Algebra

A general aim, shared with other subjects of mathematics, is to became familiar with the language and the mathematical methods, improving the capacity of reasoning, analysis, synthesis and formulation of arguments.

Other specific aims of the subject are:

- To know and to handle the concepts and the techniques of the Linear Algebra and the Euclidean Geometry that are detailed in the program.

- To apply techniques of matrix algebra.

- To solve systems of linear equations.

- Geometric interpretation of some results.

1.- (Lectures: 6 hours, seminar: 2 hours and laboratory: 4 hours).
- Matrix algebra:
Matrix. Operations with matrices. Elementary matrix. Echelon form. Rank of a matrix. Determinant of a square matrix. Properties and calculation of determinants. Inverse of a matrix.

2.-(Lectures: 3 hours, seminar: 1 hour and laboratory: 2 hours).
- Systems of linear equations:
Matrix form of a system of linear equations. Equivalent systems. Solving systems of linear equations: Gauss method and Cramer's rule.

3.-(Lectures: 6 hours, seminar: 2 hours and laboratory: 3 hours)..
- Vector spaces:
Vectorial spaces and subspaces. Linear independence. Basis and dimension.

4.-(Lectures: 5 hours, seminar: 2 hours and laboratory: 3 hours).
- Linear maps:
Linear transformations. Kernel and image. Matrix of a linear transformation. Matrix of change of basis. Rank of a linear transformation. Relation to the solution of a system of linear equations.

5.-(Lectures: 3 hours, seminar: 2 hours and laboratory: 2 hours).
- Diagonalization:
Eigenvalues and eigenvectors. Polynomial rings K [x]. Characteristic polynomial. Matrix diagonalization and similarity.

6.-(Lectures: 2 hours, seminar: 1 hour and laboratory: 1 hour).
- Scalar product and orthogonality:
Scalar product. Distances. Orthogonality. Orthogonal projection.

Basic:

-LARSON, R.; EDWARDS, B.; FOLVO, D.C., Álgebra Lineal, Pirámide, 2004.

-MERINO, L.; SANTOS, E., Álgebra Lineal con métodos elementales, Thomson, 2006.


Complementary:

-ARVESÚ, J.; MARCELLÁN, F.; SÁNCHEZ, J., Problemas resueltos de Álgebra Lineal, Thomson, 2005.

-BURGOS, J., Álgebra finita y lineal. García-Maroto Editores, 2010.

-HERNÁNDEZ, E. "Álgebra Lineal y Geometría". Addison-Wesley/Universidad Autónoma de Madrid, 1994.

To help to reach the general competences and transversals as stated in the Memory of the Title of Degree in Computer Engineering of the USC (CG8, CG9, CG10, TR1, TR2, TR3 and FB1).

The module/thematic group competences that are worked in this course are:

- To explain and argue in a clear way the hypotheses and developments used in problem solving, using appropriate terminology.
- To develop the ability to analyze and solve problems.
- Organizing and planning ability.
- To master mathematical notation, method and terminology for modeling and case study.
- To improve the communicating ability, both spoken and written.


In addition this subject will allow to reach the following specific competences:

- To know the basic concepts of Linear Algebra: linear dependence and independence, basis, changes of basis, operations and equations of subspaces, linear transformations, etc.
- To know the algorithms to reduce matrix to echelon forms and to be able to apply them to the calculation of the rank, calculation of bases, system resolution, etc.
- To understand the intimate relation among matrices, linear transformations and systems of linear equations.
- To be able to analyze if a matrix is diagonalizable and, in its case, to diagonalize it.
- To know some examples of Euclidean spaces and to handle in the real n-dimensional space the scalar product , Gram-Schmidt's method and the orthogonal projection to solve some geometric problems.

The general methodological indications established in the Memory of Title of Degree in Computer Engineering of the USC will be followed.

The lectures, in big groups, will consist basically of the presentation by the teacher of the theoretical concepts, some examples and the showing of results that are more useful to the comprehension of the subject (working on competences CG8, CG10 and TR3).

The seminar interactive classes that will serve to illustrate the theoretical contents, will consist in the resolution of questions and exercises by the teacher with the participation of students (working on competences CG9 and TR3).

In the laboratory interactive classes, there will be a major implication of the student, giving priority to a more active and personalized pedagogy, and they will be devoted to solving problems, under the supervision of the teacher, which will also contribute to the acquisition of practical skills and to the illustration of the theoretical contents (working on competences CG9, CG10, TR1,TR2 and FB1).

Assigment proposals will be done (either individual and / or in group) throughout the four-month period (working on competences CG9, CG10, TR1 and TR2).

In the tutorials in very limited groups, a personalized follow-up of student learning and out-of-class work will be carried out (working on competences CG8, TR1 and TR2).

We will open a course in the Virtual Campus in which, in addition to having various support materials, we will keep a daily record of what is treated in each class meeting, as well as exercices for work in laboratory interactive classes.

Teaching will be face-to-face. Tutorials and communication with students can be face-to-face or virtual. In the virtual case they can be asynchronous, through the virtual course forums or e-mail, or synchronous, through the MS Teams platform.

Student assessment (also in the case of repeat students) will be based on a final theoretical - practical test (F) and a continuous assessment of the work done throughout the four-month period (C).

In the final test, the competences CG8, CG9, TR1, TR3 and FB1 will be evaluated.

For the latter assessment it will be taken into account both the controls (3) that should be done in class and the assignments requested by the teacher as well as the participation in classes and tutorials corresponding to courrent academic year. The attained grade (C) will be mantained during the academic year.

Therefore, the competences CG8, CG9, CG10, TR1, TR2, TR3 and FB1 will be evaluated.

The student will have the right to realize, in the days specified in the School exam calendar, the theoretical-practical examination held in January and, in case of not passing the course, the examination to be held in June.

The final mark will be calculated by the formula:

Final mark=70%F+30%C

It will be considered to be "no shown (No presentado)" the student who does not attend neither one of the two final examinations.

For cases of fraudulent action in exercises or tests, the provisions of the Regulations for assessing students' academic performance and reviewing grades will apply.

• The final theoretical-practical exam will be written and face-to-face.

• In the case of impossibity to be face-to-face, authorship checks will be virtual through the MS Teams platform.

ON-SITE WORK AT CLASSROOM:

- Lectures: 25 hours
- Learning based on problems in limited groups: 10 hours
- Practical meetings in limited groups: 15 hours
- Tutorials in very small groups: 3 hours
- Activities of assessment: 5 hours.

- Total on-site work: 58 hours

STUDENT PERSONAL WORK (NON ON-SITE): 92 hours

TOTAL: 150 hours (6 credits ECTS)

Continuous attendance to classes.

So as the classes are useful it is necessary to study the subject explained day by day.

It is essential that students come to classes in limited groups ,previously working the exercises proposed for every session. For this it is necessary to have enough knowledge on the theory that allows to approach the above mentioned problems.


The bibliography books are to complement the classes, in them it could be found both the results explained at class and an important source of examples and exercises.

The course will be taught primarily in Spanish.