Linear Algebra and Systems of Linear Equations

2021 ◽  
pp. 235-263
Author(s):  
Qingkai Kong ◽  
Timmy Siauw ◽  
Alexandre M. Bayen
Keyword(s):  
Linear Algebra ◽  
Linear Equations ◽  
Systems Of Linear Equations

An attempt to reduce learning difficulties in Linear Algebra courses

2016 ◽  
Vol 8 (2) ◽  
pp. 156
Author(s):  
Marta Graciela Caligaris ◽  
María Elena Schivo ◽  
María Rosa Romiti
Keyword(s):  
Linear Algebra ◽  
Linear Equations ◽  
Learning Difficulties ◽  
Vector Spaces ◽  
Analytic Geometry ◽  
Teaching Activities ◽  
Interactive Tools ◽  
Systems Of Linear Equations

In engineering careers, the study of Linear Algebra begins in the first course. Some topics included in this subject are systems of linear equations and vector spaces. Linear Algebra is very useful but can be very abstract for teaching and learning.In an attempt to reduce learning difficulties, different approaches of teaching activities supported by interactive tools were analyzed. This paper presents these tools, designed with GeoGebra for the Algebra and Analytic Geometry course at the Facultad Regional San Nicolás (FRSN), Universidad Tecnológica Nacional (UTN), Argentina.

The SVD-Fundamental Theorem of Linear Algebra

2006 ◽  
Vol 11 (2) ◽  
pp. 123-136 ◽  
Author(s):  
A. G. Akritas ◽  
G. I. Malaschonok ◽  
P. S. Vigklas
Keyword(s):  
Linear System ◽  
Linear Algebra ◽  
Fundamental Theorem ◽  
Gaussian Elimination ◽  
Linear Equations ◽  
Orthogonal Complement ◽  
Schmidt Method ◽  
Orthonormal Bases ◽  
Svd Analysis ◽  
Systems Of Linear Equations

Given an m × n matrix A, with m ≥ n, the four subspaces associated with it are shown in Fig. 1 (see [1]). Fig. 1. The row spaces and the nullspaces of A and AT; a1 through an and h1 through hm are abbreviations of the alignerframe and hangerframe vectors respectively (see [2]). The Fundamental Theorem of Linear Algebra tells us that N(A) is the orthogonal complement of R(AT). These four subspaces tell the whole story of the Linear System Ax = y.  So, for example, the absence of N(AT) indicates that a solution always exists, whereas the absence of N(A) indicates that this solution is unique. Given the importance of these subspaces, computing bases for them is the gist of Linear Algebra. In “Classical” Linear Algebra, bases for these subspaces are computed using Gaussian Elimination; they are orthonormalized with the help of the Gram-Schmidt method. Continuing our previous work [3] and following Uhl’s excellent approach [2] we use SVD analysis to compute orthonormal bases for the four subspaces associated with A, and give a 3D explanation. We then state and prove what we call the “SVD-Fundamental Theorem” of Linear Algebra, and apply it in solving systems of linear equations.

scholarly journals PENGEMBANGAN BAHAN AJAR ALJABAR LINEAR ELEMENTER BERDASARKAN KEMAMPUAN KONEKSI MATEMATIS

2020 ◽  
Vol 11 (2) ◽  
pp. 217
Author(s):  
DONA FITRIAWAN
Keyword(s):  
Linear Algebra ◽  
Linear Equations ◽  
Teaching Material ◽  
Teaching Materials ◽  
Analysis Techniques ◽  
Stage Of Development ◽  
Systems Of Linear Equations ◽  
Development Approach ◽  
Mathematical Connection ◽  
Expert Validation

The purpose of this study is to develop: 1. elementary linear algebra teaching materials based on mathematical connection skills; 2. syllabus and lecture plan; 3. test mathematical connection skills. This type of research is a research and development approach whose research design consists of four stages, namely defining, planning, developing, and dissiminating. Data analysis techniques in this study describe narratively the steps in developing teaching materials. Based on the results of the analysis of the data obtained that: 1) the stages of developing teaching materials starting from the stages of defining, designing, until the first stage of development, namely expert validation. From this stage of development a revised elementary linear algebra teaching material has been produced based on input from three validators. Teaching materials compiled consist of four materials, namely systems of linear equations, matrices, inverses, and matrix determinants; 2) based on the opinions of three experts, elementary linear algebra teaching materials that have been compiled are classified as valid and good in terms of accuracy of contents, digestibility, use of language, so that they can be used to develop mathematical connection skills.AbstrakTujuan dari penelitian ini adalah untuk mengembangkan: 1. bahan ajar aljabar linier dasar berdasarkan keterampilan koneksi matematika; 2. silabus dan rencana kuliah; 3. menguji keterampilan koneksi matematika. Jenis penelitian ini adalah pendekatan penelitian dan pengembangan yang desain penelitiannya terdiri dari empat tahap, yaitu mendefinisikan, merencanakan, mengembangkan, dan menyebarluaskan. Teknik analisis data dalam penelitian ini menggambarkan secara naratif langkah-langkah dalam mengembangkan bahan ajar. Berdasarkan hasil analisis data diperoleh bahwa: 1) tahap pengembangan bahan ajar mulai dari tahap pendefinisian, perancangan, hingga tahap pertama pengembangan, yaitu validasi ahli. Dari tahap pengembangan ini bahan ajar aljabar linier revisi telah dihasilkan berdasarkan masukan dari tiga validator. Bahan ajar yang disusun terdiri dari empat bahan, yaitu sistem persamaan linear, matriks, invers, dan determinan matriks; 2) berdasarkan pendapat tiga ahli, bahan ajar aljabar linier dasar yang telah disusun diklasifikasikan sebagai valid dan baik dalam hal keakuratan isi, kecernaan, penggunaan bahasa, sehingga dapat digunakan untuk mengembangkan keterampilan koneksi matematis.Kata Kunci: bahan ajar, kemampuan koneksi matematis

On Some Properties of Semi-rings

2018 ◽  
pp. 35-50
Author(s):  
A. I. Belousov
Keyword(s):  
Linear Equations ◽  
Oriented Graph ◽  
Theoretical Computer Science ◽  
Alternative Methods ◽  
Main Role ◽  
Cost Matrix ◽  
Sequential Elimination ◽  
The Matrix ◽  
Systems Of Linear Equations ◽  
The Cost

The main objective of this paper is to prove a theorem according to which a method of successive elimination of unknowns in the solution of systems of linear equations in the semi-rings with iteration gives the really smallest solution of the system. The proof is based on the graph interpretation of the system and establishes a relationship between the method of sequential elimination of unknowns and the method for calculating a cost matrix of a labeled oriented graph using the method of sequential calculation of cost matrices following the paths of increasing ranks. Along with that, and in terms of preparing for the proof of the main theorem, we consider the following important properties of the closed semi-rings and semi-rings with iteration.We prove the properties of an infinite sum (a supremum of the sequence in natural ordering of an idempotent semi-ring). In particular, the proof of the continuity of the addition operation is much simpler than in the known issues, which is the basis for the well-known algorithm for solving a linear equation in a semi-ring with iteration.Next, we prove a theorem on the closeness of semi-rings with iteration with respect to solutions of the systems of linear equations. We also give a detailed proof of the theorem of the cost matrix of an oriented graph labeled above a semi-ring as an iteration of the matrix of arc labels.The concept of an automaton over a semi-ring is introduced, which, unlike the usual labeled oriented graph, has a distinguished "final" vertex with a zero out-degree.All of the foregoing provides a basis for the proof of the main theorem, in which the concept of an automaton over a semi-ring plays the main role.The article's results are scientifically and methodologically valuable. The proposed proof of the main theorem allows us to relate two alternative methods for calculating the cost matrix of a labeled oriented graph, and the proposed proofs of already known statements can be useful in presenting the elements of the theory of semi-rings that plays an important role in mathematical studies of students majoring in software technologies and theoretical computer science.

scholarly journals A Globally Convergent Parallel SSLE Algorithm for Inequality Constrained Optimization

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Zhijun Luo ◽  
Lirong Wang
Keyword(s):  
Constrained Optimization ◽  
Optimization Problems ◽  
Linear Equations ◽  
Coefficient Matrix ◽  
Descent Direction ◽  
Constrained Optimization Problems ◽  
Inequality Constrained Optimization ◽  
Globally Convergent ◽  
Variable Distribution ◽  
Systems Of Linear Equations

A new parallel variable distribution algorithm based on interior point SSLE algorithm is proposed for solving inequality constrained optimization problems under the condition that the constraints are block-separable by the technology of sequential system of linear equation. Each iteration of this algorithm only needs to solve three systems of linear equations with the same coefficient matrix to obtain the descent direction. Furthermore, under certain conditions, the global convergence is achieved.

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