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n-Linear Algebra of Type II - Cover:Layout 1 1/12/2009 3:16 PM Page 1 n-LINEAR ALGEBRA OF TYPE II W. B. Vasantha Kandasamy e-mail:[email protected] web: http://mat.iitm.ac.in/~wbv www.vasantha.in Florentin Smarandache e-mail:[email protected] INFOLEARNQUEST Ann Arbor 2008 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/ Peer reviewers: Professor Diego Lucio Rapoport Departamento de Ciencias y Tecnologia Universidad Nacional de Quilmes Roque Saen Peña 180, Bernal, Buenos Aires, Argentina Dr.S.Osman, Menofia University, Shebin Elkom, Egypt Prof. Mircea Eugen Selariu, Polytech University of Timisoara, Romania. Copyright 2008 by InfoLearnQuest and authors Cover Design and Layout by Kama Kandasamy Many books can be downloaded from the following Digital Library of Science: http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm ISBN-10: 1-59973-031-6 ISBN-13: 978-1-59973-031-8 EAN: 9781599730318 Standard Address Number: 297-5092 Printed in the United States of America 2 CONTENTS Preface 5 Chapter One n-VECTOR SPACES OF TYPE II AND THEIR PROPERTIES 7 1.1 n-fields 7 1.2 n-vector Spaces of Type II 10 Chapter Two n-INNER PRODUCT SPACES OF TYPE II 161 Chapter Three SUGGESTED PROBLEMS 195 3 FURTHER READING 221 INDEX 225 ABOUT THE AUTHORS 229 4 PREFACE This book is a continuation of the book n-linear algebra of type I and its applications. Most of the properties that could not be derived or defined for n-linear algebra of type I is made possible in this new structure: n-linear algebra of type II which is introduced in this book. In case of n-linear algebra of type II, we are in a position to define linear functionals which is one of the marked difference between the n-vector spaces of type I and II. However all the applications mentioned in n-linear algebras of type I can be appropriately extended to n-linear algebras of type II. Another use of n-linear algebra (n-vector spaces) of type II is that when this structure is used in coding theory we can have different types of codes built over different finite fields whereas this is not possible in the case of n-vector spaces of type I. Finally in the case of n-vector spaces of type II we can obtain n- eigen values from distinct fields; hence, the n-characteristic polynomials formed in them are in distinct different fields. An attractive feature of this book is that the authors have suggested 120 problems for the reader to pursue in order to understand this new notion. This book has three chapters. In the first chapter the notion of n-vector spaces of type II are introduced. This chapter gives over 50 theorems. Chapter two introduces the notion of n-inner product vector spaces of type II, n-bilinear forms and n-linear functionals. The final chapter 5 suggests over a hundred problems. It is important that the reader should be well versed with not only linear algebra but also n- linear algebras of type I. The authors deeply acknowledge the unflinching support of Dr.K.Kandasamy, Meena and Kama. W.B.VASANTHA KANDASAMY FLORENTIN SMARANDACHE 6 Chapter One n-V S T II ECTOR PACES OF YPE AND T P HEIR ROPERTIES In this chapter we for the first time introduce the notion of n- vector space of type II. These n-vector spaces of type II are different from the n-vector spaces of type I because the n-vector spaces of type I are defined over a field F where as the n-vector spaces of type II are defined over n-fields. Some properties enjoyed by n-vector spaces of type II cannot be enjoyed by n- vector spaces of type I. To this; we for the sake of completeness just recall the definition of n-fields in section one and n-vector spaces of type II are defined in section two and some important properties are enumerated. 1.1 n-Fields In this section we define n-field and illustrate it by examples. DEFINITION 1.1.1: Let F = F (cid:2) F (cid:2) … (cid:2) F where each F is 1 2 n i a field such that F (cid:3) F or F (cid:3) F if i (cid:4) j, 1 (cid:5) i, j (cid:5) n, we call F i j j i a n-field. 7 We illustrate this by the following example. Example 1.1.1: Let F = R (cid:2) Z (cid:2) Z (cid:2) Z be a 4-field. 3 5 17 Now how to define the characteristic of any n-field, n (cid:6) 2. DEFINITION 1.1.2: Let F = F (cid:2) F (cid:2) … (cid:2) F be a n-field, we 1 2 n say F is a n-field of n-characteristic zero if each field F is of i characteristic zero, 1 (cid:5) i (cid:5) n. Example 1.1.2: Let F = F (cid:2) F (cid:2) F (cid:2) F (cid:2) F (cid:2) F , where F 1 2 3 4 5 6 1 = Q( 2 ), F = Q( 7 ), F = Q( 3, 5), F = Q( 11), F = 2 3 4 5 Q( 3, 19) and F = Q( 5, 17); we see all the fields F , 6 1 F , …, F are of characteristic zero thus F is a 6-field of 2 6 characteristic 0. Now we proceed on to define an n-field of finite characteristic. DEFINITION 1.1.3: Let F = F (cid:2) F (cid:2) … (cid:2) F (n (cid:6) 2), be a n- 1 2 n field. If each of the fields F is of finite characteristic and not i zero characteristic for i = 1, 2, …, n then we call F to be a n- field of finite characteristic. Example 1.1.3: Let F = F (cid:2) F (cid:2) F (cid:2) F = Z (cid:2) Z (cid:2) Z (cid:2) 1 2 3 4 5 7 17 Z , F is a 4-field of finite characteristic. 31 Note: It may so happen that in a n-field F = F (cid:2) F (cid:2) … (cid:2) F , 1 2 n n (cid:6) 2 some fields F are of characteristic zero and some of the i fields F are of characteristic a prime or a power of a prime. j Then how to define such n-fields. DEFINITION 1.1.4: Let F = F (cid:2) F (cid:2) … (cid:2) F be a n-field (n (cid:6) 1 2 n 2), if some of the F’s are fields of characteristic zero and some i of the F’s are fields of finite characteristic i (cid:4) j, 1 (cid:5) i, j (cid:5) n then j we define the characteristic of F to be a mixed characteristic. 8 Example 1.1.4: Let F = F (cid:2) F (cid:2) F (cid:2) F (cid:2) F where F = Z , 1 2 3 4 5 1 2 F = Z , F = Q( 7 ), F = Q( 3, 5) and F = Q( 3, 23, 2) 2 7 3 4 5 then F is a 5-field of mixed characteristic; as F is of 1 characteristic two, F is a field of characteristic 7, F , F and F 2 3 4 5 are fields of characteristic zero. Now we define the notion of n-subfields. DEFINITION 1.1.5:Let F = F (cid:2) F (cid:2) … (cid:2) F be a n-field (n (cid:6) 1 2 n 2). K = K (cid:2) K (cid:2) … (cid:2) K is said to be a n-subfield of F if each 1 2 n K is a proper subfield of F, i = 1, 2, …, n and K (cid:3) K or K (cid:3) i i i j j K if i (cid:4) j, 1 (cid:5) i, j (cid:5) n. i We now give an example of an n-subfield. Example 1.1.5: Let F = F (cid:2) F (cid:2) F (cid:2) F where 1 2 3 4 F = Q( 2, 3), F = Q( 7, 5), 1 2 (cid:7) (cid:8) (cid:7) (cid:8) Z [x] Z [x] F = (cid:9) 2 (cid:10) and F = (cid:9) 11 (cid:10) 3 (cid:9)(cid:12) x2 (cid:11)x(cid:11)1 (cid:10)(cid:13) 4 (cid:9)(cid:12) x2 (cid:11)x(cid:11)1 (cid:10)(cid:13) be a 4-field. Take K = K (cid:2) K (cid:2) K (cid:2) K = (Q( 2 ) (cid:2) Q( 7 ) 1 2 3 4 (cid:2) Z (cid:2) Z (cid:14) F = F (cid:2) F (cid:2) F (cid:2) F . Clearly K is a 4-subfield 2 11 1 2 3 4 of F. It may so happen for some n-field F, we see it has no n-subfield so we call such n-fields to be prime n-fields. Example 1.1.6: Let F = F (cid:2) F (cid:2) F (cid:2) F = Z (cid:2) Z (cid:2) Z (cid:2) 1 2 3 4 7 23 2 Z be a 4-field. We see each of the field F’s are prime, so F is a 17 i n-prime field (n = 4). DEFINITION 1.1.6: Let F = F (cid:2) F (cid:2) … (cid:2) F be a n-field (n (cid:6) 1 2 n 2) if each of the F’s is a prime field then we call F to be prime i n-field. It may so happen that some of the fields may be prime and others non primes in such cases we call F to be a semiprime n- 9