Linear Geometry

1.-Define linear varieties as an abstraction of the notions of line and plane from elementary geometry. Study the problems of incidence, parallelism, and relative positions of linear varieties.

2.-Define the concept of affine reference and introduce coordinates. Solve classical geometric incidence problems by introducing coordinates. Calculate the linear equations of a linear variety.

3.-Study affinities and the affine group.

4.-Study Euclidean spaces. Define the concept of the length of a vector. Prove the existence of orthonormal bases and learn to calculate them by various methods: Gram-Schmidt, diagonalization by congruence, spectral theorem. Classify orthogonal transformations in the plane and in three-dimensional space.

5.-Study Euclidean affine spaces. Use the structure of a Euclidean vector space to define geometric concepts such as perpendicularity and distance between linear varieties. Introduce references and rectangular coordinates. Calculate the distance between linear varieties in a Euclidean affine space. Study the movements between Euclidean affine spaces. Learn to classify movements by giving their geometric elements and, conversely, obtain the equation of a given movement in geometric terms.

6.-Study geometric loci in the Euclidean affine plane, such as: the circle, the ellipse, the hyperbola, and the parabola. Define the concepts of affine conics and quadrics. Calculate the reduced equation of a real conic or quadric and the principal axes. Classify a real or complex conic or quadric by its metric invariants and by its affine invariants.

1.AFFINE SPACE.

1.1. Affine space over a vector space. Linear varieties. Incidence and parallelism. Relative positions. (8 lecture hours)

1.2. Affine references: coordinates. Change of coordinates. Equations of linear varieties. (5 lecture hours)

1.3. Affine applications. Affinities. Affine group. Determination of an affinity. Equation of an affinity. (6 lecture hours)


2.EUCLIDEAN SPACES.

2.1. Lengths. Orthonormal bases. Gram-Schmidt orthogonalization method. Vector product. (3 lecture hours)

2.2. Orthogonal transformations: Classification. (7 lecture hours)


3.EUCLIDEAN AFFINE SPACES.

3.1. Perpendicularity and distances. (2 lecture hours)

3.2. Rectangular references: rectangular coordinates. (1 lecture hour)

3.3. Movements: classification. (3 lecture hours)


4.CONICS AND QUADRICS.

4.1. Geometric loci in the Euclidean affine plane: circle, ellipse, parabola, and hyperbola. (1 lecture hour)

4.2. Metric classification of real conics and quadrics. (4 lecture hours)

4.3. Affine conics and quadrics: affine classification of conics and quadrics. (2 lecture hours)

Basic Bibliography:

De Burgos J. Algebra lineal y geometría cartesiana. Ed. MacGraw-Hill, Madrid, 1999.

Castellet, M., & Llerena, I. (1991). Álgebra lineal y geometría. Barcelona: Editorial Reverté. ISBN: 978-84-291-5009-4.

Golovina L. I; Álgebra lineal y algunas de sus aplicaciones. Ed. Mir, 1980.

Hernández, E. Álgebra y geometría. Ed. Addison Wesley, Madrid, 1994.

Hernández, E.; Vazquez, M. J.; Zurro M. A., Álgebra lineal y geometría. Ed. Pearson, Madrid, 2012.


Supplementary Bibliography:

Gruenberg, K. W.; Weir, A. J. Linear Geometry. Graduate texts in Mathematics. Ed.
Springer-Verlag, New York, 1977.

Kostrikin, A. I.; Manin Yu. I., Linear algebra and geometry. Ed. Gordon and Breach, New York, 1989.

Snapper, E., Troyer, R. J. Metric affine geometry . Aademic Press, Inc, London, 1971.

Acquire the competencies listed in the USC Mathematics degree syllabus, and more specifically, the following: CG3, CG4, CG5, CE1, CE3, CE4, CT1, CT2, CT3, and CT5.

The lectures and interactive classes will be held in person. The weekly distribution of the course will be as follows: 3 hours of lectures per group and 1 hour of an interactive laboratory class for each of the 6 groups into which the course is divided.

The lectures will focus on presenting the fundamental content of the course. The theoretical exposition will be supplemented with examples, and some of the problems previously assigned to students will also be solved.

The interactive laboratory classes will serve to illustrate the theoretical and practical content of the course. In the very small group tutorials, students will receive support in discussing problems related to the exercises and in resolving any doubts concerning the course material.

A virtual course will be available, including detailed notes covering all the course material.

Throughout the course, students will be expected to solve exercises and actively participate in the interactive laboratory classes. Tutorials can be held in person or via Teams, the virtual campus, or email.

The evaluation system will be coordinated across both groups of the course.

The assessment criteria involve a combination of continuous assessment and a final exam. The final exam will take place on the date set by the Faculty of Mathematics and will be the same for all students enrolled in the course.

Continuous assessment will consist of the individual completion of one test, which will be the same for all groups.

The final grade will be calculated based on the continuous assessment (CA) and the final written exam (FE), using the formula:
Final Grade = MAX{30% CA + 70% FE, FE}

The grade obtained from the continuous assessment will apply to both exam sessions within the same academic year (second semester and July). If a student does not attend the final exam in either session, the grade will be recorded as "Not Presented" (NP), even if the student completed the continuous assessment.

In cases of fraudulent completion of exercises or tests, the provisions of the Regulations on the Evaluation of Students' Academic Performance and Grade Review will apply.
Article 16. Fraudulent completion of exercises or tests:
The fraudulent completion of any exercise or test required for course assessment will result in a failing grade for the corresponding exam session, regardless of any disciplinary proceedings that may be initiated. Fraudulent acts include, among others, the submission of plagiarized work or work obtained from publicly accessible sources without adequate reworking or reinterpretation and without proper citation of authors and sources.

Lecture classes: 42 hours

Laboratory classes: 14 hours

Small group tutorials: 2 hours

Student's personal work time (non-presential): 92 hours

Total: 150 hours

Study regularly using the course notes available on the virtual platform, and work through the questions and exercises posted there. Make use of the tutorials as soon as you encounter any difficulties.