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N-LINEAR ALGEBRA of TYPE II 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] 2008 n-LINEAR ALGEBRA OF TYPE II W. B. Vasantha Kandasamy Florentin Smarandache 2008 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-VECTOR SPACES OF TYPE II AND THEIR PROPERTIES 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 = F1 ∪ F2 ∪ … ∪ Fn where each Fi is a field such that Fi ⊄ Fj or Fj ⊄ Fi if i ≠ j, 1 ≤ i, j ≤ n, we call F a n-field. 7 We illustrate this by the following example. Example 1.1.1: Let F = R ∪ Z3 ∪ Z5 ∪ Z17 be a 4-field. Now how to define the characteristic of any n-field, n ≥ 2. DEFINITION 1.1.2: Let F = F1 ∪ F2 ∪ … ∪ Fn be a n-field, we say F is a n-field of n-characteristic zero if each field Fi is of characteristic zero, 1 ≤ i ≤ n. Example 1.1.2: Let F = F1 ∪ F2 ∪ F3 ∪ F4 ∪ F5 ∪ F6, where F1 = Q( 2 ), F2 = Q( 7 ), F3 = Q( 3 , 5 ), F4 = Q( 11 ), F5 = Q( 3 , 19 ) and F6 = Q( 5, 17 ); we see all the fields F1, F2, …, F6 are of characteristic zero thus F is a 6-field of characteristic 0. Now we proceed on to define an n-field of finite characteristic. DEFINITION 1.1.3: Let F = F1 ∪ F2 ∪ … ∪ Fn (n ≥ 2), be a n- field. If each of the fields Fi is of finite characteristic and not 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 = F1 ∪ F2 ∪ F3 ∪ F4 = Z5 ∪ Z7 ∪ Z17 ∪ Z31, F is a 4-field of finite characteristic. Note: It may so happen that in a n-field F = F1 ∪ F2 ∪ … ∪ Fn, n ≥ 2 some fields Fi are of characteristic zero and some of the fields Fj are of characteristic a prime or a power of a prime. Then how to define such n-fields. DEFINITION 1.1.4: Let F = F1 ∪ F2 ∪ … ∪ Fn be a n-field (n ≥ 2), if some of the Fi’s are fields of characteristic zero and some of the Fj’s are fields of finite characteristic i ≠ j, 1 ≤ i, j ≤ n then we define the characteristic of F to be a mixed characteristic. 8 Example 1.1.4: Let F = F1 ∪ F2 ∪ F3 ∪ F4 ∪ F5 where F1 = Z2, F2 = Z7, F3 = Q( 7), F4 = Q( 3, 5 ) and F5 = Q( 3, 23, 2 ) then F is a 5-field of mixed characteristic; as F1 is of characteristic two, F2 is a field of characteristic 7, F3, F4 and F5 are fields of characteristic zero. Now we define the notion of n-subfields. DEFINITION 1.1.5: Let F = F1 ∪ F2 ∪ … ∪ Fn be a n-field (n ≥ 2). K = K1 ∪ K2 ∪ … ∪ Kn is said to be a n-subfield of F if each Ki is a proper subfield of Fi, i = 1, 2, …, n and Ki ⊄ Kj or Kj ⊄ Ki if i ≠ j, 1 ≤ i, j ≤ n. We now give an example of an n-subfield. Example 1.1.5: Let F = F1 ∪ F2 ∪ F3 ∪ F4 where F1 = Q( 2, 3 ), F2 = Q( 7, 5 ), ⎛⎞Z[x] ⎛⎞Z[x] F = ⎜⎟2 and F = ⎜⎟11 3 ⎜⎟2 4 ⎜⎟2 ⎝⎠xx1++ ⎝⎠xx1++ be a 4-field. Take K = K1 ∪ K2 ∪ K3 ∪ K4 = (Q( 2) ∪ Q( 7) ∪ Z2 ∪ Z11 ⊆ F = F1 ∪ F2 ∪ F3 ∪ F4. Clearly K is a 4-subfield 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 = F1 ∪ F2 ∪ F3 ∪ F4 = Z7 ∪ Z23 ∪ Z2 ∪ Z17 be a 4-field. We see each of the field Fi’s are prime, so F is a n-prime field (n = 4). DEFINITION 1.1.6: Let F = F1 ∪ F2 ∪ … ∪ Fn be a n-field (n ≥ 2) if each of the Fi’s is a prime field then we call F to be prime 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 field. If all the fields Fi in F are non prime i.e., are not prime fields then we call F to be a non prime field. Now in case of n- semiprime field or semiprime n-field if we have an m-subfield m < n then we call it as a quasi m-subfield of F m < n. We illustrate this situation by the following example. Example 1.1.7: Let Zx2 [ ] Zx7 [ ] Zx3 [ ] F = Q ∪ Z7 ∪ ∪ ∪ ; xx12 + + x12 + x93 + be a 5-field of mixed characteristic; clearly F is a 5-semiprime field. For Q and Z7 are prime fields and the other three fields are non-prime; take K = K1 ∪ K2 ∪ K3 ∪ K4 ∪ K5 = φ ∪ φ ∪ Z2 ∪ Z7 ∪ Z3 ⊂ F = F1 ∪ F2 ∪ F3 ∪ F4 ∪ F5. Clearly K is a quasi 3- subfield of F. 1.2 n-Vector Spaces of Type II In this section we proceed on to define n-vector spaces of type II and give some basic properties about them. DEFINITION 1.2.1: Let V = V1 ∪ V2 ∪ … ∪ Vn where each Vi is a distinct vector space defined over a distinct field Fi for each i, i = 1, 2, …, n; i.e., V = V1 ∪ V2 ∪ … ∪ Vn is defined over the n- field F = F1 ∪ F2 ∪ … ∪ Fn. Then we call V to be a n-vector space of type II. We illustrate this by the following example. Example 1.2.1: Let V = V1 ∪ V2 ∪ V3 ∪ V4 be a 4-vector space defined over the 4-field Q( 2) ∪ Q( 3) ∪ Q( 5) ∪ Q( 7). V is a 4-vector space of type II. Unless mention is made specifically we would by an n-vector space over F mean only n-vector space of type II, in this book. 10 Example 1.2.2: Let F = F1 ∪ F2 ∪ F3 where F1 = Q( 3) × Q( 3 ) × Q( 3 ) a vector space of dimension 3 over Q( 3 ). F2 = Q( 7 ) [x] be the polynomial ring with coefficients from Q( 7), F2 is a vector space of infinite dimension over Q( 7) and ⎪⎧⎡⎤ab ⎪⎫ F3 = ⎨⎢⎥a,b,c,d∈ Q( 5)⎬ , ⎩⎭⎪⎣⎦cd ⎪ F3 is a vector space of dimension 4 over Q( 5 ) . Thus F = F1 ∪ F2 ∪ F3 is a 3-vector space over the 3-field Q( 3 ) ∪ Q( 7 ) ∪ Q( 5 ) of type II.
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