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Educational guide | |||||||||||||||||||||||||||||||||||||||
IDENTIFYING DATA | 2024_25 | |||||||||||||||||||||||||||||||||||||||
Subject | LINEAR ALGEBRA AND GEOMETRY | Code | 00708001 | |||||||||||||||||||||||||||||||||||||
Study programme |
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Descriptors | Credit. | Type | Year | Period | ||||||||||||||||||||||||||||||||||||
6 | Basic Training | First | First |
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Language |
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Prerequisites | ||||||||||||||||||||||||||||||||||||||||
Department | MATEMATICAS |
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Coordinador |
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darim@unileon.es mjpism@unileon.es |
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Lecturers |
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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 |
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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 |
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Apply knowledge of Linear Algebra and Geometry. | A18145 |
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Understand how to develop algorithms and rudiments of numerical algorithms. | A18145 |
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Understand advanced mathematical concepts, and the ability to read and understand engineering mathematics texts. | A18145 |
C5 |
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Ability to transmit information, ideas, problems and solutions to both specialized and non-specialized audiences. | B5645 |
C4 |
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Learn autonomously but guided. | C5 |
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Solve problems with initiative, decision making, showing creativity, and critical and self-critical reasoning. | B5635 B5644 |
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Work in a multilingual and multidisciplinary environment. | B5641 |
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Analyze linear mathematical problems in Engineering and synthesize resolution methods. | B5643 |
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Interpret results of Linear Algebra and Geometry with initiative, creativity, and critical and self-critical reasoning. | A18145 |
B5643 B5644 |
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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 |
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 |
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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 |
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Complementary |
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Recommendations |
Subjects that are recommended to be taken simultaneously | ||
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Other comments | |
It is recommended to be fluent with the contents related to Linear Algebra and Geometry taught in High School. |