he/she will need in hs/her studies. In particular, through guided exercises he/she will be able to face in autonomy new topics, understanding the requirements necessary for their comprehension.

## Course Structure

During the lessons topics and concepts will be proposed in a formal way, together with meaningful examples, applications and exercises. A tutor will carry classroom exercises. The student will be sollicited to carry out exercises autonomously, even during the lessons.

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:

- Generalities on set theory and operations. Maps between sets, image and inverse image, injective and surjective maps, bijective maps. Sets with operation, gropus, rings, fields.
- Vectors in the ordinary space. Sum of vectors, product of a number and a vector. Scalar product, vector product. Components of vectors and operations with components.
- Complex numbers, operations and properties. Algebraic and trigonometric form of complex numbers. De Moivre formula. nth root of complex numbers.
- Vector spaces and properties. Examples. Subspaces. Intersection, union and sum of subspaces. Linear independence. Generators. Base of a vector space, completion of a base. Steinitz Lemma*, dimension of a vector space. Grassmann formula*. Direct sum.
- Generalities on matrices. Rank. Reduced matrix and reduction of a matrix. Elementary matrices. Product of matrices. Linear systems. Rouchè-Capelli theorem. Solutions of linear systems. Homogeneous systems and space of solutions.
- Determinants and properties. Laplace theorems*. Inverse of a square matrix. Binet theorem*. Cramer thoerem. Kronecker theorem*.
- Linear maps and properties. Kernel and image. Injective and surjcetive maps. Isomorphisms. L(V,W) and isomomorphism with k^{m,n}. Study of a linear map. Base change.
- Eigenvalues, eigenvectors and eigenspaces of an endomorphism. Characteristic polynomial. Dimension of eigenspaces. Independence of eigenvectors. Simple endomorphisms and diagonalization of matrices.

Geometry

- Linear geometry on the plane. Cartesian coordinates and homogeneous coordinates. Lines and their equations. Intersection of lines. Angular coefficient. Distances. Pencils of lines.
- Linear geometry in the space. Cartesian coordinates and homogeneous coordinates. Planes and their equation. Lines and their representation. Ideal elements. Angular properties of lines and planes. Distances. Pencils of planes.
- Change of coordinates in the plane, rotations and translations. Conics and associated matrices, orthogonal invariants. Reduced equations, reduction of a conic in canonic form. Classification of irreducible conics. Study of equations in canonic form. Circle. Tangent lines. Pencils of conics.
- Quadrics in the space and associated matrices. Irreducible quadrics. Vertices and dengerate quadrics. Cones and cylinders. Reduced equations, reduction in canonic form. Classification of non degenerate quadrics. Sections of quadrics with lines and planes. Lines and tangent planes.

The proofs of the theorem signed with * can be ometted.

## Textbook Information

- P. Bonacini, M. G. Cinquegrani, L. Marino. Algebra lineare: esercizi svolti. Cavallotto Edizioni, Catania, 2012.
- P. Bonacini, M. G. Cinquegrani, L. Marino. Geometria analitica: esercizi svolti. Cavallotto Edizioni, Catania, 2012.
- S. Giuffrida, A. Ragusa: Corso di Algebra Lineare. Il Cigno Galileo Galilei, Roma, 1998.
- Lezioni di Geometria. Spazio Libri, Catania, 2000.

More didactic material is available at https://studium.unict.it/ and https://algebralineare-geometria.webnode.it/

## Course Planning

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

1 | Introduction to set theory. | Testo 3: cap 1,3 - Testo 1: cap. 1 |

2 | Introduction to fields and vector spaces. | Testo 3: cap 1,3 - Testo 1: cap. 1 |

3 | Determinant of a matrix. Rank and reduction of a matrix. Resolution of a linear system. | Testo 3: cap 1,3 - Testo 1: cap. 1 |

4 | Vector spaces. Generators and linear independence. Subspaces. Base and components with respect to a base. Dimension of a vector space. | Testo 3: cap. 2 - Testo 1: cap. 2 |

5 | Sum and intersection of vector spaces. Extracting a base from a set of generators and expanding a linearly independent set to a base. | Testo 3: cap. 2 - Testo 1: cap. 2 |

6 | Linear applications and their assignment. Studying a linear application. Computation of images and inverse images. | Testo 3: cap. 4 - Testo 1: cap. 3,4,5 |

7 | Base change matrices and similar matrices. Operations with linear applications. | Testo 3: cap. 4 - Testo 1: cap. 3,4,5 |

8 | Eigenvalues, eigenvectors and eigenspaces. Characteristic polynomial. Algebraic and geometric multiplicity of an eigenvalue. Endomorphisms and diagonalization. | Testo 3: cap. 5 - Testo 1: cap. 6,7,8 |

9 | Applications under conditions. Restrictions and extensions of linear applications. | Testo 3: cap. 5 - Testo 1: cap. 6,7,8 |

10 | Generalities on vector calculus. Cartesian coordinates and homogeneous coordinates. Assignment of lines and planes and their equations. Points at infinity. Intersections. Parallelism and orthogonality. Pencils of lines and planes. Distances. | Testo 4: cap. 1,2,3 - Testo 2: cap. 1 |

11 | Conics and associated matrices. Change of coordinates in the plane, orthogonal invariants and reduced equations of a conic. Classification of conics. Circles. Tangent lines. Pencils of conics. | Testo 4: cap. 2 - Testo 2: cap. 2 |

12 | Complete study of conics. Conics under conditions. | Testo 4: cap. 2 - Testo 2: cap. 2 |

13 | Quadrics and associated matrices. Irreducible quadrics. Vertices of a quadrics and degenerate quadrics. Conic at infinity. Cones and cylinders. Reduced equations of a quadric. Classification of non degenerate quadrics. | Testo 4: cap. 5 - Testo 2: cap. 3 |

14 | Tangency. Conic sections of a quadric. Spheres. | Testo 4: cap. 5 - Testo 2: cap. 3 |

## Learning Assessment

### Learning Assessment Procedures

The examination is written and oral. The written examination, which usually lasts 3 hours, is compulsory to take the oral examination.

Exminations may also be carried out on line, should the conditions require it. In such a case, the length of the written examination might change.

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

Linear Algebra exercises:

1. study of a linear application with parameter, determining kernel and image.

2. study of the diagonalization of an endomorphism with parameters, determining, if possible, a bases of eigenvectors.

3. inverse image of a vector, resolution of a linear system with parameter, inverse image of a vector space, image of a vector space.

4. exercises on vector spaces and their dimension, direct sum, operations on linear applications, induced linear applications, restrictions and extensions.

Geometry exercise:

1. linear geometry exercises in the 3-dimensional space: parallelism and orthogonality, distances, orthogonal projections, angles.

2. pencil of conics, complete study of a conic, conics under conditions.

3. classification of quadrics with parameter, quadrics under conditions, intersection of quadrics with planes.

**VERSIONE IN ITALIANO**