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The main originality of this course consists in that all discussed problems of linear algebra are solved using a single algorithm, that gives the orthogonal subspace of a linear subspace and its complementary subspace. This permits analyzing all problems from the orthogonality point of view, which is very reach. For example, the problem of determining whether or not a vector belongs to a subspace or the intersection of two subspaces are solved by looking to them as orthogonalization problems. The algorithm permits inverting a matrix, calculating its determinant or determining its rank very easily. In addition the problems of updating inverses and determinants when changing a row reduce to a single step of the algorithm. The compatibility of systems of equations and the obtention of all its solutions or detecting infeasibility are also a direct application of the algorithm. In addition, all subsystems of a given linear system can be solved without extra calculations. Finally, some examples of illustrative applications are given
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In this first block the presentation of the G9 course will be made, exposing the block structure of the same and its sequential planning.
An algebra course based on orthogonality.
Enrique Castillo
An introduction to linear algebra
In this first block we will study the orthogonalization algorithm, which although designed to obtain the subspace orthogonal to a given subspace and its complement subspace, will be used later to solve all the linear algebra problems that we will see in this course.
The orthogonalization algorithm
Orthogonal subspaces and complements
Elemental transformations of matrices
This second block is focused on the algebraic applications of the algorithm, which include: the calculation of matrix inverses and their updating when changing rows, the calculation of the determinant, the calculation of the rank of a matrix and a base of a linear subspace, the membership of a vector to a linear subspace and the intersection of subspaces.
How to obtain the inverse of a matrix
Rank of a matrix
How to determine if a vector belongs to a linear subspace
Intersection of two subspaces
This block focuses on linear systems of equations, 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.
Homogeneous linear systems of equations
Solving complete linear systems of equations
Compatibility of a linear system of equations
In order to motivate and illustrate the power of the orthogonalization algorithm and algebraic applications, this block is dedicated to the presentation of applications to engineering, such as supply networks, traffic networks, information networks, etc. inclined with masses and pulleys, and electrical circuits.
Water supply network example
Examples of linear systems of equations
The water supply problem
This block focuses on providing 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.
Exam model 2015
Exam model 2016
Exam model 2017
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.
How to use the orthogonalization and the dual cone gamma algorithms
Bibliography
Examples of data files
Orthogonalization algorithm
Here are all slideshow from Algebra I
Slideshows