30544 - ALGEBRA AND GEOMETRY
Course taught in English
Go to class group/s: 27
Class-group lessons delivered on campus
This course covers linear algebra, emphasizing topics useful in other disciplines. Linear algebra is a branch of mathematics that studies linear spaces, linear maps, and matrices. The concepts of linear algebra are extremely useful in physics, economics and social sciences, natural sciences, and computational sciences.
Subspaces. Intersection, sum. Direct sum.
Span, linear independence, and bases.
Dimension of a finite-dimensional vector space.
Nullspace, range and rank of a linear map.
Matrix of a linear map.
Invertible linear maps.
Eigenvalues and eigenvectors.
Inner product spaces.
Orthonormal bases and the Gram-Schmidt procedure.
Orthogonal projections; applications.
Self-adjoint and normal operators; spectral theorem.
Operators on complex vector spaces.
Characteristic polynomials and minimal polynomial.
- Know the fundamental notions and results of linear algebra: vector spaces, linear maps, matrices, and their properties.
- Express these notions in a conceptually and formally correct way, using adequate definitions, theorems, and proofs.
- Understand the language and the formal aspects of mathematics as an axiomatic-deductive system.
- Apply the fundamental results of linear algebra to the modelling of systems and to solve computational problems.
- Actively search for deductive ideas that are fit to prove possible links between the properties of mathematical objects and to solve assigned problems.
- Formulate simple problems through rigorous mathematical models, which can be analyzed with the help of matrices and linear tools.
- Face-to-face lectures
- Online lectures
- Exercises (exercises, database, software etc.)
- Individual assignments
Online lectures have the same conceptual role as face-to-face lectures. The actual blend of face-to-face lectures and online lectures will mainly depend on external constraints.
Exercise sessions (again: both face-to face and online) are dedicated to the application of the main theoretical results obtained during lectures to problems and exercises of various nature.
Individual assignments have the precise aim of calling for an active involvement of students.
Students will presumably be evaluated on the basis of written and/or oral exams, also depending on external constraints. The exam will cover all content taught in the course both abstract/conceptual (statements and proofs of known results) and operational (exercises including the proof of simple new results).
Written exams consist of open and closed questions. The open questions aim to test the ability to apply the knowledge of the concepts and the methods concerning linear algebra to solve problems.
Closed answer questions will test the understanding and the knowledge of the main definitions and results of the basic geometry and algebra concepts.
The general written exam is worth 100% of the final grade (31/30).
In case of major external contingencies the written exams may be replaced by oral exams which will mimic as much as possible the structure of the written exams itself.
Sterling K. Berberian