scholarly journals Linear Representation of Vector Group, Quick Solutions to Maximal Independent Group and Linear Equations in Linear Algebra

2017 ◽  
Author(s):  
Jian-xue YAN ◽  
Yun-qiu WANG
Keyword(s):  
Linear Algebra ◽  
Linear Equations ◽  
Linear Representation ◽  
Independent Group ◽  
Vector Group

scholarly journals Division And Combination In Linear Algebra

2019 ◽  
Vol 17 ◽  
pp. 39-146
Author(s):  
Liang Fang ◽  
Rui Chena
Keyword(s):  
Linear Algebra ◽  
Linear Correlation ◽  
Linear Equations ◽  
Linear Representation ◽  
Deep Levels ◽  
High Dimensions ◽  
Matrix Operation ◽  
The Relationship

In this paper, the relationship between matrix operation, linear equations, linear representation of vector groups and linear correlation is discussed, and the idea of division and combination in linear algebra is discussed to help learners understand the connections between various knowledge points of linear algebra from multiple angles, deep levels, and high dimensions.

Numerical Linear Algebra

1966 ◽  
Vol 9 (05) ◽  
pp. 757-801 ◽  
Author(s):  
W. Kahan
Keyword(s):  
Eigenvalue Problem ◽  
Linear Algebra ◽  
Linear Equations ◽  
Numerical Linear Algebra ◽  
System Of Linear Equations ◽  
Image Position

The primordial problems of linear algebra are the solution of a system of linear equations and the solution of the eigenvalue problem for the eigenvalues λk, and corresponding eigenvectors of a given matrix A.

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.

scholarly journals USING GPU NVIDIA FOR LINEAR ALGEBRA PROLEMS

2019 ◽  
pp. 144-157
Author(s):  
A. Myasishchev ◽  
S. Lienkov ◽  
V. Dzhulii ◽  
I. Muliar
Keyword(s):  
Linear Algebra ◽  
Linear Equations ◽  
Matrix Multiplication ◽  
Performance Comparison ◽  
Double Precision ◽  
Graphics Processors ◽  
Performance Study ◽  
Software Module ◽  
Maximum Acceleration ◽  
Computational Procedures

Research goals and objectives: the purpose of the article is to study the feasibility of graphics processors using in solving linear equations systems and calculating matrix multiplication as compared with conventional multi-core processors. The peculiarities of the MAGMA and CUBLAS libraries use for various graphics processors are considered. A performance comparison is made between the Tesla C2075 and GeForce GTX 480 GPUs and a six-core AMD processor. Subject of research: the software is developed basing on the MAGMA and CUBLAS libraries for the purpose of the NVIDIA Tesla C2075 and GeForce GTX 480 GPUs performance study for linear equation systems solving and matrix multiplication calculating. Research methods used: libraries were used to parallelize the linear algebra problems solution. For GPUs, these are MAGMA and CUBLAS, for multi-core processors, the ScaLAPACK and ATLAS libraries. To study the operational speed there are used methods and algorithms of computational procedures parallelization similar to these libraries. A software module has been developed for linear equations systems solving and matrix multiplication calculating by parallel systems. Results of the research: it has been determined that for double-precision numbers the GPU GeForce GTX 480 and the GPU Tesla C2075 performance is approximately 3.5 and 6.3 times higher than that of the AMD CPU. And the GPU GeForce GTX 480 performance is 1.3 times higher than the GPU Tesla C2075 performance for single precision numbers. To achieve maximum performance of the NVIDIA CUDA GPU, you need to use the MAGMA or CUBLAS libraries, which accelerate the calculations by about 6.4 times as compared to the traditional programming method. It has been determined that in equations systems solving on a 6-core CPU, it is possible to achieve a maximum acceleration of 3.24 times as compared to calculations on the 1st core using the ScaLAPACK and ATLAS libraries instead of 6-fold theoretical acceleration. Therefore, it is impossible to efficiently use processors with a large number of cores with considered libraries. It is demonstrated that the advantage of the GPU over the CPU increases with the number of equations.

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 PEBBLE GAMES AND LINEAR EQUATIONS

2015 ◽  
Vol 80 (3) ◽  
pp. 797-844 ◽  
Author(s):  
MARTIN GROHE ◽  
MARTIN OTTO
Keyword(s):  
Linear Algebra ◽  
Linear Equations ◽  
Graph Isomorphism ◽  
Detailed Account ◽  
Base Level ◽  
Basic Colour ◽  
Higher Dimensional ◽  
Rational Arithmetic ◽  
Pebble Game ◽  
Fractional Isomorphism

AbstractWe give a new, simplified and detailed account of the correspondence between levels of the Sherali–Adams relaxation of graph isomorphism and levels of pebble-game equivalence with counting (higher-dimensional Weisfeiler–Lehman colour refinement). The correspondence between basic colour refinement and fractional isomorphism, due to Tinhofer [22; 23] and Ramana, Scheinerman and Ullman [17], is re-interpreted as the base level of Sherali–Adams and generalised to higher levels in this sense by Atserias and Maneva [1] and Malkin [14], who prove that the two resulting hierarchies interleave. In carrying this analysis further, we here give (a) a precise characterisation of the level k Sherali–Adams relaxation in terms of a modified counting pebble game; (b) a variant of the Sherali–Adams levels that precisely match the k-pebble counting game; (c) a proof that the interleaving between these two hierarchies is strict. We also investigate the variation based on boolean arithmetic instead of real/rational arithmetic and obtain analogous correspondences and separations for plain k-pebble equivalence (without counting). Our results are driven by considerably simplified accounts of the underlying combinatorics and linear algebra.

Solution of Linear Equations With Coefficients and Right-Hand Members in An Arithmetic Sequence

1977 ◽  
Vol 70 (2) ◽  
pp. 170-172
Author(s):  
Murli M. Gupta
Keyword(s):  
General Solution ◽  
Linear Algebra ◽  
Linear Equations ◽  
Arithmetic Sequence ◽  
Right Hand

A general solution of a problem in linear algebra.

scholarly journals Semisymmetric Zp-covers of the C20 graph

2021 ◽  
Vol 31 (2) ◽  
pp. 286-301
Author(s):  
A. A. Talebi ◽  
◽  
N. Mehdipoor ◽  
Keyword(s):  
Linear Algebra ◽  
Homology Group ◽  
Invariant Subspaces ◽  
Linear Representation ◽  
Matrix Groups ◽  
Semisymmetric Graph ◽  
Vertex Set ◽  
Prime Fields ◽  
Abelian Covering

A graph X is said to be G-semisymmetric if it is regular and there exists a subgroup G of A:=Aut(X) acting transitively on its edge set but not on its vertex set. In the case of G=A, we call X a semisymmetric graph. Finding elementary abelian covering projections can be grasped combinatorially via a linear representation of automorphisms acting on the first homology group of the graph. The method essentially reduces to finding invariant subspaces of matrix groups over prime fields. In this study, by applying concept linear algebra, we classify the connected semisymmetric zp-covers of the C20 graph.

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

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