| Study programme competencies |
| Code | Study programme competences |
| A1 | Capacidad para plantear y resolver los problemas matemáticos que puedan plantearse en el ejercicio de la profesión. En particular, conocer, entender y utilizar la notación matemática, así como los conceptos y técnicas del álgebra y del cálculo infinitesimal, los métodos analíticos que permiten la resolución de ecuaciones diferenciales ordinarias y en derivadas parciales, la geometría diferencial clásica y la teoría de campos, para su aplicación en la resolución de problemas de Ingeniería Civil. |
| B2 | Resolver problemas de forma efectiva. |
| B3 | Aplicar un pensamiento crítico, lógico y creativo. |
| C1 | Reciclaje continúo de conocimientos en el ámbito global de actuación de la Ingeniería Civil. |
| C4 | Entender y aplicar el marco legal de la disciplina. |
| C6 | Compresión de la necesidad de analizar la historia para entender el Presente. |
| C8 | Facilidad para la integración en equipos multidisciplinares. |
| Learning aims | |||
| Learning outcomes | Study programme competences | ||
| To know and to understand the basic theory of linear algebra required in civil engineering , especially the geometric applications of vector spaces. | A1 |
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| Know, understand and manage elementary mathematical notation. | A1 |
B3 |
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| Learn to express with precision and rigor. | A1 |
C1 |
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| Learn to use the basic techniques of mathematical reasoning. | A1 |
B2 B3 |
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| Understanding the importance of justifying the thesis and results in science | A1 |
B3 |
C4 C6 |
| Develop critical thinking and analytical skills . | A1 |
B2 B3 |
C4 C8 |
| Learn to pose and solve mathematical problems of Linear Algebra. | A1 |
B2 B3 |
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| Contents |
| Topic | Sub-topic |
| Bilinear maps and homogenous tensors. | 1. Bilinear maps and quadratic forms. 1.1 Bilinear maps. 1.2 Bilinear forms. 1.3 Quadratic forms. 1.4 Real quadratic forms. 2. Homogenous tensors and duality. 2.1 Duality. 2.2 Homogenous tensor. 2.3 Operations with homogenous tensors. 2.4 Simmetry and skewsimmetry. |
| Euclidean vectorial spaces. | 1. Introduction to euclidean spaces. 1.1 Scalar product. 1.2 Norm of a vector. Properties. 1.3 Angle between two vectors. 2. Orthogonality. 2.1 Orthogonal vectors. 2.2 Orthogonal systems. Gram-Schmidt method. 2.3 Singularties of orthonormal basis. 2.4 Orthogonal projection. 2.5 Symmetric endomorphisms. 3. Orthogonal maps. 3.1 Definition. 3.2 Properties. 3.3 Eigenvalues and eigenvectors of an orthogonal map. 3.4 Orientation of a basis 3.5 Inverse and direct orthogonal maps. 3.6 Classiication of orthogonal maps in two and three dimensions. 4. Vectorial product and triple product. 4.1 Definition. 4.2 Properties. |
| Affine geometry. | 1. Affine space. 1.1 Definition and properties. 1.2 System of reference. 1.3 Affine varieties. 1.4 Pencils of affine varietes. 1.5 Distances and angles between affine varieties. 1.6 Affine transformations. 2. Projective space. 2.1 Introduction. 2.2 Homogeneous coordinates. 2.3 Proper points and points at infinity. 2.4 Reference change in homogeneous coordinates. 2.5 Equations of affine varieties in homogeneous coordinates. |
| Conics and quadric surfaces. | 1. Conics. 1.1 Definition and equations. 1.2 Intersections of a conic and a line. 1.3 Polarity. 1.4 Important potins and lines of a conic. 1.5 Description of nondegenerated conics: ellipse, parabola e hyperbola. 1.6 Change of reference. 1.7 Classification of conics. Reduced equation. 1.8. Pencils of conics. 2. Quadric surfaces. 2.1 Definition and equations. 2.2 Intersections of a quadric surface and a line. 2.3 Polarity. 2.4 Change of reference. 2.5 Important potins, lines and planes of a quadric surface. 2.6 Classification of quadric surfaces. Reduced equation. 2.7 Description of quadric surfaces of rank 3 and 4. |
| Planning | ||||
| Methodologies / tests | Competencies | Ordinary class hours | Student’s personal work hours | Total hours |
| Guest lecture / keynote speech | A1 B2 B3 C1 | 27 | 32 | 59 |
| Seminar | A1 B2 B3 | 27 | 33 | 60 |
| Mixed objective/subjective test | A1 B2 B3 | 3 | 3 | 6 |
| Workbook | A1 B2 B3 | 0 | 10 | 10 |
| Problem solving | A1 B2 B3 | 0 | 10 | 10 |
| 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 |
| Methodologies | Description |
| Guest lecture / keynote speech | New mathematic concepts will be developed from examples familiar for the students, or explaining the questions are wished to be solved with them; from this their common caracthers will be abstracted causing its more accuracy deffinition. The theory which allows to solve the questions described at the beginning will be developed after. Students participation is desirable, sharing their doubts or comments as the class progresses. |
| Seminar | Simultaneously to the theoretical development of the matter collections of exercics are given. The goal is allowing students to practise the knowledge adquierd at theorical classes. At seminars the most important problems will be discussed. |
| Mixed objective/subjective test | Exam where concepts are methods of the subjets are evaluated. |
| Workbook | Before the beginning of each item, some notes about the contents are avaiable for the students. The notes are intended as a complement of teacher's explanations. A previous reading of students familiarize them with an outline of what they will study. |
| Problem solving | Each student must solve individually some of the proposed problems. |
| Personalized attention |
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| Assessment | |||
| Methodologies | Competencies | Description | Qualification |
| Problem solving | A1 B2 B3 | Each student must solve individually some of the proposed problems. |
20 |
| Mixed objective/subjective test | A1 B2 B3 | Exam where concepts are methods of the subjets are evaluated. |
80 |
| Assessment comments | |||
| Sources of information |
| Basic |
Juan de Burgos (2000). Álgebra Lineal. McGraw-Hill
Fuentes, Salete y Cruces (1980). Álgebra vectorial y Tensorial. ETSICCP Madrid
F. Granero (1992). Álgebra y Geometría Analítica. McGraw-Hill
Luis Fuentes García (2005-). Apuntes y ejemplos (http://caminos.udc.es/info/asignaturas/101/index.html). A Coruña
A. de la Villa (1994). Problemas de Álgebra. CLAGSA
Anzola, Caruncho y Pérez-Canales (1981). Problemas de Álgebra (Tomos 6,7). Madrid |
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| Complementary |
J. Rojo (2001). Álgebra lineal. McGraw-Hill
S.I. Grossman (1995). Álgebra lineal. McGraw-Hill
M. Castellet e I. Llerena (1991). Álgebra lineal y geometría. Reverté
J. Rojo e I. Martín (1994). Ejercicios y problemas de álgebra. McGraw-Hill
M. García Galludo y otros (1984). Problemas de álgebra y analítica. Madrid
F. González Posada (1971). Problemas de estructuras algebraicas tensoriales. Madrid |
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| 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 | |