Algebra and Geometry 390-FS1-1AZG
Profile of studies: general academic
Mode of studies: full-time
Type of course: compulsory
Field and discipline of science: Field of exact and natural sciences, discipline: physical sciences.
Level of education: first-cycle studies
Year of study/semester: Year 1 / Semester 2
ECTS credits: 5
Student workload balance:
- attendance in lectures (30 hours),
- attendance in classes (30 hours),
- attendance in laboratory classes (30 hours),
- attendance in consultations (15 hours),
- student’s independent work at home (20 hours),
Quantitative indicators:
- student workload related to classes requiring the direct participation of the teacher – 4.2 ECTS;
- student workload related to independent work – 0.8 ECTS.
Rules for the use of artificial intelligence (AI):
During classes, the use of AI systems is permitted in the following scope:
1. Machine translation of source texts from foreign languages.
2. Searching for and organizing scientific sources.
3. Creating simulations and modelling physical phenomena discussed during lectures.
The use of AI systems during the exam is not allowed.
In the event of a violation of the above rules, the student may be held accountable under separate disciplinary regulations.
Topics covered in the lectures:
1) The field of complex numbers
2) Matrices, operations on matrices, types of matrices, and systems of linear equations
3) The determinant as a criterion of invertibility and Cramer’s systems
4) Vector spaces, theory and basic examples
5) Basis and dimension of a space
6) Linear mappings and matrices of mappings, transition transformations
7) Euclidean spaces and their properties, Gram–Schmidt orthogonalization
8) Unitary spaces, theorems on the scalar product and norm, the Schwarz inequality, orthogonalization, and the theorem on orthogonal decomposition<
9) Gram–Schmidt orthogonalization applied to polynomials
10) Self-adjoint mappings, eigenvalues, eigenspaces
11) Spectral decomposition of self-adjoint and normal mappings
12) Pseudo-orthogonal spaces
13) Multilinear mappings, foundations of tensor calculus
14) Homomorphisms of SU(2), SO(3)
Type of course
Mode
Course coordinators
Term 2024: | Term 2025: |
Learning outcomes
Knowledge: The graduate knows and understands:
KP6_WG2 - advanced knowledge of the elements of higher mathematics and mathematical methods used in physics;
Skills: The graduate is able to:
KP6_UW6 - learn independently, finding necessary information in professional literature, databases, and other sources, and critically evaluate information from unverified sources;
KP6_UO1 - organize their own work and that of their team;
KP6_UU1 - pursue lifelong learning and inspire and organize the learning process of others.
Social competencies: The graduate is ready to:
KP6_KK1 - critically evaluate their knowledge and the content they receive;
KP6_KK2 - recognize the importance of knowledge in solving cognitive and practical problems;
KP6_KK3 - collaborate with experts when they encounter difficulties in solving problems independently.
KP6_KO1 - fulfilling social obligations and negating disinformation within the scope of acquired knowledge.
Assessment criteria
Assessment methods: written tests, ongoing assessment of class preparation, evaluation of activity in class, and assessment of the correctness of problem solutions.
The grade depends on the degree to which the learning outcomes have been achieved, in particular:
- computational and formal correctness of solutions,
- ability to apply the concepts and theorems of linear algebra and analytic geometry,
- ability to solve standard computational and theoretical problems,
- ability to justify successive steps of a solution,
- active participation in classes.
The grade is determined according to the following scale:
0%–50% – fail
51%–60% – satisfactory
61%–70% – satisfactory plus
71%–80% – good
81%–90% – good plus
91%–100% – very good
Bibliography
1) Paweł Urbański, ALGEBRA dla studentów fizyki, skrypt Katedra MMF, Uniwersytet Warszawski, Warszawa 1997
2) Bolesław Gleichgewicht, Algebra, PWN 1975,
3) Maria Moszyńska, Joanna Święcicka, Geometria z algebrą liniową, PWN 1987
4) A. Białynicki - Birula, Algebra liniowa z geometrią, PWN 1988
5) A.Mostowski, M.Stark, Algebra liniowa, PWN 1988
Additional information
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: