| 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 solve problems that may appear in an engineering context. | A1 |
B1 B2 B3 B5 |
C5 C6 |
| To think in a logic, critic and creative way. | B2 B3 B5 B6 |
C2 C5 C6 |
|
| 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 |
| Objective test | A1 A5 B1 B2 B3 B4 B5 B6 C1 C2 C4 C5 C6 | 5 | 0 | 5 |
| ICT practicals | A1 A5 B2 B3 B4 B5 B6 C1 C2 C4 C5 | 10 | 10 | 20 |
| Problem solving | A1 A5 B1 B2 B3 B4 B5 B6 C2 C4 C5 C6 | 20 | 28 | 48 |
| 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 |
| 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 |
| ICT practicals | This methodology allows students to learn effectively, through practical activities (demonstrations, simulations, etc.) the theory of a specific field through the use of information and communication technology. This type of training is geared towards the implementation of learning in which you can combine various methodologies / tests by using electronic tools such as a calculator, computer, etc. The students developed practical oriented tasks on a specific topic, with the support and supervision of the professors. These practices can be done individually or in groups. |
| Problem solving | Technic by means of which one has to solve a specific problematic situation related to the contents of the subject. |
| Personalized attention |
|
|
| Assessment | |||
| Methodologies | Competencies | Description | Qualification |
| Objective test | A1 A5 B1 B2 B3 B4 B5 B6 C1 C2 C4 C5 C6 | Written exam will be used to assess learning of the contents of the subject. The exam consists of four parts, the first one will be performed in the planned period for partial exams and will include lessons 1 and 2. This part will be eliminatory and retrievable and will be weighted as 40% of the final grade. The second part will be developed throughout the course by making working groups, students being graded by a test assessing gained competences. This part will be weighted as 15% of the final grade. The third part will be performed during the usual period of final exams and will assess the first, second and third parts. Its total weight is 90%. The fourth part will consist of a test related to the software MAXIMA, where the student will show his/her capability in solving problems about the contents of the subject aided by the computer. This test is not retrievable: mark will be saved until the second opportunity. Its weight is 10% of the final grade.. |
100 |
| Assessment comments | |||
|
- Students will be necessarily assessed of lessons 4 and 5 of the final exam. Furthermore, a minimum of the 40% of the grade is required to pass the subject. Remark: grades obtained in the first and the second part will be saved until the second opportunity if half of the grade is got. This is independently applied to each part involved in the assessment. |
|||
| 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 |
|
|
|
| Complementary | |
|
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 | ||
|
| Subjects that are recommended to be taken simultaneously | |
|
| Subjects that continue the syllabus | ||
|
| Other comments | |