# LINEAR ALGEBRA AND GEOMETRY

Degree course:
Corso di First cycle degree in MATHEMATICS
Academic year when starting the degree:
2021/2022
Year:
1
Academic year in which the course will be held:
2021/2022
Course type:
Basic compulsory subjects
Credits:
9
Period:
First Semester
Standard lectures hours:
80
Detail of lecture’s hours:
Lesson (56 hours), Exercise (24 hours)
Requirements:

No prerequisite is needed

Final Examination:
Orale

The examination will take place on two levels:

1) A written examination, with a duration of 3 hours, in which students solve some problems, whose difficulty is akin to that of the assigned exercises. The goal of the written examination is to verify that the students can apply the theoretical, abstract results to deduce properties of vector spaces, linear maps, and geometric entities in concrete situations. Passing the written examination requires a mark of 18/30 and is necessary to be admitted to the oral examination. The mark of the written examination remains valid for all the subsequent rounds, within the academic year. Students which obtain a mark of at least 16/30 will also be provisionally admitted to the oral examination. In order to formalize the admission, these students will be required, during a preliminary oral discussion, to defend the written examination.

2) A traditional oral examination, during which the performance of the written examination is discussed; moreover, students are required to explain basic notions, to illustrate the proofs of the main theorems, and to analyze concrete examples.

The final mark, to be expressed over 30 points, is determined by the outcome of the oral examination, which can confirm that of the written examination, or increment it, or cause the failure, in case the exhibited competences should not be considered as sufficient.

Assessment:
Voto Finale

The goal of this course is to introduce the main concepts and results on the algebra of linear spaces of finite dimension over the real and complex numbers, with applications to the study of the Euclidean geometry of linear entities and to the classification of quadratic entities. Because of its abstract nature, linear algebra is notoriously thought of as a tough branch of mathematics. The applications to analytic geometry, as well as the abundance of concrete examples, will lend clarity to it, allow a development of intuition and familiarize students to the peculiar reasonings of this discipline. At the end of the course, the student should be able:

1) to define the main entities and to present the basic results of this branch;

2) to analyze these results on concrete examples in order to deduce the main properties of linear spaces, of linear maps among these spaces, and of the linear and geometric entities;

3) to carry out, in an autonomous way, simple reasonings in order to deduce the validity of general properties of abstract vector spaces, and of linear and quadratic entities.

The course is essentially divides into three parts, which are mutually connected.

A) Algebra of linear entities.

C) Elements of Euclidean analytic geometry.

More in detail, the lecture will cover the following topics:

- Vector spaces and subspaces

- Generating systems, linear independence, and bases

- Operations on subspaces and Grassmann's formula

- Linear maps, kernel, and range

- Duality in finite dimensional vector spaces

- Linear maps and matrices: the representation theorem

- Base change in endomorphisms, and similar matrices

- Systems of linear equations

- Determinants

- Canonical form of endomorphisms

- Inner product spaces and isometries

- Self-adjoint and normal endomorphisms: the spectral theorem

- Quadratic forms and their classification

- Affine spaces, affine references and maps

- Euclidean affine spaces

- Conics in the Euclidean plane

The Linear Algebra and Geometry course is a very standard one. By following the lectures, students will be able to use any textbook.
Concretely, we will refer to the following textbooks:

1) Marco Abate, Geometria. McGraw-Hill (1996)

2) Michèle Audin, Geometry. Springer (2003)

3) Sheldon Axler, Linear Algebra Done Right. 3rd edition. Springer (2015)

Further material:

1) Slides of the lectures
2) Assignments

The teaching modalities will strongly depend on how the Covid-19 emergency will evolve. At the moment, the most likely scenario is that of hybrid teaching, with lectures in presence and online.

The teaching modality consists of:

1) theoretical lectures, where the instructor will provide the students with the key notions of the course.

2) Weekly exercise assignments, to apply the theoretical content of the lecture. The exercises can be worked out individually, or in small groups.

3) Problem sessions, during which the solutions of the assigned exercises will be discussed, ideally by the students, but it any case under the guidance of the instructor.

4) Multiple choice quizzes, on the e-learning page of the course; these are intended to be a way to quickly monitor the effective understanding of the notions acquired during the lectures.

5) The creation of a course glossary; it is a reasoned list, a sort of Wikipedia, of concepts and objects which are introduced during the lecture. It is drafted, on a voluntary base, from students of the lecture to their benefit and that of their colleagues.