Educational guide
IDENTIFYING DATA 2024_25
Subject LINEAR ALGEBRA AND GEOMETRY Code 00708001
Study programme
0708 - GRADO EN INGENIERÍA MECÁNICA
Descriptors Credit. Type Year Period
6 Basic Training First First
Language
Castellano
Prerequisites
Department MATEMATICAS
Coordinador
ARIAS MOSQUERA , DANIEL
E-mail darim@unileon.es
mjpism@unileon.es
Lecturers
ARIAS MOSQUERA , DANIEL
PISABARRO MANTECA , MARÍA JESÚS
Web http://
General description This subject provides the basic theoretical/practical knowledge of linear algebra (systems of linear equations, matrices, determinants, vector spaces, linear maps, diagonalization), as well as those related to affine and Euclidean geometry and the study of conics and quadrics. Some applications of these contents to engineering are studied.
Tribunales de Revisión
Tribunal titular
Cargo Departamento Profesor
Presidente MATEMATICAS FRANCISCO IRIBARREN , ARACELI DE
Secretario MATEMATICAS ARANA SUAREZ , MARIA VICTORIA
Vocal MATEMATICAS SANTAMARIA SANCHEZ , RAFAEL
Tribunal suplente
Cargo Departamento Profesor
Presidente MATEMATICAS GOMEZ PEREZ , JAVIER
Secretario MATEMATICAS SAEZ SCHWEDT , ANDRES
Vocal MATEMATICAS GARCIA FERNANDEZ , ROSA MARTA

Competencias
Code  
A18145
B5635
B5641
B5643
B5644
B5645
C1 CMECES1 That students have demonstrated possession and understanding of knowledge in an area of study that is based on general secondary education, and is usually found at a level that, although supported by advanced textbooks, also includes some aspects that involve knowledge from the cutting edge of their field of study
C4 CMECES4 That students can transmit information, ideas, problems and solutions to both a specialised and non-specialised audience
C5 CMECES5 That students have developed those learning skills necessary to undertake further studies with a high degree of autonomy

Learning aims
Competences
Solve mathematical problems that may arise in engineering. C1
Apply knowledge of Linear Algebra and Geometry. A18145
Understand how to develop algorithms and rudiments of numerical algorithms. A18145
Understand advanced mathematical concepts, and the ability to read and understand engineering mathematics texts. A18145
 C5
Ability to transmit information, ideas, problems and solutions to both specialized and non-specialized audiences. B5645
 C4
Learn autonomously but guided. C5
Solve problems with initiative, decision making, showing creativity, and critical and self-critical reasoning. B5635
B5644
Work in a multilingual and multidisciplinary environment. B5641
Analyze linear mathematical problems in Engineering and synthesize resolution methods. B5643
Interpret results of Linear Algebra and Geometry with initiative, creativity, and critical and self-critical reasoning. A18145
 B5643
B5644
Communicate in writing knowledge and reasoning derived from your individual work in a clear and concrete way. B5635
B5645
 C4

Contents
Topic Sub-topic
Topic I: SYSTEMS OF LINEAR EQUATIONS. MATRICES. DETERMINANTS. Sub-topic 1: PRELIMINARIES.
Numerical sets. Fields. Complex numbers.

Sub-topic 2: SYSTEMS OF LINEAR EQUATIONS.
System of linear equations (SLE). Solution of a SLE. Elementary operations. Gaussian elimination method. Discussion of a SLE. Simultaneous SLEs.

Sub-topic 3: MATRICES.
Definition of matrices and special types of matrices. Operations and properties of matrices. Matrices and SLEs. Elementary matrices. Rank of a matrix. Rouché-Frobenius theorem. Inverse matrix (Hermite method).

Sub-topic 4: DETERMINANTS.
Determinant of a matrix. Properties of the determinants. Laplace's rule. Calculation of the inverse of a matrix by determinants. SLEs and determinants (Cramer's rule). Operational comparison between Cramer's rule and the Gaussian elimination method.
Topic II: VECTOR SPACES. Sub-topic 1: VECTOR SPACES.
Definition of vector space. Properties. Definition of vector subspace. System of generators, set of linearly independent vectors and basis of a vector subspace. Dimension of a vector space. Parametric and implicit equations of a vector subspace. Coordinates of a vector in a base. Base change matrix.

Sub-topic 2: LINEAR MAPS.
Definition of linear map. Matrix expression of a linear map. Matrix of the composition of linear maps. Base changes in linear maps. Characterization of injective, suprajective and bijective maps with respect to the range of the matrix. Rotation matrices.
Topic III: MATRIX DIAGONALIZATION. Sub-topic 1: MATRIX DIAGONALIZATION.
Introduction to the problem. Characteristic polynomials, eigenvalues and eigenvectors of a matrix/endomorphism. Algebraic and geometric multiplicity of an eigenvalue. Theorem for the characterization of diagonalizable matrices.
Topic IV: AFFINE AND EUCLIDEAN GEOMETRY. Sub-topic 1: AFFINE SPACE.
Definition of affine space and affine subspace/manifold. Dimension of an affine subspace. Affine subspace generated by a set. Parallelism between affine subspaces. Reference systems. Coordinates of a point in a reference. Reference changes.

Sub-topic 2: AFFINE EUCLIDEAN SPACE.
Definition of bilinear form. Matrix of a bilinear form with respect to a base. Base change in bilinear forms. Orthogonal vectors with respect to a bilinear form. Diagonalization of bilinear forms. Inveriants of a bilinear form. Sylvester's theorem. Scalar product in a vector space. Euclidean vector space. Norm of a vector. Orthonormal base. Scalar product and vector product. Euclidean affine space. Orthonormal reference. Distance between points and between varieties.
Topic V: CONICS AND QUADRICS. Sub-topic 1: CONICS.
Equation of a conic. Matrix expression of a conic. Reference change matrix in conics. Conic reduction method. Classification of conics. Graphic representation of a conic.

Sub-topic 2: QUADRICS.
Equation of a quadric. Matrix expression of a quadric. Reference change matrix in quadrics. Quadric reduction method. Classification of quadrics.

Planning
Methodologies  ::  Tests
  Class hours Hours outside the classroom Total hours
Lecture 24 30 54
 
Problem solving, classroom exercises 24 48 72
 
Tutorship of group 6 9 15
 
Practical tests 6 3 9
 
(*)The information in the planning table is for guidance only and does not take into account the heterogeneity of the students.

Methodologies
Methodologies   ::  
  Description
Lecture Sessions aimed at explaining theoretical aspects and algorithmic techniques in the context of the contents proposed for the subject.
Problem solving, classroom exercises Sessions aimed at applying the theoretical aspects and algorithmic techniques of each block in solving problems.
Tutorship of group The student will have the help of the teacher to resolve any possible doubts regarding the subject.

Personalized attention
 
Lecture
Problem solving, classroom exercises
Tutorship of group
Practical tests
Description
It is advisable to use individualized tutoring sessions with the teacher, where the specific learning difficulties of each student can be addressed. To request a tutoring session, the student must send an email to the teacher.

Assessment
  Description Qualification
Practical tests Se llevará a cabo una evaluación continua del trabajo realizado por el alumno a través de la valoración de dos pruebas escritas, cada una con un peso relativo del 50% de la calificación final. Cada prueba tendrá un 50% del valor de la calificacion final
 
Other comments and second call
Primera convocatoria:

Aproximadamente a la mitad del periodo de clases se realizará una primera prueba escrita en la que se evaluarán los conocimientos adquiridos en la primera mitad del semestre. Su peso sobre la calificación final de la asignatura será de un 50%.

Casi al final del periodo de clases se llevará a cabo una segunda prueba escrita en la que se evaluarán los conocimientos adquiridos en la segunda mitad del semestre. Su peso sobre la calificación final de la asignatura será de otro 50%.

En la fecha destinada a la segunda prueba escrita, los alumnos tendrán la oportunidad de recuperar la primera prueba escrita.

Segunda convocatoria:

El alumno podrá optar por repetir en el examen de recuperación una o ambas pruebas escritas (en caso de no recuperar la parte correspondiente a uno de los bloques, se conservará la nota obtenida en esa parte en la primera convocatoria).

Convocatoria de diciembre:

La calificación en dicha convocatoria se obtendrá exclusivamente de la evaluación de un único examen escrito acerca de los contenidos de la materia.

 

Material no permitido durante el desarrollo de las pruebas de evaluación:

Durante el desarrollo de las pruebas de evaluación queda terminantemente prohibida la tenencia y el uso de dispositivos móviles y/o electrónicos. La simple tenencia de dichos dispositivos así como de apuntes, libros, carpetas o materiales diversos no autorizados durante las pruebas de evaluación, supondrá la retirada inmediata del examen, su expulsión del mismo y su calificación como suspenso, comunicándose la incidencia a la Autoridad Académica del Centro para que realice las actuaciones previstas en las Pautas de Actuación en los Supuestos de Plagio, Copia o Fraude en Exámenes o Pruebas de Evaluación, aprobadas por la Comisión Permanente del Consejo de Gobierno de 29 de enero de 2015.

Sources of information
Access to Recommended Bibliography in the Catalog ULE

Basic J. Burgos , Álgebra Lineal, McGraw Hill. ,
J. Aversú y otros,, Problemas Resueltos de Álgebra Lineal, Thomson ,
F. Puerta, Álgebra Lineal, Universidad Politècnica de Barcelona,
E. Hernández, MªJ. Vázquez, Mª A. Zurro, Álgebra Lineal y Geometría, Pearson,
E. Hernández , Álgebra y Geometría, Addison Wesley.,
G. Strang , ÁlgebraLineal y sus Aplicaciones, Addison Wesley Iberoamericana.,
M. Carriegos, R. Santamaría, Geometría 201, Universidad de León,

Complementary S. Lang, Álgebra Lineal, Addison-Wesley Iberoamericana,
G.A. Jennings, Modern Geometry with Applications, Springer,


Recommendations

Subjects that are recommended to be taken simultaneously
DIFFERENTIAL AND INTEGRAL CALCULUS / 00708002

 
Other comments
It is recommended to be fluent with the contents related to Linear Algebra and Geometry taught in High School.