| Study programme competencies |
| Code | Study programme competences |
| A1 | Skill for the resolution of the mathematical problems that can be formulated in the engineering. Aptitude for applying the knowledge on: linear algebra; geometry; differential geometry; differential and integral calculation; differential equations and in partial derivatives; numerical methods; algorithmic numerical; statistics and optimization |
| A5 | Have a capacity for the space vision and knowledge of the techniques of graphic representation, so much for traditional methods of metric geometry and descriptive geometry, as through the applications of design assisted by computer |
| B1 | That the students proved to have and to understand knowledge in an area of study what part of the base of the secondary education, and itself tends to find to a level that, although it leans in advanced text books, it includes also some aspects that knowledge implicates proceeding from the vanguard of its field of study |
| B2 | That the students know how to apply its knowledge to its work or vocation in a professional way and possess the competences that tend to prove itself by the elaboration and defense of arguments and the resolution of problems in its area of study |
| B3 | That the students have the ability to bring together and to interpret relevant data (normally in its area of study) to emit judgments that include a reflection on relevant subjects of social, scientific or ethical kind |
| B4 | That the students can transmit information, ideas, problems and solutions to a public as much specialized as not specialized |
| B5 | That the students developed those skills of learning necessary to start subsequent studies with a high degree of autonomy |
| B6 | Be able to carrying out a critical analysis, evaluation and synthesis of new and complex ideas. |
| C1 | Using the basic tools of the technologies of the information and the communications (TIC) necessary for the exercise of its profession and for the learning throughout its life. |
| C2 | Coming across for the exercise of a, cultivated open citizenship, awkward, democratic and supportive criticism, capable of analyzing the reality, diagnosing problems, formulating and implanting solutions based on the knowledge and orientated to the common good. |
| C4 | Recognizing critically the knowledge, the technology and the available information to solve the problems that they must face. |
| C5 | Assuming the importance of the learning as professional and as citizen throughout the life. |
| C6 | Recognizing the importance that has the research, the innovation and the technological development in the socioeconomic and cultural advance of the society. |
| Learning aims | |||
| Learning outcomes | Study programme competences | ||
| To familiarize ourselves with mathematical lenguage, in particular with the algebraic one. | A1 A5 |
B2 B3 B5 B6 |
C4 |
| To understand the main ideas in posing mathematical problems, making use of algebraic tools. | A1 |
B1 B2 B3 B4 B5 B6 |
C2 C4 |
| To be able to use the bibliographical references and other computer tools, such as mathematical software, to find out the appropriate information to solve a given problem. | A1 |
B2 B3 B4 |
C1 |
| To know the main characteristics of a space endowed with an algebraic structure, mainly the vector space structure. | A1 |
B2 B3 |
C4 C5 |
| To understand the equivalence between the matrix concept and the linear map concept, knowing the consequences of this relationship. | A1 |
B2 |
C4 C5 |
| To know and understand the concepts of paths and surfaces in Euclidean space. To understand the geometrical and physical meaning of derivatives and integrals applied to these mathematical objects. | A1 A5 |
B2 B6 |
C4 C5 C6 |
| Contents |
| Topic | Sub-topic |
| Vector spaces | Euclidean spaces R^2 and R^3. Operations: sum, product by real numbers. Vector subspaces. Direct sum. Linear combination, span. Linear independence. System of generators. Basis and dimension. Theorem of the basis. Coordinates, change of coordinates. Applications to systems of linear equations. |
| Linear maps | Correspondences. Maps. Linear maps. Properties of linear maps. Matrix associated to a linear map. Applications to systems of linear equations. |
| Diagonalization of endomorphisms | Invariant subspaces. Eigenvalues and eigenvectors. Diagonalizable endomorphisms. |
| Integrals over paths | Paths in R^2 and R^3. Parametrizations. Path integrals of scalar functions. Line integrals of vector fields. Gradient vector fields. Green's Theorem. |
| Integrals over surfaces | Parametrized surfaces. Surface integrals. Rotational and divergence. Stokes's Theorem. Divergence Theorem. |
| Appendix: the free software program MAXIMA | Practical sessions with the free software program MAXIMA |
| Planning | ||||
| Methodologies / tests | Competencies | Ordinary class hours | Student’s personal work hours | Total hours |
| Guest lecture / keynote speech | A1 A5 B3 B4 B5 B6 C2 C4 C5 C6 | 30 | 45 | 75 |
| Workshop | A1 A5 B2 B3 B4 B5 B6 C1 C2 C4 C5 | 10 | 10 | 20 |
| Objective test | A1 A5 B1 B2 B3 B4 B5 B6 C1 C2 C4 C5 C6 | 5 | 0 | 5 |
| Collaborative learning | A1 B3 B4 B5 B6 C2 C4 C5 | 4 | 12 | 16 |
| Problem solving | A1 A5 B1 B2 B3 B4 B5 B6 C2 C4 C5 C6 | 16 | 16 | 32 |
| Personalized attention | 2 | 0 | 2 | |
| (*)The information in the planning table is for guidance only and does not take into account the heterogeneity of the students. | ||||
| Methodologies |
| Methodologies | Description |
| Guest lecture / keynote speech | Oral exhibition complemented with the use of audiovisual means and some questions headed to the students, with the purpose to transmit knowledges and facilitate the learning |
| Workshop | Formative modality oriented to the application of learnings in which one can combine different methodologies (exhibitions, simulations, debates, problems solving, practical guided, etc) through which the students manage tasks essentially practical on a specific subject, with support and guide of the lecturer. |
| Objective test | Written exam used for the evaluation of the learning, whose distinctive stroke is the possibility to determine if the answers given are or no correct. It constitutes an instrument of measure, elaborated rigorously, that allows to evaluate knowledges, capacities, skills, performance, aptitudes, attitudes, etc |
| Collaborative learning | Modalidade didáctica que fomenta a aprendizaxe centrada no alumno baseando o traballo en pequenos grupos, onde os estudantes desenvolven actividades para mellorar a comprensión dunha materia ou dun tema específico da mesma. Cada membro do grupo é responsable da súa aprendizaxe pero tamén de axudar aos seus compañeiros a aprender xa que o éxito na actividade dependerá de todos os membros do grupo. Os elementos esenciais desta técnica son responsabilidade individual, interdependencia positiva, interacción cara a cara, traballo en equipo e proceso de grupo. |
| Problem solving | Technic by means of which one has to solve a specific problematic situation related to the contents of the subject. |
| Personalized attention |
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| Assessment | |||
| Methodologies | Competencies | Description | Qualification |
| Objective test | A1 A5 B1 B2 B3 B4 B5 B6 C1 C2 C4 C5 C6 | Written exam to assess the knowledge of the subject by the students. Three exams will be performed, the first one in the reserved period for the partial exams, and will involve all the issues studied until the celebration of the exam. The second (and final) exam will be carried out in the period of final exams. The weight of both exams will be the 75% of the final qualification. The third exam will consist of a computer exam with the program MAXIMA, where the students must show their capacity for solving problems using the MAXIMA software. The weight of this third part will be the 10% of the final qualification. |
85 |
| Collaborative learning | A1 B3 B4 B5 B6 C2 C4 C5 | This is an activity that is based on active learning in groups. This will be graded up to a 15% of the final grade. Those studendts that obtain a minimum of half of the maximum grade are (optionally) excluded to be evaluated of this part in the final exam. The grade will be valid for the two oportunities. | 15 |
| Assessment comments | |||
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| Sources of information |
| Basic |
Larson, R., Edwards, B.H., Calvo, D. C. (2004). Álgebra lineal. Pirámide Ediciones
Burgos, J. (1993). Álgebra lineal. McGrawHill
Grossman, S. I. (1995). Álgebra Lineal con Aplicaciones. Mcgraw-Hill
Lay, D. C. (2007). Álgebra lineal y sus aplicaciones. Addison-Wesley
Granero Rodríguez, F. (1991). Álgebra y Geometría Analítica. Mcgraw-Hill
Hwei P. Hsu (1987). Análisis Vectorial. Addison-Wesley
Marsden, J., Tromba, A. (2004). Cálculo Vectorial. Addison-Wesley
Larson, R., Hostetler, R., Edwards, B. (1999). Cálculo y Geometría Analítica, Vol. 2. McGraw-Hill
Ladra, M., Suárez, V., Torres, A. (2003). Preguntas test de Álgebra Lineal y Cálculo Vectorial. E. U. Politéctica
Villa Cuenca, A. (1994). Problemas de Álgebra. CLAGSA |
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| Complementary | |
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The following webpages may be of interest for students: http://www.cds.caltech.edu/~marsden/books/Vector_Calculus.html This webpage contains complement material to the reference Marsden-Tromba from the bibliography, one can download as slides different parts of the book. http://demonstrations.wolfram.com/index.html This webpage from Wolfram Research has computer programs developed in Mathematica. They can be useful for visualization of concepts and techniques explained during the course. http://193.144.60.200/elearning/ This webpage contains several applets created with Geogebra (free software), that the student can find useful to visualize contents of the course. |
| Recommendations | ||
| Subjects that it is recommended to have taken before | ||
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| Subjects that are recommended to be taken simultaneously | |
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| Subjects that continue the syllabus | ||
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| Other comments | |