In this paper an approximate and exact methods for solving a system of linear equations with an arbitrary number of equations and the same number of unknowns is presented. All the mentioned problems can be avoided by the proposed methods.
It is possible to define an algorithm that does not solve the system of equations in the usual mathematical way, but still finds its exact solution in the exact number of steps already defined. The methods consist of simple computations that are not cumulative. At the same time, the number of operations is acceptable even for a relatively large number of equations and unknowns. In addition, the algorithms allows the process to start from an arbitrary initial n-tuple and always leads to the exact solution if it exists.
Aleksa Srdanov was born in 1958 in Ruma, the Republic of Serbia. He graduated from the Faculty of Mathematics in Beogard in 1981 and worked in scence until 2010. His narrow interests are: number theory, multidimensionality, artificial intelligence and philosophy of natural sciences. His work is based on the finding and using some invariant properties of the subject of research and therefore without a doubt, bit can be said that his approach is cybernetic. In the theory of numbers, he worked on the problem of onumber of partitions and the general form of the partition function. This monograph is a result of his investigation of the invariant properties of all dimensional spaces. He has also published in philosophy and works related to problems of infinity, as well as problems of undecidability.
Aleksandra M. Jankovic was born in Pozarevac, Serbia, in 1983. He studied at University of Kragujevac, Mathematical Science and completed doctoral thesis in Mechanical Engineering, University of Belgrade. He has published 12 papers and one monograph, and is currently working at Technical College of Pozarevac.
The workshop on Algorithms in Algebraic Geometry that was held in the framework of the IMA Annual Program Year in Applications of Algebraic Geometry by the Institute for Mathematics and Its Applications on September 18-22, 2006 at the University of Minnesota is one tangible indication of the interest. One hundred ten participants from eleven countries and twenty states came to listen to the many talks; discuss mathematics; and pursue collaborative work on the many faceted problems and the algorithms, both symbolic and numberic, that illuminate them.
This volume of articles captures some of the spirit of the IMA workshop.
The book uses classical and ad hoc tools, but a unifying theme is the elementary bidiagonal factorization, which has emerged as the single most important tool for this particular class of matrices. Recent work has shown that bidiagonal factorizations may be viewed in a succinct combinatorial way, leading to many deep insights. Despite slow development, bidiagonal factorizations, along with determinants, now provide the dominant methodology for understanding total nonnegativity. The remainder of the book treats important topics, such as recognition of totally nonnegative or totally positive matrices, variation diminution, spectral properties, determinantal inequalities, Hadamard products, and completion problems associated with totally nonnegative or totally positive matrices. The book also contains sample applications, an up-to-date bibliography, a glossary of all symbols used, an index, and related references.
The solution of large and sparse linear systems is the most time-consuming part for most of the scientific computing simulations. Indeed, mathematical models become more and more accurate by including a greater volume of data, but this requires the solution of larger and harder algebraic systems. In recent years, research has focused on the efficient solution of large sparse and/or structured systems generated by the discretization of numerical models by using iterative solvers.
The third edition contains major improvements and revisions throughout the book. More than 300 new exercises have been added since the previous edition. Many new examples have been added to illustrate the key ideas of linear algebra. New topics covered in the book include product spaces, quotient spaces, and dual spaces. Beautiful new formatting creates pages with an unusually pleasant appearance in both print and electronic versions.
No prerequisites are assumed other than the usual demand for suitable mathematical maturity. Thus the text starts by discussing vector spaces, linear independence, span, basis, and dimension. The book then deals with linear maps, eigenvalues, and eigenvectors. Inner-product spaces are introduced, leading to the finite-dimensional spectral theorem and its consequences. Generalized eigenvectors are then used to provide insight into the structure of a linear operator.
Your hands-on guide to real-world applications of linear algebra
Does linear algebra leave you feeling lost? No worries —this easy-to-follow guide explains the how and the why of solving linear algebra problems in plain English. From matrices to vector spaces to linear transformations, you'll understand the key concepts and see how they relate to everything from genetics to nutrition to spotted owl extinction.Line up the basics — discover several different approaches to organizing numbers and equations, and solve systems of equations algebraically or with matrices Relate vectors and linear transformations — link vectors and matrices with linear combinations and seek solutions of homogeneous systems Evaluate determinants — see how to perform the determinant function on different sizes of matrices and take advantage of Cramer's rule Hone your skills with vector spaces — determine the properties of vector spaces and their subspaces and see linear transformation in action Tackle eigenvalues and eigenvectors — define and solve for eigenvalues and eigenvectors and understand how they interact with specific matrices
Open the book and find:Theoretical and practical ways of solving linear algebra problems Definitions of terms throughout and in the glossary New ways of looking at operations How linear algebra ties together vectors, matrices, determinants, and linear transformations Ten common mathematical representations of Greek letters Real-world applications of matrices and determinants
The book is intended for advanced undergraduates in math, applied math, engineering, or science disciplines, as well as for researchers and professionals looking for an introduction to a subject they missed or overlooked in their education.?