Algebra II
Code: 100144 ECTS Credits: 6| Degree | Type | Year |
|---|---|---|
| 2500097 Physics | FB | 1 |
Contact
- Name:
- Eduardo Gallego Gómez
- Email:
- eduardo.gallego@uab.cat
Teachers
- Maria Rosa Camps Camprubi
- Enric Nart Viñals
Teaching groups languages
You can view this information at the end of this document.
Prerequisites
It is advisable to understand the main notions of the course Algebra I, specially those of Linear Algebra.
Objectives and Contextualisation
Competences
- Develop strategies for analysis, synthesis and communication that allow the concepts of physics to be transmitted in educational and dissemination-based contexts
- Use critical reasoning, show analytical skills, correctly use technical language and develop logical arguments
- Use mathematics to describe the physical world, selecting appropriate tools, building appropriate models, interpreting and comparing results critically with experimentation and observation
Learning Outcomes
- Argue with logical rigor.
- Describe and use tensioners and calculate the effect of coordinate changes.
- Diagonalise endomorphisms and bilinear forms.
- Express definitions and theorems rigorously.
- Transmit orally and in writing, in a clear manner, the logical-mathematical reasoning that leads to problem resolution.
- Use critical reasoning, show analytical skills, correctly use technical language and develop logical arguments
- Work with Euclidean and Hermitian metrics and their associated geometries.
Content
1. Diagonalization of matrices and endomorphisms
2. Bilinear forms
2.1 Symmetric bilinear forms over real vector spaces. Euclidian inner podruct.
2.2 Hermitian forms.
2.3 Minkowski product.
2.4 Orthogonal diagonalization of symmetric matrices: the Spectral Theorem. el Teorema espectral.
3. Lineal Geometry.
4. Multilinear Algebra
4.1 Dual space.
4.2 Tensors.
Activities and Methodology
| Title | Hours | ECTS | Learning Outcomes |
|---|---|---|---|
| Type: Directed | |||
| Lectures | 29 | 1.16 | 1, 2, 3, 4, 6, 7 |
| Problem sessions | 21 | 0.84 | 1, 2, 3, 4, 5, 6, 7 |
| Type: Autonomous | |||
| Solving problems | 45 | 1.8 | 1, 2, 3, 4, 5, 6, 7 |
| Studying theoretical concepts | 38 | 1.52 | 1, 2, 3, 4, 5, 6, 7 |
The objectives will be achieved indirectly in the following way:
1. Learning the techniques of diagonalization of matrices and endomorphisms.
2. Learning the algebraic foundations of Euclidean geometry and, more generally, the symmetrical bilinear forms on the real ones.
3. Learning the algebraic foundations of Minkowski's geometry
4. Learning the techniques of multilinear algebra and, in particular, working with tensors.
And all this accompanied by the development of logical reasoning, which is encouraged by teaching the demonstrations of many of the theorems of the course.
Annotation: Within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.
Assessment
Continous Assessment Activities
| Title | Weighting | Hours | ECTS | Learning Outcomes |
|---|---|---|---|---|
| A mid-term exam. | 40% | 2 | 0.08 | 1, 3, 4, 5, 6, 7 |
| Final exam | 45% | 2.5 | 0.1 | 1, 2, 3, 4, 5, 6, 7 |
| Resit exam | 85% | 2.5 | 0.1 | 1, 2, 3, 4, 5, 6, 7 |
| Submission of exercise sets | 15% | 10 | 0.4 | 1, 2, 3, 4, 5, 6, 7 |
40% of the final score will be obtained after the completion of a partial test. Passing this test does not eliminate matter from the final exam.
45% of the final score will be obtained from the final exam
The remaining 15% will be calculated based on the submission of exercice sets.
Students who do not pass the subject after the final exam may submit to a resit exam, which will be worth 85% of the grade. Grade from submissions of exercices has no resit.
Only those students that have been submitted to the partial and final exams can do the resit exam.
After the final exam Honors may be already awarded.
Unique assessment
Students who have accepted the single assessment modality will have to take a final test which will consist of a theory and problem test. These tests will take place on the same day, time and place as the tests of the second part of the continuous assessment modality. It will also be necessary to hand in the proposed problems to the students following the continuous assessment.
The student's grade will be the weighted average of the previous activities, where the exam will give 85% of the grade and the assignments 15%.
If the final grade does not reach 5, the student has another opportunity to pass the subject through the remedial exam that will be held on the date set by the degree coordinator. The part of the note corresponding to the deliveries is not recoverable.
Bibliography
R. Camps, E. Nart, G. Solanes, X. Xarles, Àlgebra lineal i multilineal.
Complementary bibliography
Lectures
1. F. Cedó i A. Reventós, Geometria plana i àlgebra lineal, Manuals de la UAB, 39, 2004
2. A. Kostrikin and Y. Manin, Linear Algebra and Geometry, Gordon and Breach Science Publishers, Amsterdam, 1989.
Problemes
1. F. Cedó i V. Gisin, Àlgebra Bàsica, Manuals de la UAB, 1997.
2. J. García Lapresta, M. Panero, J. Martínez, J. Rincón y C. Palmero, Tests de Álgebra lineal, Editorial AC, Madrid, 1992.
3. J. Rojo y I. Martín, Ejercicios y Problemas de Álgebra Lineal, Mc. Graw-Hill, Madrid 1994.
4. A. de la Villa, Problemas de Algebra, CLAGSA, Madrid, 1994
Software
You can use the software:
- Sagemath: https://www.sagemath.org
- Maxima: https://maxima.sourceforge.io
- WxMaxima: https://wxmaxima-developers.github.io/wxmaxima/index.html
but it is not part of the assessable course content.
Language list
| Name | Group | Language | Semester | Turn |
|---|---|---|---|---|
| (PAUL) Classroom practices | 1 | Catalan | second semester | morning-mixed |
| (PAUL) Classroom practices | 2 | Catalan | second semester | afternoon |
| (TE) Theory | 1 | Catalan | second semester | morning-mixed |
| (TE) Theory | 2 | Catalan | second semester | afternoon |