Universidad de Jaén

Unit 1. The ring of polynomials.

The ring of polynomials. Divisibility. Division algorithm. Factorization of polynomials. Greatest common divisor and least common multiple.

Unit 2. Symmetric group.

Group and subgroups. Permutations, cycles and transpositions. Cycle decomposition for permutations. Signature of a transposition. The alternating subgroup.

Unit 3. Systems of linear equations. Matrices and determinants.

Systems of linear equations. Gauss-Jordan elimination method. Matrices and systems of linear equations. Hermite normal form. Rank of a matrix. Rouché-Fröbenius theorem. Elementary matrices. Inverse matrices. Determinants and some applications.

Unit 4. Vector spaces and inner product space.

Vector spaces. Linearly independent and dependent sets. Bases and dimension. Vector subspaces and operations with subspaces. Inner product and length. Orthogonal vectors. 

Unit 5. Linear maps. Diagonalization.

Linear maps. Null space and range. Matrix expression of a homomorphism. Classification of a linear map. Diagonalization of an endomorphism. Applications.

Unit 6. Introduction to the graph theory.

Basics of graphs. Subgraphs. Isomorphic graphs. Connected graphs. Geodesic. Euler and Hamilton graphs. Trees. Bipartite graphs. Planar graphs. Coloring of planar graphs. Directed graphs.

PRACTICAL CONTENTS

Practice 1. The polynomial ring.

Practice 2. Groups and subgroups.

Practice 3. The symmetric group.

Practice 4. Graphs I. Definition, implementation and representation.

Practice 5. Graphs II. Regular graphs, complete graphs, subgraphs and bipartite graphs.

Practice 6. Graph III. Paths and cycles.

Practice 7. Graphs IV. Graph coloring, planar graphs and trees.

Practice 8. Elementary matrices. Hermite normal form.

Practice 9. Systems of linear equations. Matrices and determinants.

Practice 10. Vector spaces.

Practice 11. Inner product space.

Practice 12. Linear maps.

Practice 13. Diagonalization.

Lecture (Theory): there will be two one-hour-long sessions each week during all the semester. These sessions will be devoted to regular lectures and problem solving, using a video projector and computer-generated slides. The lecturer makes their notes available online after a lecture through his institutional web which can help to supplement the notes you take during the class. You have to attend lectures in order to make better use of the course. These lectures will be taught in Spanish.

Lecture (Practice): They will be two hours long, and will be held weekly during the semester. In these classes, students will solve, with the help of a computer, problems related to content of the subject. In this classes it also will be presented in a more practical way, those theoretical contents of the subject which will not be exposed in the theory lectures. Finally, teacher will solve on the blackboard, and if it is possible, he will use the computer, exercises of the subjects that has previously been proposed to the students for the work at home. You may be asked to work individuals. In most cases, you will be asked to `write up` your work and this will be assessed with the mark contributing to your overall exam result. You have to attend your practical classes in order to make better use of the course.  These lectures will be taught in Spanish.

Students with special educational needs should contact the Student Attention Service (Servicio de Atención y Ayudas al Estudiante) in order to receive the appropriate academic support

• To pass the course is necessary to obtain a score of 5 out of 10 points between the weighted average of theory and practical part of the course; It will also be essential to have obtained a minimum of 4 out of 10 points in each block (theoretical and practical part). When someone will not reach the minimum in a block, the rating which it will reflect in the minutes will be less than 4 out of 10.
• The block "Theoretical concepts of the matter" will be assessed through a final exam in each call, the weight of the block of 70%. However, students who would attend class actively will have the opportunity to take part in a system of continuous assessment, which will be held in period class and will be complemented by the final exam, in which will be essential to obtain a minimum of 4 out of 10 on average between questions that will be answered in the examination.
• The block "Computer practices" will be assessed through a final exam for each call. However, students who actively attend all practical classes of the course, optionally and voluntarily, could assess this block through a system of continuous assessment, which will take place in the class period. The total weight of this block will be 20%. 
  The students must be submitted to each practical exam with all proposed activities during the practical classes well resolved and printed in paper.
• In the exams no electronic devices, notes, books or any other medium that allows the storage or transmission of data will be allowed. In case of default we will act in accordance with regulations.
• The scores for the practice or the theory exam, exceeding 5 out of 10 points, if the subject has not been approved, will remain in each of the official announcements of the academic year.
• If the percentage allocated to the final exam, depending on attendance, continuous assessment and earlier work by each student, is equal to or greater than 70%. The grades obtained by students who exceed 5 out of 10, in the process of continuous assessment in paragraphs "Theoretical concepts of matter" and "Computer practices" will be kept in each of the official announcements of the academic year. However, since the maximum score that students can get in the sum of these items does not exceed 30% of the total grade, students who do not present the final exam of theory and also the practices appear as absent in corresponding to this call record. If the percentage allocated to the final exam, depending on attendance, continuous assessment and earlier work by each student, is less than 70%. The grades obtained by students who exceed 5 out of 10, in the process of continuous assessment in paragraphs "Theoretical concepts of matter" and "Computer practices" will be kept in each of the official announcements of the academic year. But according to Article 18 of the Rules of Academic System and Student Assessment at the University of Jaen, is considered out a call, this call is understood to be the ordinary call of course, considering calls for the same criteria as other in the previous section, that is, the final grade for the course will be "not attend" for all students who do not make the final exam of theory and also the practices in that call, although it has done some previous work, continuous assessment or attended some lectures or practices.