Linear Algebra and Geometry I26645
- Centre
- Faculty of Science and Technology
- Degree
- Double Degree in Physics and Electronic Engineering
- Academic course
- 2024/25
- Academic year
- 1
- No. of credits
- 12
- Languages
- Spanish
- Basque
- Code
- 26645
TeachingToggle Navigation
Teaching guideToggle Navigation
Description and Contextualization of the SubjectToggle Navigation
In this course, students will become familiar with basic concepts of Linear Algebra and some of their applications. Student will also be introduced to the management of mathematical language and the most common demonstration techniques.
In Degree in Mathematics, this subject shares a module with Linear Algebra and Geometry II, which is studied in the second year of the Degree. Both subjects have as common goal the understanding of the main concepts of Linear Algebra and Affine and Euclidean Geometries and their use to solve linear problems through matrices and geometric problems on planes and spaces. Likewise, both courses intend for the student to acquire basic and horizontal training in these subjects to allow them to understand and apply such knowledge and skills in multiple interrelated directions. Also, the contents studied in both will be used in both mandatory and optional higher-level courses.
In Degree in Physics, Degree in Electronic Engineering and Double Degree in Physics and Electronic Engineering, Linear Algebra and Geometry I, Differential and Integral Calculus I, Vector and Complex Analysis and Mathematical Methods comprise the Mathematics module. The central goal of this module is the acquisition of mathematical tools to allow students to focus on the physical aspects in other modules in the respective curricula. Likewise, students will learn to appreciate mathematical abstraction and conceptual rigour.
Skills/Learning outcomes of the subjectToggle Navigation
SPECIFIC COMPETENCIES
Know how to solve linear equation systems.
Understand the concept of vector space and the basic concepts related to vector spaces (subspaces and quotient spaces, basis and spanning set, linear transformations).
Know how to diagonalize matrices and compute the Jordan form of a matrix.
Know how to orthogonalize a vector system in an euclidean space.
Know how to diagonalize a quadratic form.
Work with points, vectors, distances and angles in affine and euclidean spaces.
Use references systems, subspaces and affine transformations.
Solve geometric problems of the plane and the spaces.
Classify isometries in the plane and the space, giving its type and characteristic elements.
LEARNING OUTCOMES
Solve linear equation systems.
Compute the Jordan form of a matrix.
Compute an orthogonalization of a vector system in an euclidean space.
Diagonalizing a quadratic form.
Work with points, vectors, distances and angles in affine and euclidean spaces.
Use references systems, subspaces and affine transformations.
Theoretical and practical contentToggle Navigation
UNIT 1. VECTOR SPACES.
Vector space. Vector subspaces. Basis and dimension of a vector space. Change of basis.
UNIT 2. LINEAR TRANSFORMATIONS.
Linear transformations. Kernel and Range of a linear transformation. Isomophisms of Vector spaces. Matrix of a linear transformation.
UNIT 3. SYSTEMS OF LINEAR EQUATIONS AND DETERMINANTS.
Rank of a matrix. Elementary transformations and the computation of the rank of a matrix. System of linear equations. Rouché-Frobenius Theorem. The symmetric group. Determinant of a matrix. Cramer's Rule.
UNIT 4. DIAGONALIZATION OF ENDOMORPHISMS FROM V INTO V.
f-invariant subspaces. Eigenvalues and eigenvectors. Characteristic polynomial. Diagonalization. Introduction to Jordan canonical form.
UNIT 5. BILINEAR AND QUADRATIC FORMS.
Bilinear forms. Associated matrix of a bilinear form. Orthogonality. Non-degenerated forms. Orthogonal basis. Sylvester's law of inertia. Quadratic forms.
UNIT 6. EUCLIDEAN SPACES.
Inner product and norm. Orthonormality. Orthogonal subspaces. Some special endomorphisms. Isometries.
UNIT 7. AFFINE GEOMETRY
Affine structure of R^n. Affine subspaces. Intersection and parallelism. Affine reference system.
UNIT 8. EUCLIDEAN GEOMETRY
Euclidean affine structure of R^n. Perpendicularity. Distances and angles. Euclidean affine geometry of the plane and the space.
UNIT 9. GEOMETRIC TRANSFORMATIONS.
Affine transformations. Translations. Homotecies. Symmetries. Proyections. Rotations. Movements and similarities. Movements in the plane and the space.
UNIT 10. INTRODUCTION TO CONICS AND QUADRATICS.
Geometric elements of the conics. Reduction equations of the conics. Reduction equations of the quadratics.
MethodologyToggle Navigation
Using the lecture methodology, the theoretical sessions will be presented in the master sessions, following the basic references contained in the Bibliography and the mandatory material. These lectures will be complemented with problem-solving classes in the practical classroom. These will be proposed to the students to solve questions in which the knowledge acquired in the theoretical classes is applied. Finally, in the seminar sessions, students will take a more active role and develop issues and representative examples of the content of the subject.
Assessment systemsToggle Navigation
- Continuous Assessment System
- Final Assessment System
- Tools and qualification percentages:
- See Guidelines and resignation (%): 100
Ordinary Call: Orientations and DisclaimerToggle Navigation
A final written examination will be taken on the subject taught in class on the date set in the official examination calendar of the Faculty corresponding to the regular May-June evaluation. This exam will be on the second of the dates assigned in the May-June calendar for the course. This examination will evaluate the level of acquisition of all the skills associated with the subject.
In addition, in order for students to be able to measure their progress in learning the subject, two partial exams are scheduled to take place in the official exam period in January and May-June, respectively. Both partial exams will be written. The first of the partial exams will cover the content explained in the first term of the course (weeks 1-15). The second partial exam will evaluate the acquisition of the competences associated to the content explained during the second term (weeks 16-30) and will take place on the first of the dates assigned to the course in the official May-June exam calendar. Students who pass one of the two partial exams or both partial exams will not have to take the exam on the content they have passed in the final exam of the ordinary evaluation.
CONTINUOUS EVALUATION:
PERCENTAGES OF THE MARKS
Written exam: 80%-100%
Oral exhibitions: 0%-5%
Submitted exercises and problems: 0%-15%
To apply the mentioned percentages the minimum mark in the written exam would be 4 over 10.
NON-CONTINUOUS EVALUATION: Final written exam 100%
Extraordinary Call: Orientations and DisclaimerToggle Navigation
A final written examination will be taken on the subject taught in class (weeks 1-30) on the date set in the official examination calendar of the Faculty corresponding to the extraordinary evaluation.
Final written exam: 100%
Compulsory materialsToggle Navigation
Classroom notes. Exercise and problem sheets.
BibliographyToggle Navigation
Basic bibliography
M. CASTELLET e I. LLERENA, Álgebra Lineal y Geometría, Reverté, 2000.
M. EIE, S. CHANG, A first course in linear algebra, World Scientific, 2016.
E. HERNÁNDEZ, M.J. VÁZQUEZ y M.A. ZURRO, Álgebra Lineal y Geometría, Pearson, 2012.
P. PETERSEN, Linear algebra, Springer-Verlag, 2012.
A. SHELDON, Aljebra Lineala ondo egina, Euskal Herriko Unibertsitateko Argitalpen Zerbitzua, UPV/EHU, 2017.
A. SHELDON, Linear Algebra Done Right, Springer International Publishing, 2015.
G. STRANG, Introduction to Linear Algebra, 5th ed. Wellesley-Cambridge Press, 2016.
A. VERA y P. ALEGRIA, Problemas de Geometría Analítica y Formas Bilineales. Murcia,1993.
A. VERA y J.M. ARREGI, Aljebra Lineala eta Geometria I, Ed. AVL, Bilbao 1998.
A. VERA, J.L. HERNANDO y F.J. VERA, Problemas de Algebra I, Ed. Ellacuria, Bilbao 1986.
A. VERA y F.J. VERA, Introducción al Álgebra. Ed. Ellacuria, Bilbao 1984.
In-depth bibliography
R. BENAVENT, Cuestiones sobre Álgebra Lineal, Paraninfo, 2011.
J. DE BURGOS, Álgebra lineal y Geometría cartesiana, MacGraw-Hill, 2006.
J. DE BURGOS, Test y Problemas Álgebra, García-Maroto Editores, 2011.
W. H. GREUB, Linear Algebra, Springer-Verlag, 1981.
I.M. GUELFAND, Lecciones de Álgebra Lineal, Servicio Editorial de la Universidad del País Vasco, 1986.
E. HERNÁNDEZ, Álgebra y Geometría, Addison Wesley, 1999.
J. IKRAMOV, Problemas de Álgebra Lineal, Mir, 1990.
I.V. PROSKURIAKOV, Problemas de Álgebra Lineal, Mir, 1986.
Web addresses
https://ocw.ehu.eus/file.php/133/algebra/Course_listing.html
http://ocw.ehu.es/course/view.php?id=212
http://ocw.ehu.es/course/view.php?id=43
https://ocw.ehu.eus/course/view.php?id=343
http://ocw.ehu.es/ciencias-experimentales/introduccion-al-algebra-lineal/Course_listing
http://math.about.com/od/linearalgebra/Linear_Algebra_Help_and_Tutorials.htm
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