Avaliación do alumnado matriculado a tempo parcial: Dependendo das particularidades de cada caso concreto e as posibilidades do profesorado encargado do grupo ao que estea asignado un estudante matriculado a tempo parcial, axustaranse as probas da avaliación continua para que o devandito estudante poida obter a mesma cualificación que un estudante de matrícula ordinaria.

Na oportunidade adiantada a decembro, o exame cualificarase sobre dez puntos, sendo necesario obter polo menos un cinco para aprobar a materia.

Evaluation
of the student registered in time partial: Depending on the
particularities of every specific case and the possibilities of the
teaching staff put in charge of the group to the that it is a student
registered in time partial assigned, the tests of the continuous
evaluation will be adjusted so that this student can obtain the same
qualification as a student of ordinary registration.

In the
opportunity advanced to December, the examination will be qualified
on ten points, being necessary to obtain at least one five to approve
the matter.

Learning outcomes	Study programme competences
Acquire basic concepts from elementary number theory.	A1 A3
Interpret and apply the acquired knowledge from elementary number theory to Cryptography.	A1 A3	B3
Know some basic concepts of Linear Algebra: systems of linear equations, vectorial spaces, matrices and linear maps.	A1
Use methods from linear algebra as a tool for modeling and solving processes related to computer science.	A1	B6	C6
Know the definitions and basic principles from Coding Theory that are related to Linear Algebra.	A1
Simulate coding and decoding processes using matricial techniques.	A1	B6	C6
Learn how to use mathematical language in a proper way to express ideas.	A1		C1
Develop the capacities of abstraction, concretion, concision, imagination, intuition, reasoning, criticism, objectivity, synthesis and accuracy; put all of them in practice either in the academic or the professional life for facing and solving problems successfully.		B3	C7
Apply basic concepts from the subject and relate to algorithmic and computational concepts in the light of the mathematical ones.	A1		C6
Adquirir ferramentas e destrezas para resolver os problemas de forma axeitada. Expresar e interpretar de forma precisa os resultados obtidos. Verificar o resultado e, en caso de obter unha incongruencia, revisar o proceso para detectar o erro cometido. Acquire tools and skills for solving problems in a proper way. Express and interprete results in a rigorous way. Check the result and, in case of an incongruence, revise the process to detect the error.	A1	B6	C1 C7

Topic	Sub-topic
Chapter 1: Modular arithmetic: application to Cryptography.	Basic concepts from elementary number theory. Euclides' algorithm. Prime numbers. Linear diophantine equations. Congruences. Modular arithmetic. Definition of cryptosystem. Classical cryptography. Symmetrical and asymmetrical cryptography. Examples of cryptosystems. Numeration systems. Divisibility criteria.
Chapter 2: Systems of Linear Equations, Matrices and Determinants.	Definition and properties of systems of linear equations. Echelon row form of system. Gauss method. Matrices. Operations with matrices. Invertible matrix. Determinant of a square matrix, properties. Cramer's rule.
Chapter 3: Vector Spaces.	Definition and properties of a vector space. Bases and coordinates. Dimension. Rank of a set of vectors and matrix rank. Computation of the rank. Change of basis. Rouché-Frobenius theorem.
Chapter 4. Linear maps.	Definición e propiedades das aplicacions lineais. Núcleo e imaxe de unha aplicación lineal. Matriz asociada a unha aplicación lineal. Teorema da dimensión. Definition and properties of linear maps. Kernel and image of a linear map. Matrix associated to a linear map. Dimension theorem.
Chapter 5. Linear Codes	Definition of linear codes. Parameters of a linear code. Hamming distance and Hamming weight. Generator matrix and parity-check matrix of a code. Error correction in linear codes. Binary Hamming codes.

Methodologies / tests	Competencies	Ordinary class hours	Student’s personal work hours	Total hours
Guest lecture / keynote speech	A1 A3 C6 C7	30	37.5	67.5
Laboratory practice	A1 B3 B6 C1 C6	20	30	50
Collaborative learning	A1 B3 C1 C7	10	17.5	27.5

Personalized attention		5	0	5

(*)The information in the planning table is for guidance only and does not take into account the heterogeneity of the students.

Methodologies	Description
Guest lecture / keynote speech	Across the virtual platform of the university, will put itself the arrangement of the student body the detailed information of the contents of every subject in order to that every pupil/to is configured, according to his criterion and needs, the material adequate for the follow-up and understanding of the matter, for that will be able to make use of the recommended bibliography and/or available material in the net. The theoretical and practical classes will keep coping in a simultaneous way in the classroom, carrying out exercises after the theoretical explanations, to a rhythm adequate for the total understanding of the ones constrained in order to achieve the suggested goals. A presentation of the formal techniques will be looked for by examples, with emphases in specific calculations and in the algorithmic nature of some of them. It will be intended that the pupils are capable of obtaining conclusions of the result incubated, attempting to motivate to the pupils so that they take part and they are capable of inferring conclusions that can be more or less evident.
Laboratory practice	A bulletin of exercises related to the theoretical contents explained in the classes of theory will facilitate it itself the pupils to the beginning of every subject. In these sessions it is aimed for: I) encouraging the pupil through the resolution of exercises, with the help of the professor, to reinforce the understanding of the incubated concepts, II) to encourage the resolution reasoned of the exercises, preventing the use of "recipes". Depending on the subject and on the available resources, works will be formulated with computer programs that reinforce the concepts worked in theoretical classes and of exercises. Together with the bulletin of exercises the goals or results of learning that they must obtain when finishing the subject will be indicated to the students.
Collaborative learning	Throughout the course they will dedicate themselves ten hours approximately so that the pupils formulate the doubts on the concepts, exercises and procedures seen in the sessions of theory and problems. Likewise small projects or the resolution of exercises in reduced groups of pupils can be formulated.

Methodologies	Competencies	Description	Qualification
Guest lecture / keynote speech	A1 A3 C6 C7	Ao final do curso realizarase unha proba escrita. Esta proba inclúe: - Preguntas curtas que permitan valorar se o alumno comprendeu os conceptos teóricos básicos. - Exercicios cun grao de dificultade similar aos realizados en clase e aos presentados nas coleccións de exercicios propostos. Avaliarase o dominio dos conceptos teóricos da materia, a comprensión destes e a súa aplicación na resolución de exercicios. Así mesmo, valorarase a claridade, a orde e a presentación dos resultados expostos. A presentación á proba final do curso supón que o estudante completou o proceso de avaliación continua. Para aprobar a materia é necesario obter máis de tres puntos dos oito posibles na proba escrita. At the end of the course a written test will be carried out. This test includes: - Short questions which they allow valuing if the pupil understood the basic theoretical concepts. - Exercises with a degree of difficulty similar to the ones carried out in class and to the ones presented in the collections of suggested exercises. The domination of the theoretical concepts of the matter, the understanding of these and his application in the resolution of exercises will be evaluated. Likewise, it will value itself the clarity, the order and the presentation of the exposed results. The presentation to the final test of the course means that the student completed the process of continuous evaluation. To approve the matter it is necessary to obtain more than three points of the eight possible ones in the written test.	80
Laboratory practice	A1 B3 B6 C1 C6	Ao longo do curso realizarase unha avaliación dos distintos temas onde se formularán preguntas sobre as definicións dos conceptos introducidos, cuestións e exercicios similares aos do correspondente boletín. Valorarase a resposta correcta ás cuestións e exercicios formulados así como a presentación e claridade da exposición realizada. Poderase valorar unha actitude participativa do alumnado na resolución das cuestións formuladas durante as prácticas e nas titorías en grupo reducido. A nota obtida neste apartado será a mesma nas dúas oportunidades da convocatoria do curso académico. Throughout the course an evaluation of the different subjects where questions will be formulated on the definitions of the introduced concepts, questions and exercises similar to those of the corresponding bulletin will be carried out. It will value itself the correct answer to the questions and formulated exercises as well as the presentation and clarity of the exhibition carried out. A participative attitude of the student body will be able to be valued in the resolution of the questions formulated during the practices and in the tuitions in reduced group. The note obtained in this section will be it in the two opportunities of the call of the academic course.	20
Collaborative learning	A1 B3 C1 C7	Valorarase a participación activa do alumnado nas sesións. The active participation of the student body will be valued in the sessions.	0

Assessment comments
Avaliación do alumnado matriculado a tempo parcial: Dependendo das particularidades de cada caso concreto e as posibilidades do profesorado encargado do grupo ao que estea asignado un estudante matriculado a tempo parcial, axustaranse as probas da avaliación continua para que o devandito estudante poida obter a mesma cualificación que un estudante de matrícula ordinaria. Na oportunidade adiantada a decembro, o exame cualificarase sobre dez puntos, sendo necesario obter polo menos un cinco para aprobar a materia. Evaluation of the student registered in time partial: Depending on the particularities of every specific case and the possibilities of the teaching staff put in charge of the group to the that it is a student registered in time partial assigned, the tests of the continuous evaluation will be adjusted so that this student can obtain the same qualification as a student of ordinary registration. In the opportunity advanced to December, the examination will be qualified on ten points, being necessary to obtain at least one five to approve the matter.

Basic	Grossman, S. I. (1996). Álgebra lineal con aplicaciones. McGraw-Hill Interamericana México. Merino, L. y Santos, E. (2006). Álgebra Lineal con Métodos Elementales. Thomson. Lay, D. C. (2007). Algebra Lineal y sus Aplicaciones. Prentice Hall Rosen, K. H. (2003). Discrete Mathematics and Its Applications. McGraw-Hill Grossman, S. I. (1994). Elementary Linear Algebra with Applications. Wiley Cameron, P. J. (1998). Introduction to Algebra. Oxford University Press, Oxford. Lay, D. C. (2011). Linear Algebra and Its Applications. Pearson Biggs, N. L. (1994). Matemática Discreta. Madrid, Vicens Vives. Rosen, K. H. (2004). Matemática Discreta y sus aplicaciones. McGraw-Hill Interamericana.

Complementary	Nakos, G. y Joyner, D. (1999). Álgebra lineal con aplicaciones. Thomson. Hernández, E. (1994). Álgebra y Geometría. Addison-Wesley. Lidl, R. y Pilz, G. (1998). Applied Abstract Algebra. Nueva York, Springer. Rojo, J. y Martín, I. (2005). Ejercicios y problemas de Álgebra Lineal. McGraw-Hill. Torrecilla Jover, B. (1999). Fermat. El Mago de los Números. Nivola. Van Lint, J. H. (1999). Introduction to Coding Theory. Berlín, Springer. Nakos, G. y Joyner, D. (1998). Linear Algebra with Applications. Brooks Cole Publising Singh, S. (2000). Los Códigos Secretos. Debate

Other comments