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A Practical Approach to LINEAR ALGEBRA A Practical Approach to LINEAR ALGEBRA Prabhat Choudhary 'Oxford Book Company Jaipur. India ISBN: 978-81-89473-95-2 First Edition 2009 Oxford Book Company 267, 10-B-Scheme, Opp. Narayan Niwas, Gopalpura By Pass Road, Jaipur-3020 18 Phone: 0141-2594705, Fax: 0141-2597527 e-mail: [email protected] website: www.oxfordbookcompany.com © Reserved Typeset by: Shivangi Computers 267, lO-B-Scheme, Opp. Narayan Niwas, Gopalpura By Pass Road, Jaipur-3020 18 Printed at : Rajdhani Printers, Delhi All Rights are Reserved. No part ofthis publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic. mechanical, photocopying, recording, scanning or otherwise, without the prior written permission of the copyright owner. Responsibility for the facts stated, opinions expressed, conclusions reached and plagiarism, if any, in this volume is entirely that of the Author, according to whom the matter encompassed in this book has been origmally created/edited and resemblance with any such publication may be incidental. The Publisher bears no responsibility for them, whatsoever. Preface Linear Algebra has occupied a very crucial place in Mathematics. Linear Algebra is a continuation of classical course in the light of the modem development in Science and Mathematics. We must emphasize that mathematics is not a spectator sport, and that in order to understand and appreciate mathematics it is necessary to do a great deal of personal cogitation and problem solving. Scientific and engineering research is becoming increasingly dependent upon the development and implementation of efficient parallel algorithms. Linear algebra is an indispensable tool in such research and this paper attempts to collect and describe a selection of some of its more important parallel algorithms. The purpose is to review the current status and to provide an overall perspective of parallel algorithms for solving dense, banded, or block-structured problems arising in the major areas of direct solution of linear systems, least squares computations, eigenvalue and singular value computations, and rapid elliptic solvers. There is a widespread feeling that the non-linear world is very different, and it is usually studied as a sophisticated phenomenon of Interpolation between different approximately-Linear Regimes. Prabhat Choudhary Contents Preface v 1. Basic Notions 1 .2:' Systems of Linear Equations 26 ,3. Matrics', 50 4. Determinants 101 5. Introduction to Spectral Theory 139 6. Inner Product Spaces 162 7. Structure of Operators in Inner Product Spaces 198 .. 8. Bilinear and Quadratic Forms 221 : 9. Advanced Spectral Theory 234 10. Linear Transformations 252 Chapter 1 Basic Notions VECTOR SPACES A vector space V is a collection of objects, called vectors, along with two operations, addition of vectors and multiplication by a number (scalar), such that the following properties (the so-called axioms of a vector space) hold: The first four properties deal with the addition of vector: 1. Commutativity: v + w = w + v for all v, W E V. 2. Associativity: (u + v) + W = u + (v + w) for all u, v, W E V. 3. Zero vector: there exists a special vector, denoted by 0 such that v + 0 = v for all v E V. 4. Additive inverse: For every vector v E V there exists a vector W E V such that v + W = O. Such additive inverse is usually denoted as -v. The next two properties concern multiplication: 5. Multiplicative identity: 1v = v for all v E V. 6. Multiplicative associativity: (a~)v = a(~v) for all v E Vand all E scalars a, ~. And finally, two distributive properties, which connect multiplication and addition: 7. a(u + v) = au + av for all u, v E Vand all sCfllars a. 8. (a + ~)v = av + ~v for all v E Vand all scalars a, ~. Remark: The above properties seem hard to memorize, but it is not necessary. They are simply the familiar rules of algebraic manipulations with numbers. The only new twist here is that you have to understand what operations you can apply to what objects. You can add vectors, and you can multiply a vector by a number (scalar). Of course, you can do with number all possible manipulations that you have learned before. But, you cannot multiply two vectors, or add a number to a vector. Remark: It is not hard to show that zero vector 0 is unique. It is also easy to show that