Linear and Multilinear Algebra

- Deepen the understanding of the structure of linear applications.
- Learn the Jordan canonical form of an endomorphism: existence, computation, and utility.
- Study the structure of Euclidean and Hermitian vector spaces.
- Classify orthogonal, symplectic, and Hermitian geometries.
- Classify quadratic forms.
- Learn about tensors and their basic applications.

1. Polynomials: divisibility; fundamental theorem of algebra; irreducibility of polynomials. (2 lecture hours and 1 problem session)

2. Multilinear applications and determinants. (5 lecture hours and 2 problem sessions)

3. Structure of a linear application: eigenvalues and eigenvectors of a linear application; diagonalizable applications; minimal polynomial of an endomorphism; Cayley-Hamilton theorem; Jordan form; invariant subspaces. (10 lecture hours and 4 problem sessions)

4. Bilinear and quadratic forms: metric structures in vector spaces and Sylvester's theorem; orthogonality; real and complex spectral theorem; isometries; quadratic forms; symplectic forms. (16 lecture hours and 5 problem sessions)

5. Tensors and tensor algebra: covariant and contravariant tensors; exterior product; tensor product of vector spaces; tensor algebra and exterior algebra. (9 lecture hours and 2 problem sessions)

BASIC:

Axler, S., Linear algebra done right.
Springer, 1995.

Castellet, M.; Llerena, I., Álgebra lineal y geometría.
Ed. Reverté, Barcelona, 1991.

Hernandez, E., Álgebra y geometría.
Ed. Addison Wesley, Madrid, 1994.

COMPLEMENTARY:

Aroca Hernández Ros, J.M.; Fernández Bermejo, M.J. Algebra Lineal y Geometría. Secretariado de Publicaciones, Universidad de Valladolid. 1988.

Artin, E., Álgebra geométrica.
Ed. Limusa, México, 1992.

De Burgos, J., Álgebra lineal y geometría cartesiana.
Ed. MacGraw-Hill, Madrid, 1999.

Godement, R., Álgebra.
Ed. Tecnos, Madrid, 1967.

Gruenberg, K.W.; Weir, A.J., Linear Geometry.
Springer-Verlag, Berlin, 1977.

Hernandez, E., Álgebra y geometría.
Ed. Addison Wesley, Madrid, 1994.

Kostrikin, A. I.; Manin, Yu. I., Linear algebra and geometry.
Ed. Gordon and Breach, N. York, 1981.

Contribute to achieving the general, specific, and transversal competencies outlined in the Degree in Mathematics program at USC, particularly the following:
- Written and oral communication of knowledge, methods, and general ideas related to linear and multilinear algebra (CG4).
- Use tools for searching bibliographic resources on the course topics, including internet access. Manage these resources in different languages, especially in English (CT1, CT5).
- Use software programs to solve problems and implement algorithms (CE9).
- Be able to formulate hypotheses and draw conclusions using well-reasoned arguments, identifying logical errors and fallacies in reasoning (CG2, CE4).

Specific competencies of the course:
- Recognize if a matrix is diagonalizable or triangularizable. Be able to calculate the Jordan canonical form of an endomorphism and apply it to the classification of endomorphisms.
- Distinguish the different types of metric vector spaces. Be able to calculate orthogonal bases in a real orthogonal or complex Hermitian geometry.
- Work with tensors and apply their properties.

The course will be conducted over a semester, specifically 14 weeks, with a weekly schedule of three theoretical-practical hours and one seminar hour for each of the groups into which each course is divided.

Lecture sessions will involve the instructor explaining the main results, which can be found in the course notes and recommended bibliography. For each topic, an exercise bulletin will be proposed, which will be mainly worked on during the lab classes.

The lecture and interactive teaching will be conducted in person. Tutorials can be held either in person or virtually. Communication with students can also be done through virtual classroom forums and email, in addition to face-to-face interactions.

The evaluation system will be coordinated for the two groups of the course.

The evaluation criteria will include continuous assessment combined with a final exam. This exam will take place on the date set by the Faculty of Mathematics. The final exam will not be necessarily the same in both groups, but coordination is granted, both in the contents and in the exam.

The continuous assessment will be coordinated between the two groups, although the tests will be different. It will consist of two activities. The sum of the percentages is 110, aimed at encouraging student participation in continuous assessment and ensuring they remain engaged even if they perform poorly on one test.

- One or two midterm exams, which will represent 70% of the grade.
- Monitoring of learning throughout the course through group tasks to be completed with notes, either during (interactive) sessions or outside them. These tasks will represent 40% of the continuous assessment.

The final grade is calculated using the continuous assessment (CA) and the final written exam (FE). The final grade is obtained using the formula: MAX{35% CA + 65% FE, FE}

The grade obtained in the continuous assessment applies to both opportunities within the same academic year (first semester and July). If the student does not take the final exam in either opportunity, they will receive a grade of "Not Presented" even if they participated in the continuous assessment.

In the case of fraudulent activities during exercises or tests, the provisions of the Regulations for the Evaluation of Academic Performance of Students and Grade Review will apply: Article 16. Fraudulent activities during exercises or tests: Engaging in fraudulent activities during any exercise or test required for the evaluation of a subject will result in a failing grade for the corresponding exam session, regardless of any disciplinary process that may be pursued against the offending student. Fraudulent activities include, but are not limited to, submitting plagiarized work or work obtained from publicly accessible sources without reworking or reinterpretation and without citing the authors and sources.

Expositive Classes: 42 hours.
Laboratory Classes: 14 hours
Tutorials for very little groups : 2 hours.
Evaluation Activities: 4 hours
Time of Homework for the student: 88 hours
Total: 150 hours

Regular attendance to classes. Individual or collectively work each and every of the questions indicated at classes.
To take advantage of the tutorials, to ask questions about theory and exercises in the interactive classes.