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.

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.

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.

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%

Classroom notes. Exercise and problem sheets.

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.