The objective of the subject is to deepen into some of the topics of linear algebra and geometry that were treated more superficially in the course Linear Algebra and Geometry I (canonical forms, affine, euclidean and proyective geometry, conics and quadrics). Both subjects belong to the same module and have as a common objective to learn main concepts of both linear algebra and affine and euclidean geometry. The aim is to use such knowledge to solve both linear problems using matrices and geometric problems of the plane and the space.



Another purpose of these courses is students to acquire a basic and horizontal training in these matters that allow them to understand and apply such knowledge and skills in multiple interrelated directions. Indeed, most of the contents in them will be used in higher courses.

SPECIFIC COMPETENCIES

M16CM03 - To understand the abstract concept of vector space and related basic concepts (subspaces and quotient spaces, basis and generating systems, linear aplications).

M16CM04 - To be able to diagonalize a matrix and to be able to compute the Jordan form of a matrix.

M16CM05 - To know how to orthogonalize a system of vectors in the euclidean space.

M16CM06 - To know how to diagonalize a quadratic form.

M16CM07 - Operate with points, vectors, distances and angles in the afine and euclidean space.

M16CM08 - To use, adequately, systems of references, subspaces and afine transformations.

M16CM09 - To solve, reasonably, geometric problems in the plane and space.

M16CM10 - To classify isometries in the plane and space determining their type and charactersitic elements.

M16CM11 - Understand the basics of the affine, euclidean and projective geometry.

M16CM12 - To recognise main types of homographies.

M16CM13 - To recognise conics and quadrics and to find their prominent elements.



LEARNING OUTCOMES



- To be able to diagonalize a matrix and to compute the Jordan form of a matrix.

- To know how to orthogonalize a system of vectors in the euclidean space.

- To know how to diagonalize a quadratic form.

- Operate with points, vectors, distances and angles in the afine and euclidean space.

- To use, adequately, systems of references, subspaces and afine transformations.

- To calssify isometries in the plane and space determining their type and charactersitic elements.

- To recognise main types of homographies.

- To recognise conics and quadrics and to find their prominent elements and classify them in the projective, afiine and metric spaces.

- To solve, reasonably, geometric problems in the plane and space.

- To use the suitable computing methods in each geometry.

1. QUOTIENT VECTOR SPACES: Quotient vector space. Bases and dimension. Isomorphism theorem for vector spaces.

2. TRIANGULARIZATION AND JORDAN CANONICAL FORM: Endomorphisms and triangularizable matrices. Generalized fundamental subspaces. Jordan canonical form. Cayley-Hamilton Theorem. Minimal polynomial.

3. DUAL VECTOR SPACES: Dual space. Dual bases. Dual map. Orthogonality.

4. AFFINE EUCLIDEAN SPACES: Euclidean spaces: Orthogonality and duality. Affine spaces. Affine subspaces. Affine reference frames. Barycentric coordinates. Convexity. Affine maps. Affine euclidean spaces. Orthogonal affine subspaces. Clasification of isometries.

5. PROJECTIVE SPACES: Projective space. Homogeneous coordinates. Projective subspaces. Dual projective space. Homographies. Double points and hyperplanes. Main homography types.

6. CONICS AND QUADRICS: Classification of conics and quadrics from the affine, metric and projective viewpoint. Sheaves.

The theoretical sessions will be presented in the lectures, 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 work sessions, in which the knowledge acquired in the theoretical classes will be applied. Finally, in the seminar sessions, students will take a more active role and will develop and discuss representative examples/exercises of the contents of the subject. In order the discussion to be more productive, those exercises will be given to the students in advance so that they can work on them before the seminar.

• Continuous Assessment System
• Final Assessment System
• Tools and qualification percentages:
  • See GUIDELINES (%): 100

Final written exam: %80-%100

Individual and/or group tasks: 0-%20



If any student renounces the continuous evaluation method, the final written exam of the usual call

counts the 100% of the final mark.

Same percentages as in the ordinary call. The qualification of the students who have not previously passed the individual and/or group tasks part will depend solely on the written exam of the extraordinary call. Similarly, if any student renounces the continuous evaluation method, the final written exam of the extraordinary call counts the 100% of the final mark.

Basic bibliography

M. CASTELLET e I. LLERENA, Álgebra Lineal y Geometría, Reverté, 2000.

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.

In-depth bibliography

W. H. GREUB, Linear Algebra, Springer-Verlag, 1981.
S. LANG, Linear Algebra 3rd. ed., Springer-Verlag, 1987.
R. H. WASSERMAN. Tensors & Manifolds, Oxford University Press, 1992.