Course: 2024/2025

Linear Geometry

(18259)

Requirements (Subjects that are assumed to be known)

Fundamentals of Algebra,
Linear Algebra of the bachelor's degree in Applied Mathematics and Computing (a Linear Algebra course from an engineering degree is not enough),
Differential Calculus

Skills and learning outcomes

CB1. Students have demonstrated possession and understanding of knowledge in an area of study that builds on the foundation of general secondary education, and is usually at a level that, while relying on advanced textbooks, also includes some aspects that involve knowledge from the cutting edge of their field of study.
CB2. Students are able to apply their knowledge to their work or vocation in a professional manner and possess the competences usually demonstrated through the development and defence of arguments and problem solving within their field of study.
CB3. Students have the ability to gather and interpret relevant data (usually within their field of study) in order to make judgements which include reflection on relevant social, scientific or ethical issues.
CB4. Students should be able to communicate information, ideas, problems and solutions to both specialist and non-specialist audiences.
CB5. Students will have developed the learning skills necessary to undertake further study with a high degree of autonomy.
CG1. Students are able to demonstrate knowledge and understanding of concepts in mathematics, statistics and computation and to apply them to solve problems in science and engineering with an ability for analysis and synthesis.
CG2. Students are able to formulate in mathematical language problems that arise in science, engineering, economy and other social sciences.
CG5. Students can synthesize conclusions obtained from analysis of mathematical models coming from real world applications and they can communicate in verbal and written form in English language, in an clear and convincing way and with a language that is accessible to the general public.
CG6. Students can search and use bibliographic resources, in physical or digital support, as they are needed to state and solve mathematically and computationally applied problems arising in new or unknown environments or with insufficient information.
CE1. Students have shown that they know and understand the mathematical language and abstract-rigorous reasoning as well as to apply them to state and prove precise results in several areas in mathematics.
CE3. Students have shown that they understand the fundamental results from linear algebra, linear geometry and discrete mathematics.
RA1. Students must have acquired advanced cutting-edge knowledge and demonstrated indepth understanding of the theoretical and practical aspects of working methodology in the area of applied mathematics and computing."
RA5. Students must know how to communication with all types of audiences (specialized or not) their knowledge, methodology, ideas, problems and solutions in the area of their field of study in a clear and precise way.

Description of contents: programme

1. Bilinear and quadratic forms
2. Positive definite matrices
3. Least squares problems
4. Eigenvalues and eigenvectors: diagonalization of matrices and Schur's triangularization
5. The Jordan canonical form
6. Normal matrices and their spectral theorem
7. The singular value decomposition
8. Affine spaces and their applications
9. Conic sections and quadric surfaces

Learning activities and methodology

LEARNING ACTIVITIES AND METHODOLOGY
THEORETICAL-PRACTICAL CLASSES. [44 hours with 100% classroom instruction, 1.76 ECTS]
Knowledge and concepts students must acquire. Student receive course notes and
will have basic reference texts to facilitate following the classes and carrying
out follow up work. Students partake in exercises to resolve practical problems
and participate in workshops and evaluation tests, all geared towards
acquiring the necessary capabilities.
TUTORING SESSIONS. [4 hours of tutoring with 100% on-site attendance, 0.16 ECTS]
Individualized attendance (individual tutoring) or in-group (group tutoring) for students with a teacher.
STUDENT INDIVIDUAL WORK OR GROUP WORK [98 hours with 0 % on-site, 3.92 ECTS]
FINAL EXAM. [4 hours with 100% on site, 0.16 ECTS]
Global assessment of knowledge, skills and capacities acquired throughout the course.
METHODOLOGIES
THEORY CLASS. Classroom presentations by the teacher with IT and audiovisual
support in which the subject`s main concepts are developed, while providing
material and bibliography to complement student learning.
PRACTICAL CLASS. Resolution of practical cases and problems, posed by the
teacher, and carried out individually or in a group.
TUTORING SESSIONS. Individualized attendance (individual tutoring sessions) or
in-group (group tutoring sessions) for students with a teacher as tutor.

Assessment System

- % end-of-term-examination 50
- % of continuous assessment (assigments, laboratory, practicals...) 50

Calendar of Continuous assessment

Extraordinary call: regulations

Basic Bibliography

- B. Noble, J.W. Daniel. Applied Linear Algebra. Prentice-Hall Int.. 1988
- C.D. Meyer. Matrix Analysis and Applied Linear Algebra. SIAM. 2000
- D.C. Lay, S.R. Lay, J.J. McDonald. Linear Algebra and its Applications. 5th edition. Pearson, 2016
- G. Strang. Introduction to Linear Algebra. Wellesley-Cambridge Press. 2016
- O. Faugeras. Three Dimensional Computer Vision, A Geometric Viewpoint. The MIT Press. 1993
- S.R. García and R.A. Horn. A Second Course in Linear Algebra. Cambridge University Press. 2017

- David Lay, Steven Lay, and Judi McDonald · Linear Algebra and Its Applications, EBook, Global Edition : https://ebookcentral.proquest.com/lib/bibliouc3m-ebooks/detail.action?pq-origsite=primo&docID=5174425

Additional Bibliography

- E. Outerelo Domínguez y J.M. Sánchez Abril. Nociones de Geometría Proyectiva. Sanz y Torres. 2009
- P. Lancaster and M. Tismenetsky. The Theory of Matrices with Applications, 2nd edition. Academic Press, Inc.. 1985
- R.A. Horn and C.R. Johnson. Matrix Analysis, 2nd edition. Cambridge University Press. 2013

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The course syllabus may change due academic events or other reasons.