Course: 2020/2021

Linear Geometry

(18259)

Students are expected to have completed

Fundamentals of Algebra, Linear Algebra, Differential Calculus

ObjectivesFurther information on this link

1. Students have shown that they know and understand the mathematical language and the abstract-rigorous reasoning, as well as to apply them to state and prove precise results in several areas of mathematics.
2. Students have shown that they understand the fundamental results of linear algebra, matrix theory and linear geometry concerning spectral theory of matrices and linear transformations, symmetric and Hermitian matrices, affine spaces and projective geometry.
3. Students are able to use techniques from linear algebra, matrix theory and linear geometry to construct mathematical models of processes that appear in real world applications.
4. Students are able to communicate, in a precise and clear manner, ideas, problems and solutions related to linear algebra, matrix theory and linear geometry to any kind of audience (specialist or not).

Description of contents: programme

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

Learning activities and methodology

1. THEORETICAL-PRACTICAL CLASSES, where the knowledge that the students must acquire is explained and developed. Students will have basic reference texts to facilitate the understanding of the classes and the development of follow up work. The teacher and the students will solve exercises and practical problems, previously suggested by the teacher. There will be mid term tests for evaluating the competences and skills acquired by the students and for helping the students to improve their learning strategies.
2. TUTORING SESSIONS. Individualized attendance for students with a teacher for at least two hours a week.
3. STUDENT INDIVIDUAL OR GROUP WORK. Each student's individualized study, understanding of results and proofs, and exercise and problem solving is fundamental in mathematics, both for learning and for self-evaluation of acquired competences and skills. Solving exercises and problems and discussing theoretical results inside small groups of students is an excellent complementary activity for improving the learning.

Assessment System

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

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

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