LINEAR ALGEBRA AND GEOMETRY
No prerequisite is needed. It is however suggested to attend the pre-course in Mathematics.
The examination will take place on two levels:
1) A written examination, with a duration of 3 hours, in which the students are required to 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 who obtain a mark of at least 16/30 will also be provisionally admitted to the oral examination. To formalize the admission, these students will be required, during a preliminary oral discussion, to defend the written examination.
There will be a mid-term examination for students who attend the lecture. A positive mark in the mid-term will reflect in an exemption from the corresponding part of the final written examination. The admission mark to the oral examination will be the average of the two marks.
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 origin of linear algebra is the search for a formal context in which to develop the theory of linear systems of equations, whose solution has applications in a number of areas. From there, the notions of linear space and linear map stem in a natural way. Among the various applications of linear algebra there is the study of the geometry of lines, planes, and conics.
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 hopefully add clarity, allow to develop intuition, and familiarize students to the peculiar reasonings of this discipline. At the end of the course, the student should be able:
1) to remember and understand the main definitions of linear algebra and geometry;
2) to understand and analyze the basic results of this branch of knowledge;
2) to apply these results to concrete examples, also coming from the real world.
The lecture will cover the following topics:
- The Gauss method for the solution of linear systems of equations
- Vector spaces and subspaces
- Generating systems, linear independence, and bases
- Operations on subspaces and Grassmann's formula
- Affine spaces and references
- Linear maps, kernel, and range
- Linear maps and matrices: the representation theorem
- Base change in endomorphisms, and similar matrices
- Affine maps
- Determinants
- Eigenvalues, eigenvectors and diagonalizability
- Inner product spaces and isometries
- Self-adjoint and normal endomorphisms: the spectral theorem
- Quadratic forms and their classification
- Euclidean spaces and their isometries
- Conics in the Euclidean plane
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.
For office hours, please contact the instructor at his email address, giovanni.bazzoni@uninsubria.it
Professors
Borrowers
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Degree course in: Physics