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w. b. vasantha kandasamy
LINEAR ALGEBRA
AND
AMERICAN RESEARCH PRESS
2003
SMARANDACHE LINEAR ALGEBRA
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Linear Algebra and
Smarandache Linear Algebra
W. B. Vasantha Kandasamy
Department of Mathematics
Indian Institute of Technology, Madras
Chennai – 600036, India
American Research Press
2003
1
This book can be ordered in a paper bound reprint from:
Books on Demand
ProQuest Information & Learning
(University of Microfilm International)
300 N. Zeeb Road
P.O. Box 1346, Ann Arbor
MI 48106-1346, USA
Tel.: 1-800-521-0600 (Customer Service)
http://wwwlib.umi.com/bod/
and online from:
Publishing Online, Co. (Seattle, Washington State)
at: http://PublishingOnline.com
This book has been peer reviewed and recommended for publication by:
Jean Dezert, Office National d=Etudes et de Recherches Aerospatiales (ONERA), 29, Avenue de la
Division Leclerc, 92320 Chantillon, France.
M. Khoshnevisan, School of Accounting and Finance, Griffith University, Gold Coast, Queensland
9726, Australia.
Sabin Tabirca and Tatiana Tabirca, University College Cork, Cork, Ireland .
Copyright 2003 by American Research Press and W. B. Vasantha Kandasamy
Rehoboth, Box 141
NM 87322, USA
Many books can be downloaded from our E-Library of Science:
http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm
ISBN : 1-931233-75-6
Standard Address Number : 297-5092
Printed in the United States of America
2
CONTENTS
PREFACE
5
Chapter One
LINEAR ALGEBRA : Theory and Applications
1.1 Definition of linear Algebra and its properties
12
1.3 Elementary canonical forms
20
1.4 Inner product spaces
29
1.5 Operators on inner product space
33
1.6 Vector spaces over finite fields Z p
37
1.7 Bilinear forms and its properties
44
1.8 Representation of finite groups
46
1.9 Semivector spaces and semilinear algebra
48
1.10 Some applications of linear algebra
60
Chapter Two
SMARANDACHE LINEAR ALGEBRA AND ITS PROPERTIES
2.1 Definition of different types of Smarandache linear algebra with examples
65
2.2 Smarandache basis and S-linear transformation of S-vector spaces
71
2.3 Smarandache canonical forms
76
2.4 Smarandache vector spaces defined over finite S-rings Z n
86
2.6 Smarandacherepresentationof finite S-semigroup
88
2.7 Smarandache special vector spaces
99
2.8 Algebra of S-linear operators
103
2.9 Miscellaneous properties in Smarandache linear algebra
110
2.10 Smarandache semivector spaces and Smarandache semilinear algebras
119
3
1.2 Lineartransformations and linear operators
7
2.5 Smarandache bilinear forms and its properties
81
Chapter Three
SMARANDACHE LINEAR ALGEBRAS AND ITS APPLICATIONS
3.1 A smattering of neutrosophic logic using S-vector spaces of type II
141
3.2 Smarandache Markov Chains using S-vector spaces II
142
3.3 Smarandache Leontief economic models
143
3.4 Smarandache anti-linear algebra
146
Chapter Four
SUGGESTED PROBLEMS
149
REFERENCES
165
INDEX
169
4

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