Algebra II (English)

In this first block the presentation of the course will be made, exposing the block structure of the same and its sequential planning.

In this first block the polytope and cone algebraic structures are presented, showing some illustrative examples that facilitate their understanding. The concept of a dual cone of a given cone is also presented. As a fundamental element for the development of the course, the algorithm for obtaining the dual cone of a given cone is described, which although designed to obtain the dual cone, will serve as the basis for solving all the other problems that arise in the course. It is shown that the vector space is a particular case of cone and that it is enough to add one more generator to the base in order to express the vectors of the vector space as cone, that is, generated by nonnegative linear combinations. Finally, the standard form of a cone is defined as the sum of its components of vector space and acute cone, which allows expressing the cone in its minimal form.

This second block is focused on the algebraic applications of the algorithm, which include: the problem of the membership of a vector to a cone and the intersection of two cones. The first of these is a complex problem because it involves nonnegative linear combinations, that is, systems of linear inequations, whose solutions are not usually known by students or studied in standard algebra courses. The use of the algorithm, explained in the previous block, allows to solve the problem in a very elegant way and avoid having to solve these systems. The intersection of cones is another problem of some complexity, since it also implies the systems of inequations, solving it in an ingenious way. It can also be resolved by realizing that such an intersection is the dual cone of the dual of one of them in the other, which directly suggests how to solve it.

This block focuses on linear inequation systems, including homogeneous and complete systems. It also explains how all the subsystems of a given system can be solved simultaneously, as well as how to analyze the compatibility of a system, that is, whether or not it has a solution.
Content

In order to motivate and illustrate the power of the algorithm of orthogonalization and algebraic applications, this block is dedicated to the presentation of applications to engineering, such as water supply networks, although its application to other networks, such as of traffic, information, etc., is identical. It also includes applications to the problem of inclined planes with masses and pulleys, and electrical circuits.

This block provides a series of standard exams with their solutions, to facilitate that the student can check if he has understood the material explained in the different topics of the course. Correspond to the type of exam that we have used at the University of Castilla-La Mancha for 20 years and that have been shown as very satisfactory.

In order to facilitate the use of the methods described in this course and students can work not only the exercises and problems raised, but other applications, we present here a computer application that implements the orthogonalization algorithm. Finally, a list of bibliographical references is given.

Here are all slideshow from Algebra I