LINEAR ALGEBRA AND GEOMETRY
 Overview
 Assessment methods
 Learning objectives
 Contents
 Bibliography
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No prerequisite is needed
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.
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.
B) Algebra of quadratic 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
 Selfadjoint 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. McGrawHill (1996)
2) MichÃ¨le 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 Covid19 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 elearning 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.
For office hours, please contact the instructor at his email address, giovanni.bazzoni@uninsubria.it
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Degree course in: Physics