MODULE SUBJECT YEAR SEMESTER ECTS TYPE

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1 COURSE DESCRIPTION MATHEMATICS I ACADEMIC YEAR This document has been translated at the beginning of the academic year and may suffer some changes throughout the current course or next. Use it only for information or guidance. For future modifications, consult the updated Spanish version. MODULE SUBJECT YEAR SEMESTER ECTS TYPE Basic module Mathematics 1st 1 st - Fall 6 Core subject PROFESSORS Professors responsible of the lecture in this subject are full members of the Department of Applied mathematics. Group A: Ana Isabel Garralda Guillem (Theoretical lectures)/ Ana Isabel Garralda Guillem & Manuel Ruiz Galán (Practice groups) Group B: María del Carmen Serrano Pérez (Theoretical lectures)/ María del Carmen Serrano Pérez & Olga Valenzuela Cansino (Practice groups) Group C: Manuel Ruiz Galán (Theoretical lectures)/ Manuel Ruiz Galán & Olga Valenzuela Cansino (Practice groups) Group D: María del Carmen Serrano Pérez (Theoretical lectures)/ Olga Valenzuela Cansino & Daniel de la Fuente (Practice groups) Group E: Olga Valenzuela Cansino (Theoretical lectures)/ Olga Valenzuela Cansino & Philippe Bechouche (Practice groups) CONTACT DETAILS FOR CONSULTATIONS TIMETABLE OF TUTORSHIP P. Bechouche: School of Building Engineering, 5 th floor. Office room 5 phbe@ugr.es Daniel de la Fuente: Faculty of Sciences. Department of Mathematics, delafuente@ugr.es A.I. Garralda Guillem: School of Building Engineering, 5 th floor. Office room 7 agarral@ugr.es M. Ruiz Galán: School of Building Engineering, 5 th floor. Office room 27 mruizg@ugr.es M.C. Serrano Pérez: Faculty of Science, Mathematics Department, Office room 58 cserrano@ugr.es O. Valenzuela Cansino: School of Building Engineering, 5 th floor. Office room 26 olgavc@ugr.es Schedule for appointments during the course, place and procedures are to be published through the usual means used by this department of Applied Mathematics. These will be published at the beginning of the course. Página 1

2 FIELD OF STUDY Degree in Building Engineering OTHER DEGREES THIS SUBJECT IS OFFERED TO Degree in Architecture Degree in Civil Engineering PREREQUISITES AND/OR RECOMMENDATIONS (if necessary) Matrix calculation dexterity: Sum, Product, Calculate the inverse matrix of a regular matrix, and determinant of a square matrix. Affine plane and space: affine subspaces, its equations, associated problems and exercises. SHORT DESCRIPTION OF CONTENTS (ACCORDING TO THE REPORT OF DEGREE VERIFICATION) Linear algebra, analytic geometry, descriptive statistics and correlation, probability and random variables. GENERAL AND SPECIFIC COMPETENCES Ability to use applied knowledge, related to numerical and infinitesimal calculus, differential geometry and methods of statistical analysis. Ability to relate in a critical way, mathematical knowledge with Building Engineering. To be familiar with the use of computer programs in order to apply gained theory knowledge. Go in depth and integrate knowledge gathering various subjects, related to physical/structural and mathematical analysis of technical problems. Use mathematical language fluently, both oral and written, with rigorous formalization and structuring of a problem. Use an adequate terminology to discuss theory and practical problems. Ability to identify basic techniques of each problem. Use of information and communication technology in the field of the subject. Skills for teamwork GOALS (EXPRESSED AS EXPECTED RESULTS OF EDUCATION) Relate terms used in linear algebra with its definition and properties. Express a real problem in mathematical terms that set out in a common language can be solved using linear algebra or geometry. Use an adequate method to discuss and solve systems of linear equations. Find the rank of a given matrix. Apply matrices and determinants to various fields. Study the vectorial structure of R n. Recognize if the subset of vectorial space R n is its vectorial subspace. Analyze if a vector can be expressed as a linear combination of two other given vectors. Consider whether a set of vectors is linearly independent. Reason if a family of vectors is generating a vector space. Reason if a family of vectors is basis of a vectorial space. Obtain the coordinates of a vector in relation to a given basis. Study the Euclidean structure of a vectorial space R n. Obtain an orthonormal basis of a vectorial subspace of R n. Calculate the orthographic projection of a vector in a subspace. Página 2

3 Apply the results of the best approximation to the adjustment by least squares. Calculate eigenvalues and eigenvectors of a squared matrix. Study if a given matrix is diagonalizable. Carry out the process if yes Recognize if a given matrix is orthogonal. Diagonalize orthogonally real symmetric matrices. Apply the diagonalization of matrices to different areas. Study the affine structure of R 2 and R 3. Find angles and distances in the plane and in the space. Draw and find characteristic elements of a conic, given its equations in reduced form. Identify a conic from its general equation in rectangular coordinates. Identify a quadric, given by its equations in reduced form. Use graphs and numerical methods to explore, summarize and describe data Find and select information on the Internet about the implementation of linear algebra and geometry to the sector of building engineering. Use education software with application to geometry or linear algebra. Know and use appropriately the usual terminology of the subject. Know the bibliography related to the subject. DETAILED LIST OF THE SUBJECT S TOPICS The program of the course is structured around five topics, presented in a systematic way, sorted by their connections and dependencies. Topic 1. Matrices and systems of linear equations. 1.1 Matrices. Matrix calculus. 1.2 Elementary transformations of a matrix. Rank. 1.3 Regular matrices. Inverse matrix. 1.4 Determinants. 1.5 Systems of linear equations. Rouché-Fröbenius theorem. 1.6 Solving systems of linear equations: method of Gaussian elimination. Topic 2. Euclidean vectorial space R n. 2.1 Vectorial space Rn. 2.2 Linear combination. Linear Independence. 2.3 Basis and dimensions. 2.4 Change of base. 2.5 Vectorial subspaces. 2.6 Euclidean structure of R n. 2.7 Least squares approximation. Topic 3. Matrix Diagonalization 3.1 Diagonalizable matrices. Diagonalization of a matrix. 3.2 Diagonalization of real symmetric matrices. 3.3 Applications. Topic 4. Conics and quadrics. 4.1 Affine spaces R 2 and R 3. Frames of reference. 4.2 Ellipse, hyperbola and parabola defined by their metric properties. 4.3 General equation of a conic. 4.4 Reduced equation of a conic and its geometric elements. Página 3

4 4.5 Quadrics. Reduced equation. 4.6 Applications for the building sector. Topic 5. Introduction to statistics and data analysis. 5.1 Descriptive statistics. 5.2 Probability. Random variables. COMPUTER-BASED PRACTICE UNITS UNIT 1.- Introduction to Maxima UNIT 2.- Matrices and systems of linear equations UNIT 3.- Vector spaces UNIT 4.- Euclidean vector space UNIT 5.- Diagonalization and conics UNIT 6.- Introduction to statistics and data analysis BIBLIOGRAPHY Basic bibliography Alsina, C. y Trillas, E., Lecciones de Álgebra y Geometría (5ª edición), Gustavo Gili, (1991). Anderson, D.R. Estadística para la administración y empresa (2 vol.), Thomson (2001) Burgos, J. de, Álgebra Lineal, McGraw-Hill (2006). Castellano, J., Gámez, D., Garralda, A.I., Ruiz, M., Matemáticas para la Arquitectura, Proyecto Sur Ediciones, (2000). Grossman, S.I., Álgebra Lineal, (5ª edición) McGraw-Hill, México, (1996). Johnson, R. y Kuby, P., Estadística elemental. Lo esencial (3ª edición), Thomson (2006). Merino, L. M. y Santos, E., Álgebra Lineal con métodos elementales, Thompson, (2006). Additional bibliography Coquillat, F, Espacios Vectorial, Afín y Euclídeo. Metodología y Problemas, Tebar Flores (1990). Larson, R. E., Hostetler, R. P. y Edwards, B. H., Cálculo y geometría analítica. Vol. I, (8ª edición) Mc- Graw-Hill (2005). Larson, R. E., Hostetler, R. P. y Edwards, B. H., Cálculo y geometría analítica. Vol. II, (8ª edición) Mc- Graw-Hill, Madrid, (2005). Moreno Flores, J., Problemas resueltos de Matemáticas para la Edificación y otras ingenierías, Paraninfo (2011). Rojo, J. y Martín, I., Ejercicios y problemas de Álgebra Lineal, (2ª edición), McGraw-Hill, (2005). Villa, A. de la, Problemas de Álgebra, CLAGSA, (1998). RECOMMENDED WEBSITES Página 4

5 TEACHING METHODOLOGY Regarding attended activities, the professor will conduct lectures which include a breakdown of basic concepts and essential theoretical results in order to achieve specific competences of the course. Later on, the student is meant to complete the proposed activities. Additionally, there are exercises and problems aimed to fulfil and clarify the contents, which will help them solve other exercises autonomously. Some of these sessions will be held in computer rooms and supervised by the teachers. The students will make use of the knowledge gained to solve exercises with and without the computer. The software used for this purpose is MAXIMA under license of GPL. As for non-attended, individual activities, these will be completed in the frame of autonomous work by the student. PROGRAM OF ACTIVITIES The 6 ECTS points are a total of 6 x 25 = 150 hours, to be divided into 60 hours of classroom work, equivalent to 40% of them, and 90 hours of individual work. The type of activities and the corresponding schedule are detailed in the following table: TOPIC Activities to attend Autonomous Theory Exercise Practical student s work lessons lessons exams Topic 1 6 h 4 h 12 h Topic 2 10 h 6 h 25 h First exam 2 h 8 h (preparation for the exam) Topic 3 6 h 6 h 10 h Topic 4 10 h 2 h 21 h Topic 5 4 h 2 h 6 h Second exam 2 h 8 h (preparation for the exam) Total 36 h 20 h 4 h 90 h GUIDELINES OF ATTENDANCE Attendance to all lectures is highly recommended. In addition, the student must bear in mind that some activities carried out during those lectures are considered for final grading (See section below). ASSESSMENT In accordance to Grading regulations of the University of Granada ( there are two kinds of assessment: Formative (throughout the course) and Summative (on final exam). As for the formative type, there will be 2 practical exams throughout the semester, each one of them, weighted up to 3.6 points: 2.7 corresponding theoretical part and 0.9 to problems and practice using MAXIMA. Schedule and dates for the first exam is published soon enough, whereas the second will take place on the 5 th Página 5

6 of February at 9am. The 2.8 points remaining can be obtained by completing activities during the semester. These activities are published at the beginning of the course. Final mark comes out from the weighted mean of all practices. All students who cannot follow up formative assessment type for duly justified reasons still can pass the subject through final single evaluation or summative. The students ought to request this type of evaluation within the first two weeks justifying reasons why the student cannot attend lessons regularly. This type of assessment comprises only one exam, which marks ranges from 0 to 10. The date for it is settled beforehand and approved by the start of each academic year. At the same time,it contains the following parts: Paper-based theory and problems: up to 7.5 points Computer-based exam: up to 2.5 points (Máxima software) Total mark is obtained by summing the previous ones. A final grade of 5 out of 10 is the minimum accepted to pass the subject Call to final exam in September: Same guidelines described above apply for this. The updated date for Final exams in this academic year 14/15 are: - Regular call to exam: February 5 th, 2015 at 9am (See noticeboard that day for classroom details) - Additional call to exam: September 14 th, 2015 at 9am (See noticeboard that day for classroom details) RULES OF THE SUBJECT To ensure proper organization of the course, students must respect the following rules: Be strictly punctual when arriving to class. Remain absolutely silent during the development of classes, not leaving the classroom without a justified reason. Mobile phones or any other device must be turned off, both in the classroom and during exams. During exams, the following rules have to be obeyed: Once has the exam started, no student will have the right to enter the classroom. During exams all students have to carry their Identity document or passport. Use of calculator is not allowed during exam, unless some notification is given. Exams written in pencil will not be assessed. The only accepted material is black or blue pen. Página 6