"Science, Faculty of"@en . "Mathematics, Department of"@en . "DSpace"@en . "UBCV"@en . "Lim, Marion Josephine Sui Sim"@en . "2012-03-07T21:02:57Z"@en . "1967"@en . "Doctor of Philosophy - PhD"@en . "University of British Columbia"@en . "Let U be an n-dimensional vector space over an\r\nalgebraically closed field. Let [formula omitted] denote the [formula omitted]\r\nspace spanned by all Grassmann products [formula omitted].\r\nSubsets of vectors of [formula omitted] denoted by [formula omitted] and [formula omitted]\r\nare defined as follows [formula omitted]. A vector which is in [formula omitted] or is zero is called\r\npure or decomposable. Each vector in [formula omitted] is said to have\r\nrank one. Similarly each vector in [formula omitted] has rank two.\r\nA subspace of H of [formula omitted] is called a rank two subspace If [formula omitted] is contained in [formula omitted].\r\nIn this thesis we are concerned with investigating rank\r\ntwo subspaces. The main results are as follows:\r\nIf dim [formula omitted] such that every nonzero vector [formula omitted] is independent\r\nin U.\r\nThe rank two subspaces of dimension less than four\r\nare also characterized."@en . "https://circle.library.ubc.ca/rest/handle/2429/41206?expand=metadata"@en . "CHARACTERIZATION OF SUBSPACES OF RANK TWO GRASSMANN VECTORS OF ORDER TWO M A R I O N - J O S E P H I N E S U I S I M L I M B . S c , M . S c . , V i c t o r i a U n i v e r s i t y o f W e l l i n g t o n , New Z e a l a n d / 1963 A T H E S I S SUBMITTED I N : P A R T I A L F U L F I L M E N T OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n t h e D e p a r t m e n t o f MATHEMATICS We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o r e q u i r e d ^ s t a n d a r i d The U n i v e r s i t y o f B r i t i s h C o l u m b i a December 1967 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree -at the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s represen-t a t i v e s . I t i s understood tha t copying o r p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed w i t h o u t my w r i t t e n p e r m i s s i o n . Depa rtment The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date . ^ -i i . S u p e r v i s o r : D r . R. W e s t w i c k . ABSTRACT L e t U be a n n - d i m e n s i o n a l v e c t o r s p a c e o v e r a n 2 a l g e b r a i c a l l y c l o s e d f i e l d . L e t A U d e n o t e t h e ' ( 2 ) - d i m e n s i o n a l s p a c e s p a n n e d b y a l l G r a s s m a n n p r o d u c t s X^AX^ , e U . 2 2 S u b s e t s o f v e c t o r s o f A U d e n o t e d b y C ^ ( U ) a n d Cg'Cu) > a r e d e f i n e d a s f o l l o w s 2 2 C 1 ( U ) ' = {z \u00E2\u0082\u00AC A U / o =j= z = x-^Ax^; x^,Xg e U ] p 2 p C 2 ^ = {z e A U / o 4= z = z i + z 2 > zi>z2 e C l ^ U ^ a n d z 4 C * ( U ) } p A v e c t o r w h i c h i s i n C^(U) o r i s z e r o i s c a l l e d 2 p u r e o r d e c o m p o s a b l e . E a c h v e c t o r i n C ^ U ) i s s a i d t o h a v e 2 r a n k o n e . S i m i l a r l y e a c h v e c t o r i n C (U) h a s r a n k t w o . 2 A s u b s p a c e o f H o f A U i s c a l l e d a r a n k two s u b -p s p a c e I f H - [o} i s c o n t a i n e d i n C 2 ( U ) . I n t h i s t h e s i s we a r e - c o n c e r n e d w i t h i n v e s t i g a t i n g r a n k two s u b s p a c e s . The m a i n r e s u l t s a r e as f o l l o w s : I f d i m U > 6 , t h e n d i m H < n-3 . I f d i m H _> 4 , t h e n t h e r e e x i s t s a 2 - d i m e n s i o n a l s u b s p a c e o f U s u c h t h a t e v e r y n o n z e r o v e c t o r f \u00E2\u0082\u00AC H h a s t h e f o r m i i i . f = x A x f + y A y f where { x , y , x f , y f } i s i n d e p e n d e n t i n U . The r a n k two subspaces o f d i m e n s i o n l e s s t h a n f o u r a r e a l s o c h a r a c t e r i z e d . i v . TABLE OP CONTENTS PAGE INTRODUCTION 1 CHAPTER I . P r e l i m i n a r i e s 8 CHAPTER I I . R e p r e s e n t a t i o n Theorems 19 CHAPTER I I I , The r a n k two subspaces when U has d i m e n s i o n 5 32 CHAPTER IV, The ran k two subspaces when U has d i m e n s i o n 6 k]J CHAPTER V. The r a n k two subspaces when U has d i m e n s i o n 7 79 CHAPTER IV. The main r e s u l t s 92 BIBLIOGRAPHY 10J V . ACKNOWLEDGEMENTS I am indebted to my supervisor. Dr. E. Westwick, for his generous and valuable assistance in the research and writing of this paper. Much gratitude i s due to Dr B, N. Moyls, who not only read this thesis and offered helpful suggestions, but guided me into the f i e l d of Multilinear Algebra. Also, thanks i s due to Dr- J. L. Brenner who provided some excellent suggestions. I am grateful to the University of B r i t i s h Columbia and the National Research Council for their f i n a n c i a l support. Last, but not least, I wish to thank Misses Sally Bate and Doreen Mah and Mrs. Carol Gerlach for typing the thesis. INTRODUCTION The o b j e c t o f t h i s p a p e r i s t h e c h a r a c t e r i z a t i o n o f a l l s u b s p a c e s o f . G r a s s m a n n v e c t o r s o f r a n k 2 a n d o r d e r 2 , o v e r a n a l g e b r a i c a l l y c l o s e d f i e l d . I n t h i s i n t r o d u c t i o n , we s h a l l d e f i n e t h e G r a s s m a n n P r o d u c t S p a c e , s t a t e someof t h e k n o w n p r o p e r t i e s o f G r a s s m a n n p r o d u c t s , a n d g i v e a s h o r t summary o f t h e p a p e r . The d e f i n i t i o n o f t h e G r a s s m a n n P r o d u c t S p a c e we s h a l l u s e i s t h a t o f N . B o u r b a k i . Some p r e l i m i n a r y d e f i n i t i o n s a n d a p r o p o s i t i o n a r e n e e d e d . I n t h i s p a p e r , U d e n o t e s a n n - d i m e n s i o n a l s p a c e o v e r a f i e l d F . P r o p o s i t i o n 1. L e t { E ^ : 1 <_ i <_ r ] be a f a m i l y o f f i n i t e -r d i m e n s i o n a l v e c t o r s p a c e s a n d l e t TT E . be t h e C a r t e s i a n 1=1 1 p r o d u c t o f [ E i : 1 <_ i <_ r } , L e t M be a f i n i t e - d i m e n s i o n a l s p a c e w i t h t h e f o l l o w i n g p r o p e r t i e s : r ( a ) T h e r e e x i s t s a m u l t i l i n e a r t r a n s f o r m a t i o n cj> o f ir E . i n t o i = l M , s u c h t h a t M i s s p a n n e d b y ( TT E . ) . i = l r ( b ) I f f i s a n y m u l t i l i n e a r t r a n s f o r m a t i o n o f T E . i n t o i = l some v e c t o r s p a c e N , t h e n t h e r e e x i s t s a u n i q u e l i n e a r t r a n s -2. formation g of M i n t o N such that f = g- . Then every vector space having the above two properties is isomorphic to M . (p. 1 8 , [ l ] ) . Definition ' 2 . Let M be the vector space described i n Proposition 1. Then M i s the tensor'product of the family r {E,: 1 < i < r} , and. i s denoted by \u00C2\u00AE E . In the case where r r E, = ... = E = U , then \u00C2\u00AE E. i s \u00C2\u00AE U . Every vector 1 r 1=1 1 i - l r x-^\u00C2\u00AE. . . \u00C2\u00AE x r i n \u00C2\u00AE U ; x ^ \u00E2\u0082\u00AC U , i s called a pure tensor\u00C2\u00AB r Definition 3. A r-linear transformation f of \u00C2\u00AE U into a . . i = 1 vector space \u00C2\u00A5 i s called alternating i f f ( x 1 , . . . , x r ) = o for r every vector x = (x.) \u00E2\u0082\u00AC \u00C2\u00AE U having at least two members of the 1 i=l (x i: 1 < i < r} equal. (p.58, [ 1 ] ) . We note that every multilinear alternating trans-formation f i s skew-symmetric; i. e . , f (x^^y.. . , x ^ r ^ ) = sgn a . f ( x x , . . . , x r ) . (Seep. 5 8 , [ l ] ) . r , Definition 4. Let U be a vector space and \u00E2\u0082\u00AC> U be the 1=1 tensor product of U f o r r > 2 . Let N be the subspace generated by the pure tensors x-j\u00C2\u00AE. . v\u00C2\u00AEx r f o r which at l e a s t two of the {x^} are equal. Then the Grassmann Product Space r r denoted by A U , i s the quotient space of \u00C2\u00AE U of N . Thus 1=1 r r f : \u00C2\u00AE U \u00E2\u0080\u0094 \u00C2\u00BB A U such that f ( x , \u00C2\u00AE . . . \u00C2\u00AE x ) = x , A . . . A x ^ i s alternating 1=1 \u00E2\u0080\u00A2 \u00C2\u00B1 r 1 r 3. B y c o n v e n t i o n , AU i s t h e s p a c e U i t s e l f , AU i s t h e f i e l d P . ( f l ] , p . 63). r E v e r y v e c t o r o f t h e f o r m X-^A . . . A x r e AU , x ^ e U , 1 <_ i <_ r , i s c a l l e d a p u r e ( G r a s s m a n n ) v e c t o r . L e t < x 1 , . .'. j> x r > d e n o t e t h e v e c t o r s p a c e g e n e r a t e d b y t h e v e c t o r s x-^, . . \u00E2\u0080\u00A2 >^ r \u00E2\u0080\u00A2 We n o t e t h e f o l l o w i n g p r o p e r t i e s o f p u r e v e c t o r s . 4a. I f x - ^ , . . . , x r a r e l i n e a r l y d e p e n d e n t , t h e n x 1 A . . . A x r = o . I f x 1 , . . . , x r a r e l i n e a r l y i n d e p e n d e n t , t h e n x 1 A , . . A x r 4= o . I f a i s a p e r m u t a t i o n o f {1,. . . ,r] , t h e n xC T ( l ) A - \u00E2\u0080\u00A2 - A x a ( r ) = ssn a . x 1 A . . . A x r . ( f l ] , ' p . .'64). 4b. L e t x-^A. . . A x r , y-^A. . . A y ^ b e ' n o n z e r o p u r e v e c t o r s i n r AU . Then- x ^ A . . . A x r = y 1 A . . . A y r i m p l i e s < x 1 , . . . , x r > = \u00E2\u0080\u00A2 ( U ] , P . 0 5 ) . 4c. I f < x 1 , . . . , x r > = , t h e n X\u00C2\u00B1A...Axr = Y y 1 A . . . A y r f o r some y e P , ( [ l ] , p . 95) \u00E2\u0080\u00A2 D e f i n i t i o n 5. L e t z = x ^ A . . . A x r be a n o n z e r o p u r e v e c t o r i n r AU . T h e n U ( z ) i s d e f i n e d t o be t h e r - d i m e n s i o n a l s p a c e < x ^ , . . . , x > . B y p r o p e r t y 4.D, U(Z) i s w e l l - d e f i n e d . r L e t o 4= x = x 1 A . . . A x r \u00E2\u0082\u00AC AU . T h e n we w i l l s a y t h a t x d e f i n e s V i f V = U ( x ) . P r o p o s i t i o n 6. L e t , V a n d W be two s u b s p a c e s o f U o f 4. o f d i m e n s i o n r a n d s r e s p e c t i v e l y ; l e t v be a p u r e v e c t o r r s i n AU d e f i n i n g V , a n d w a p u r e v e c t o r i n AU d e f i n i n g W . I n o r d e r t h a t VfiW = o , i t i s n e c e s s a r y a n d s u f f i c i e n t r*t s t h a t VAW =[= o 5 t h e n o n z e r o p u r e ' v e c t o r VAW i n A U d e f i n e s t h e s u b s p a e e (V + W) . ( [ 1 ] , p . 9 7 ) C o r o l l a r y 7 . L e t x = x-^A. . . A x r = y-^A.,.Ay^ be a n o n z e r o p u r e r v e c t o r i n AU . L e t v be a n o n z e r o v e c t o r i n U s u c h t h a t r - t l v U(x) . T h e n VAX^A . . , A x ^ = vAy-^A. . . Ay^ =f o i n A U . r A n y v e c t o r i n AU i s s a i d t o h a v e o r d e r r . I n t h i s p a p e r , we s h a l l be m a i n l y i n t e r e s t e d i n G r a s s m a n n v e c t o r s o f o r d e r t w o , a l t h o u g h i n C h a p t e r I we w i l l f i n d i t n e c e s s a r y t o c o n s i d e r G r a s s m a n n v e c t o r s o f h i g h e r o r d e r . U n l e s s o t h e r w i s e s t a t e d , we w i l l assume t h a t F I s a n a l g e b r a i c a l l y c l o s e d f i e l d . I n o r d e r t o show t h i s c o n d i t i o n i s n o t u n r e a s o n a b l e , we w i l l show ( E x a m p l e 11.22) t h a t i f F i s n o t a l g e b r a i c a l l y c l o s e d , t h e n t h e r a n k two s u b s p a c e s ( s e e d e f i n i t i o n b e l o w ) c a n be d i f f e r e n t f r o m t h e o n e s o b t a i n e d i n t h i s p a p e r . r We d e f i n e s u b s e t s o f v e c t o r s o f AU , d e n o t e d b y C ^ ( U ) , i n d u c t i v e l y a s f o l l o w s ; C^(U) = {z e AU I o '=}= x-j^A. . . A x r w h e r e x^ e U , i = 1 , . . . , r ] c\u00C2\u00A3(U) = {z e AU | z = z 1 + z 2 w h e r e z]_ e c\u00C2\u00A3(U) , z 2 e C ^ _ 1 ( U ) k - 1 a n d z | U cT(U)} 1=1 C, (U) i s t h e s e t o f G r a s s m a s s v e c t o r s o f r a n k k a n d 5. o r d e r r . We d e f i n e t h e s e t ; 2 2 R g ( U ) = {H / H i s a s u b s p a c e c o n t a i n e d i n A U a n d H - {o} i s c o n t a i n e d i n C g ( U ) } T h e n H e R ^ u ) i s c a l l e d a r a n k two s u b s p a c e . 2 We show i n t h i s p a p e r t h a t i f H s R ^ U ) , d i m U = n } n > 6 a n d F i s a l g e b r a i c a l l y c l o s e d , t h e n d i m H <_ n-3 ( T h e o r e m VT.100). A l s o i f d i m H _> 4 , t h e n H h a s a (1,1) b a s i s ( D e f i n i t i o n 11.33). I n a d d i t i o n t o t h e s e m a i n r e s u l t s , we show t h a t i f d i m U <_ 5 , t h e n d i m H _ 3 . We a l s o o b t a i n b a s e s f o r t h e r a n k 2 s u b s p a c e s o f d i m e n s i o n t h r e e a n d l e s s t h a n t h r e e . I n C h a p t e r I , we show t h a t a n e c e s s a r y a n d s u f f i c i e n t s c o n d i t i o n f o r t h e v e c t o r ( E x.Ay.) to h a v e r a n k s i s t h a t 1=1 1 1 t h e v e c t o r s x ^ , y ^ , . . . , x s , y g be i n d e p e n d e n t ( T h e o r e m 1.13). F r o m t h i s , we o b t a i n the r e s u l t t h a t i f C 2 ( U ) 4 \u00C2\u00B0 > t h e n d i m u _> 4 . I n C h a p t e r 1I3 we o b t a i n some r e p r e s e n t a t i o n s o f a n y r a n k 2 v e c t o r b y v e c t o r s i n U , a n d a l s o r e p r e s e n t a t i o n s f o r a n y i n d e p e n d e n t p a i r o f v e c t o r s i n a ' r a n k two s u b s p a c e . We show t o o t h a t i f d i m U = 4 , t h e n e v e r y r a n k two s u b s p a c e h a s d i m e n s i o n o n e , p r o v i d e d F i s a l g e b r a i c a l l y c l o s e d . H o w e v e r , i f F i s n o t a l g e b r a i c a l l y c l o s e d , e . g . , F a R e a l s , t h e n 6. E x a m p l e 11 .22 shows t h e r e e x i s t s a 2 - d i m e n s i o n a l r a n k two s u b -s p a c e when d i m II = 4 . I n C h a p t e r I I I , we c o n s i d e r t h e c a s e d i m U = 5 . I n T h e o r e m 111.40, we show t h a t t h e r a n k 2 s u b s p a c e s h a v e d i m e n s i o n < 3 \u00E2\u0080\u00A2 T h e o r e m 111,45 g i v e s b a s e s f o r t h e 3 - d i m e n s i o n a l r a n k two s u b s p a c e s . We show a l s o t h a t f o r d i m U = n , e v e r y ( 1 , 1 ) b a s i s ( D e f i n i t i o n 11 ,33) f o r H e R | (U) h a s d i m e n s i o n < n - 3 . I n C h a p t e r I V , we c o n s i d e r t h e c a s e d i m U = 6 . We show t h a t e v e r y 3 - d i m e n s i o n a l r a n k 2 s u b s p a c e H w i t h b a s i s { f , f , f } s u c h t h a t d i m [ 1 U ( f , ) ] = 6 ( D e f i n i t i o n 1.12) x. d y 1 = 1 x h a s a b a s i s o f p a i r w i s e - P g v e c t o r s ( D e f i n i t i o n 11.19 a n d T h e o r e m I V . 6 l ) . U s i n g t h i s r e s u l t , we show H h a s a b a s i s w h i c h i s one o f t h r e e k i n d s ( T h e o r e m I V . 6 2 ) . T h e s e a r e g i v e n i n R e m a r k I V . 5 3 * T h e o r e m s IV,57> I V . 5 9 . C o m b i n i n g t h e s e r e s u l t s w i t h t h o s e o f C h a p t e r s I I a n d I I I , we show t h a t t h e r a n k two s u b s p a c e s h a v e d i m e n s i o n <: 3 ( T h e o r e m I V . 7 2 ) . I n C h a p t e r V , we c o n s i d e r t h e c a s e d i m U = 7 . We show t h a t t h e r a n k 2 s u b s p a c e s [H] h a v e d i m e n s i o n < 4 , a n d i f d i m H = 4 , H h a s a ( 1 , 1 ) b a s i s ( T h e o r e m V.85). To o b t a i n t h i s r e s u l t , we f i r s t show t h a t i f d i m H = 3 , H = < f 1 , f 2 , f ^ > a n d d i m T, U ( f . ) = 7 , t h e n H h a s a b a s i s o f p a i r w i s e - P , -i = l 1 D v e c t o r s . We f i n d t h i s b a s i s i s one o f 2 p o s s i b l e k i n d s ( T h e o r e m V . 7 3 ) ; r e p r e s e n t a t i o n s f o r t h e s e b a s i s members a r e g i v e n i n T h e o r e m s V . 7 4 , 75. U s i n g t h e a b o v e r e s u l t s a n d t h o s e o f C h a p t e r s I I t o I V , we show d i m H < 4 , a n d i f 7. dim H = 4 , i t has a (1,1) b a s i s . In Chapter VI, we obtain the main r e s u l t s . We consider f i r s t the case dim U = 8 . We note that i f i s a ^-dimensional rank. 2 subspace and i f 3 dim s U ( f . ) . = 8 , then { f - ^ f ^ f ^ } i s a (1,1) b a s i s of pairwise-Pg vectors f o r . Furthermore, i f H e R|(U) and H o (as above), then H has a (1,1) b a s i s (Theorems VI.86, 88). We then show that dim H 1 n-j5 when dim U _> 6 (Theorem VI. 100) using the above r e s u l t s . A l s o , i f dim H _> 4 , H has a (1,1) basis. 8 . CHAPTER I PRELIMINARIES The aim o f t h i s c h a p t e r i s t o . f i n d a n e c e s s a r y and s o s u f f i c i e n t c o n d i t i o n t h a t a v e c t o r ( \u00C2\u00A3 x Ay, ) e C ( U ) ; i . e . , ^ t h a t the v e c t o r has rank, s . We show i n f a c t t h a t the above c o n d i t i o n i s t h a t {x-^y-j , . . . , x g , y g ] i s i n d e p e n d e n t (Theorem 1.13)\u00E2\u0080\u00A2 I n the p r o c e s s o f o b t a i n i n g the above r e s u l t , we o b t a i n a l s o the r e s u l t t h a t i f z = z ] _ + * ' ' + z j c e C ^ ( U ) } z^ e C ^ ( U ) , i = 1, . . . , k , t h e n the space U ( z ) = U ( ) + \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 - + U ( z k ) i s w e l l -d e f i n e d (Theorem I . 1 0 ) . Thus we can a s s o c i a t e w i t h each z e C ^ ( U ) a p a r t i c u l a r sub space U ( z ) o f U . I f x^,...,x i s a b a s i s o f U , t h e n f o r any r z \u00E2\u0082\u00AC AU , we can w r i t e z = S a ( i 1 , . . . , i )x. A...Ax., K i , <\u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 F to be alternating i f i t is i=l r-liriear and p(k,,...,kr) = 0 whenever at least two members of {k,,...,kr}(5 E) are equal. ' We note that such a p is skew-symmetrie; i.e., P ( k a ( i ) > \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 j k a ( r ) ^ = sgn a.p(k1,...,kr) where a is a permutation of {l,...,r} . (See p. 5 0 , [ 5 ] . ) The following known result tells us when z e A*U is zero or has rank one. Theorem 1.2 Let x-,, . . ., x^ be a basis of U . Let z = p ( i X J . . . , i r ) x i A...Ax \u00C2\u00B1 e AU l i - r _2_ > Jj^ ) P ( \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 > ^ ( j - i 3 ^ | i + l s ' ' \u00C2\u00B0 s ^ r ^ \" We show t h i s sum i s z e r o and a p p l y Theorem 1.2. Case 1. > m o r > m f o r some t . I n e i t h e r case t h e sum i s z e r o s i n c e e v e r y term i s zero. Case 2. Not a l l o f t h e i n t e g e r s 1 s...,s are p r e s e n t i n ...,1^ ^ , o r n o t a l l o f the i n t e g e r s 1,.. ., s a r e p r e s e n t i n iQ>. .. .<. J r . A g a i n the sum i s z e r o s i n c e each term i s z e r o . Case ~}. A l l t h e i n t e g e r s 1,. .., s a r e p r e s e n t i n i 1 , . . . Ar_1 and i n J D , . . . , J r . A l s o \u00C2\u00B1 t < m ( t = 1,. .. , r - l ) and <_ m (t = o,.. . , r ) . I n t h i s c a s e , p ' ( i 1 , . . . , l r _ 1 , j u ) = v{\>> \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 ^ r . i j J j ) a n d p' (3Q>' \u00E2\u0080\u00A2 > > ' \u00E2\u0080\u00A2 \" * ^ R ) = P ( J 0 i > ' ' ^ - l ' ^ + i ' * \u00E2\u0080\u00A2 \u00C2\u00BB * J r ) > e x c e p t p o s s i b l y when i s one o f the i n t e g e r s 1,...,s . B u t i f J i s one of the i n t e g e r s l , . . . , s , the n 1 1 . p ( i 1 , . . . , i r _ 1 , J ( J ) p ( j 0 , . . . , j f j l _ 1 , J | J + 1 , . . . , d r ) =o s i n c e i n f i , , . . . , i ^ ^ j ^ ] we h a v e a r e p e t i t i o n o f . r H e n c e \u00C2\u00A3 (- l) M p' ( i , , - . \u00E2\u0080\u00A2 , ^ x ^ ) ? ' (3Q> \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 > J u - 1 * \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 * J r ) p=o r [1=0 s i n c e z e C ^ ( U ) ( T h e o r e m 1 . 2 ) . T h e r e f o r e b y T h e o r e m 1.2, z' e cJ(U) . C o r o l l a r y IA. L e t z = E p(l,,. . . , 1 )x\u00C2\u00B1 A...Ax, e C k ( U ) . 1 1 \" 1?\u00E2\u0080\u00A2\u00E2\u0080\u00A2- A a n d a p p l y Lemma 1 . 3 t o e a c h t e r m z^ . N o t e t h a t z' = (z,+ -\u00E2\u0080\u00A2'+z^.)' T h e o r e m 1 . 5 L e t U ' c U be a s u b s p a c e . T h e n c\u00C2\u00A3 ( U ' ) c c\u00C2\u00A3(U) , P r o o f : We show t h a t i f y\u00C2\u00B1 e C \u00C2\u00A3 ( U ' ) } i = 1 , . . . , k s u c h t h a t E y i e cJ(U') , t h e n E y , e G\u00C2\u00A3(U) . 1=1 1 K 1=1 1 . & L e t x-,,...,x be a b a s i s o f U ' a n d l e t 1 s x , , . . . , x be a n e x t e n s i o n o f t h i s b a s i s t o a b a s i s o f U . 12. S u p p o s e k t \u00C2\u00A3 y \u00C2\u00B1 = \u00C2\u00A3 z e CJ(U) , z \u00E2\u0082\u00AC c f ( U ) , 1=1,...,*, . 1=1 1 1=1 1 * x l C l e a r l y \u00C2\u00A3 <_ k . To show l > k , l e t Z i ' = ^ P ^ ( 1 T , . . . , 1 ) X . /-, , \ i t f o l l o w s t h a t S i n c e y i e C ^ ( U ' ) , 1 < i a , i \u00C2\u00A3 p ( J ) ( i 1 , . . . ,1 ) = o w h e n e v e r [ i ^ . . . , i r ) $ { 1 , . . \u00E2\u0080\u00A2 , s } \u00E2\u0080\u00A2 j = l L e t z ' = \u00C2\u00A3 p ( J ^ ( i , , . . . , i ) x . A...Ax. \u00E2\u0080\u00A2 J K i , < - \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 < ! \u00E2\u0080\u0094 i r \u00E2\u0080\u0094 B y Lemma 1.3, z' e c F ( U ' ) . S i n c e \u00C2\u00A3 z ' = \u00C2\u00A3 z , = \u00C2\u00A3 y , , 3 1 J = l J j = l 3 j = l 1 t h e n \u00C2\u00A3 >_ k . \u00C2\u00A5e h a v e t h e r e s u l t I t w i l l be c o n v e n i e n t t o h a v e t h e f o l l o w i n g n o t a t i o n f o r r a n k . D e f i n i t i o n 1.6 F o r z \u00E2\u0082\u00AC c\u00C2\u00A3 ( U ) , we d e f i n e R r ( z ) = k , i . e . , R r : A U - J s u c h t h a t R p ( z ) = k i f a n d o n l y i f z e c\u00C2\u00A3 ( U ) . We w i l l d r o p t h e i n d e x r when no c o n f u s i o n r e s u l t s . We s h a l l now p r o c e e d t o show t h a t i f z e c \u00C2\u00A3 ( U ) , z = y 1 - t \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 - + y k = z 1 + - \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 + z k >; y i , z i \u00E2\u0082\u00AC C ^ ( U ) , 1 < i < k , t h e n U ( y 1 ) + - \u00E2\u0080\u00A2 - + U ( y k ) = U ( z 1 ) + - \u00E2\u0080\u00A2 - + U ( z k ) . I n t h i s w a y , we c a n a s s o c i a t e w i t h z \u00E2\u0082\u00AC C ^ ( U ) a p a r t i c u l a r s u b s p a c e o f U . To t h i s e n d we n e e d some d e f i n i t i o n s a n d l e m m a s . r I f x e U , z e A U , s u c h t h a t z = \u00C2\u00A3 p ( I - . , . . . , i )x. A . - \u00E2\u0080\u00A2 Ax. , w h e r e X - , , . . . , x i s K i , < - \u00E2\u0080\u00A2 - < i k . Let x^,...,x he a hasis of U so chosen that x = x 1 , and x 2 J . . . , x g i s a basis of U( z 1)+. \u00E2\u0080\u00A2 .+TJ(zk) . Then z = \u00C2\u00A3 p ( i , ,. . . ,i )x. A... Ax, 2 ; v^ 6 c\u00C2\u00A3(U) , j = 1,...,t . Hence = v ( , E p ^ ^ C i - . , i )x A . . . A X , 3 2 \u00C2\u00A3 , t h e n k = t a n d R ( x A z ) = k T h e o r e m I . 1 0 L e t y\u00C2\u00B1 \u00E2\u0082\u00AC C^(U) , z \u00C2\u00B1 e C | ( U ) ( i = l , . . . , k ) s u c h t h a t y 1 + \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 + y k = z x + - \u00E2\u0080\u00A2 - + z k e c\u00C2\u00A3(U) . T h e n U ( y 1 ) + . . . + U ( y k ) = U ( Z l ) + - \" + U ( z k ) . P r o o f : S u p p o s e o n t h e c o n t r a r y t h a t U ( y 1 ) + \u00C2\u00AB \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 + U ( y k ) 4= u ( z 1 ) + ' * \u00E2\u0080\u00A2 + U ( z k ) . W i t h o u t l o s s o f g e n e r a l i t y , we c a n assume t h a t t h e r e e x i s t s a v e c t o r x e U ( y ^ ) s u c h t h a t x 4 U ( ) + \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 + U ( z k ) . B u t i f t h i s w e r e t h e c a s e , t h e n R ( x A ( y i + - \u00E2\u0080\u00A2 - + y k ) ) = R ( x A ( y 2 + - \u00E2\u0080\u00A2 - + y k ) ) < k - 1 , a n d R ( x A ( z 1 + - \u00E2\u0080\u00A2 \u00C2\u00AB + z k ) ) = k (Lemma 1 . 9 ) . S i n c e x A ( y ^ + .\u00E2\u0080\u00A2.+y k) = X A ( Z ^ + \u00C2\u00AB \u00E2\u0080\u00A2 - + z k ) , we h a v e a c o n t r a d i c t i o n . D e f i n i t i o n 1 .11 L e t z = z1+---+z^ e c k ( u ) \u00C2\u00BB z \u00C2\u00B1 e c i ( u ) \u00C2\u00BB i = 1 , . . . , k . T h e n U ( z ) i s d e f i n e d t o be U( z\u00C2\u00B1) + - \u00E2\u0080\u00A2 . +,U( z f c ) . B y T h e o r e m I.1 0 , U ( z ) i s w e l l - d e f i n e d . We w i s h now t o f i n d a n e c e s s a r y a n d s u f f i c i e n t 16. . S . 2 / \ c o n d i t i o n t h a t t h e v e c t o r ( \u00C2\u00A3 x . A y ) e C (U) . We prove f i r s t 1=1 1 1 s the f o l l o w i n g . Lemma 1.12 L e t \u00E2\u0082\u00AC U) , i = 1,. .. ,k , and l e t dim ( U ( z 1 ) + - \u00E2\u0080\u00A2 - + U ( z k ) ) = r k . Then R(2,+...+z k) = k . P r o o f : Suppose t h e lemma i s f a l s e . L e t k be the s m a l l e s t i n t e g e r f o r w h i c h i t f a i l s . C l e a r l y k >_ 2 . L e t zl + f z k = y l + \" ' + y i e C / U ) > ^ 1 \u00E2\u0082\u00AC c i ( u ) , 1 = 1,.. . ,_ k-1 . B u t we assumed i < k . T h e r e f o r e i = k-1 . By Lemma I.10, U(xAz 2)+\u00C2\u00AB \u00E2\u0080\u00A2 -+U(xAz k) = U(xAy 1)-f \u00E2\u0080\u00A2 \u00E2\u0080\u00A2+U(xAy k_ 1) Hence +U(z 2)+\u00C2\u00AB\u00E2\u0080\u00A2'+U(z k) = +U(y1 ) + \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 *+U(y k_ 1) . L e t x' e U ( z 1 ) , x' i n d e p e n d e n t o f x . Then a g a i n +U(z 2)+-\u00E2\u0080\u00A2 -+U(z k) = + U(y 1) + \u00C2\u00AB \u00E2\u0080\u00A2 \u00E2\u0080\u00A2+U(y k_ 1) . T a k i n g i n t e r s e c t i o n s , we o b t a i n U(z 2)+...+U(z k) = U ( y 1 ) + . . - + U ( y k _ 1 ) . By a s i m i l a r argument, 7\u00C2\u00B1 = U ( Z l ) + . . . + U ( z 1 _ 1 ) + U ( z 1 + 1 ) + - \u00E2\u0080\u00A2 - + U ( z k ) = U ( y 1 ) + . . . + U ( y k _ 1 ) . k H e n c e U(y 1)+\u00C2\u00AB\u00E2\u0080\u00A2'+ u(y k_ 1) = H Y\u00C2\u00B1 = {o} w h i c h i s i m p o s s i b l e . i = l The r e s u l t f o l l o w s . T h e o r e m I.13 I x.Ay e C 2 ( U ) i f a n d o n l y i f i = 1 1 i s f x l ' ^ \" l ^ \u00E2\u0080\u00A2 ' * *xs,ys^ ^S a n i n ( * e p e n d e n t s e * o f v e c t o r s . P r o o f : We show f i r s t t h a t t h e c o n d i t i o n i s n e c e s s a r y . S u p p o s e y g e < x 1 , y 1 , , . . , x g > , i . e . s s-1 s s-1 T h e n f = E x.Ay, = \u00C2\u00A3 X,Ay. + E a,x Ax, + E p,x Ay, i i \u00E2\u0080\u0094 . j ^ i s i = * j \u00C2\u00A3 [ x \u00C2\u00B1 A ( y i - a \u00C2\u00B1 x s ) + P i x s A y i ] . I f a\u00C2\u00B1 = o , t h e n x 1 A y \u00C2\u00B1 + ^\u00C2\u00B1^s^7\u00C2\u00B1 = ( x \u00C2\u00B1 + ^ 1 x s ) A y i \u00E2\u0080\u00A2 I f a\u00C2\u00B1 \u00C2\u00B1 o , t h e n x \u00C2\u00B1 A ( y i - a \u00C2\u00B1 x s ) + P \u00C2\u00B1 x s A y i = x . A ( y i - a.x s) + P i x s A ( y \u00C2\u00B1 = a \u00C2\u00B1 x s ) = ( x i + \u00C2\u00A5 s ) A ( y i \" a i X s } \u00E2\u0080\u00A2 A d d i n g t h e t e r m s i n f i n t h e a b o v e w a y , we h a v e t h a t R ( f ) <_ s-1 . H e n c e , i f { x 1 , y 1 , \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 5 x s , y g } i s d e p e n d e n t , t h e n R ( f ) <_ s-1 < s . T h e r e f o r e , i f R ( f ) = s , t h e n { x , , y , , . . . , x .y_} i s i n d e p e n d e n t . 1 1 S 8 C o n v e r s e l y , i f { x 1 , y 1 , . . . , x g , y g ] i s I n d e p e n d e n t , t h e n d i m [+\u00E2\u0080\u00A2\u00E2\u0080\u00A2'+ ] = 2s . S i n c e X X 8 S U ( x \u00C2\u00B1 A y i ) \u00C2\u00AB < x \u00C2\u00B1 , y i > , i = l , . . . , s ; a n d x \u00C2\u00B1 A y \u00C2\u00B1 e C^(U) i t f o l l o w s f r o m Lemma 1.12 t h a t R( 2 x.Ay.) = s . 1=1 1 i s C o r o l l a r y 1.14 Let f = S x.Ay, , and 1=1 dim < 2k ; k < s . Then R(f) <_ k-1 . Proof: . I f R(f) = k , then f = y-[+..\u00C2\u00AB+yk where [U(y()+...+U(y k) c , y^ \u00E2\u0082\u00AC cf(U) , i = l , . . . , k , By Theorem 1.13, we can see e a s i l y that dim [U(y()+.-.+U(y\u00C2\u00A3)] = 2k . But [U(y\u00C2\u00A3)+.\u00E2\u0080\u00A2-+U(y\u00C2\u00A3)] c < xl'^\"l' \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 , x s > y s > w n i c n h a s dimension < 2k . Hence H ( f) < k-1 . 19. CHAPTER I I REPRESENTATION THEOREMS T h i s c h a p t e r i s made up o f f o u r s e c t i o n s . The f i r s t p s e c t i o n i s c o n c e r n e d w i t h r e p r e s e n t a t i o n s f o r f \u00E2\u0082\u00AC C|(U) . The o t h e r t h r e e a r e concerned* w i t h r e p r e s e n t a t i o n s o f p a i r s o f 2 / \ v e c t o r s i n H e R 2 ( U ) . p As any f e C 2 ( U ) has many r e p r e s e n t a t i o n s i n p Cg(U) (we s h a l l i l l u s t r a t e l a t e r ) , i t i s e s s e n t i a l f o r us t o o b t a i n some p a r t i c u l a r r e p r e s e n t a t i o n s w hich w i l l be s i m p l e r t o use when c o n s d i e r i n g r a n k two subspaces o f d i m e n s i o n g r e a t e r t h a n one. The main r e s u l t s a r e c o n t a i n e d i n Theorems 11 .26 and 11.31 and C o r o l l a r i e s 11 .21 and I I . 3 0 . SECTION 1. R e p r e s e n t a t i o n s f o r f 6 Cg(U) . D e f i n i t i o n I I . 15 L e t f e c|(U) . L e t f = x.jAx 2 + x y \ x ^ be any r e p r e s e n t a t i o n o f f e'C ^ U ) . Then U ( f ) i s d e f i n e d t o be t h e 4 - d i m e n s i o n a l subspace . We note t h a t by D e f i n i t i o n I.11 and Theorem 1.13* U ( f ) i s w e l l - d e f i n e d and i n f a c t c o i n c i d e s w i t h t h a t i n D e f i n i t i o n I.11. We have hence u s e d the same n o t a t i o n f o r i t . 20. p I t i s easy to see that f e C 2(U) has many represent-a t i o n s ; e.g., f = x-^AXg + X ^ A X ^ = x 1 A ( x 2 + x^) + (x^ - X 1 ) A X ^ The f o l l o w i n g theorem shows that i f \u00E2\u0082\u00AC U(f) , and U' i s any complementary subspace of y-^ i n U(f) , then there i s a re p r e s e n t a t i o n f = y-j_Au + V A W where = U' . Theorem II.16 Let f e C 2(U) and l e t fy^,...,y^} be any basi s of U(f) . Then f has a representation f = y-]_Au + V A W where = 2 Proof; Since f e A , then f = \u00C2\u00A3 a, 1y lAy 1 , a, , e F l since 2 \u00C2\u00A3 a. .y.Ay. e A i s n e c e s s a r i l y of rank at most 2=. The f o l l o w i n g theorem i n d i c a t e s how any f e C 2(U) may be represented i n terms of any 2-dimensional subspace of U ( f ) . Theorem 11.17 Let f e Cg(U) and l e t x 2> be any 2-dimensional subspace of U ( f ) . Then e i t h e r ( i ) there e x i s t v,w i n U(f) such that f = yx^Ax^ + vAw , o =}= y e F 5 o r 21. ( i i ) There e x i s t v',w' i n U ( f ) such that f = x^Av' + XgAw' . Proof: Let x^,...,x^ be any basi s of U(f) . By Theorem I I . 16, f has a re p r e s e n t a t i o n f = x-^Au + vAw where = . Then we have two cases: Case 1: x-^AXgAf \u00E2\u0080\u00A2\u00E2\u0080\u00A2- o . Then \"H i s nonzero and so ctx-^ + px 2 c f o r some a, p e F , not both zero. Now a 4= \u00C2\u00B0 would imply x^ e c . This i s impossible and hence x 2 e . Therefore f has form ( i i ) . Case 2: x-^AXgA f 4= o . Then 0 i s zero, and hence = . But u = ax-j^ + b x 2 + cv + dw with b 4= \u00C2\u00B0 \u00E2\u0080\u00A2 Therefore f = bx-jAXg + [ X 1 A ( C V + dw) + vAw] . By C o r o l l a r y 1.14 and the f a c t that R(f) = 2 , [ x ^ f c v + dw) + vAw] has rank one. Hence f has form ( i ) . I f f e C 2(U) has a representation f = x i A x 2 + X ^ A X ^ , then the f o l l o w i n g theorem shows how f may be represented i n terms of any 2-dimensional subspace of U ( f ) which i n t e r s e c t s and . Theorem II.18 Let f = x 1 A x 2 + x^Ax^ e C 2 ( U ) . Let V be a 2-dimensional subspace of U(f) such that V fi and V n are both nonzero. Then f o r any basi s {v -^Vg} of V , there i s a representation f = v-jAu + VgAw , where = . Proof: v-^AVgAf = o . We have Case 1 of Theorem I I . 17 and the r e s u l t f ollows. 22. SECTION 2. P a i r s o f R a n k 2 V e c t o r s . I n t h i s s e c t i o n , we a r e I n t e r e s t e d i n i n d e p e n d e n t p a i r s o f v e c t o r s { f^,fg} i n H \u00E2\u0082\u00AC R g ( U ) . We s h a l l o b t a i n r e p r e s e n t a t i o n s f o r ^1^2 i n ^2^ U^ ' T o d o s o > w e ^ l - 1 -c o n s i d e r t h e s p a c e [ U ( f 1 ) + U ( f 2 ) ] . D e f i n i t i o n 11.19 P k ( U ) = { i t l i t 2 } / f 1 \u00C2\u00BB f 2 e \u00C2\u00B0 2 ( U ) a n d d i m [ U ( f 1 ) + U ( f 2 ) ] = k } {f^,. ..,?\u00E2\u0080\u00A2\u00C2\u00A3} a r e p a i r w i s e - P ^ i f e a c h p a i r f f , , f .} e P V ( U ) j i + i > 1 < i\u00C2\u00BB3 < k' \u00E2\u0080\u00A2 I n t h e c a s e w h e r e k' i s 2 j we s a y t h a t { f-^fg} i s a P ^ - p a i r . L e t f e c J ( U ) . S i n c e d i m U ( f ) = 4 , t h e n d i m U _> 4 f o r C 2 ( U ) t o be n o n - t r i v i a l . A l s o d i m U ( f \u00C2\u00B1 ) = 4 , i = 1,2, a n d s o d i m [ U ( f 1 ) + U ( f g ) ] < 8 . We n e e d t h e r e -f o r e c o n s i d e r o n l y P ^ - p a i r s , w h e r e 4 <_ k <_ 8 . We c o n s i d e r f i r s t a P ^ - p a i r . T h e o r e m 11.20 L e t H e Rg (U ) a n d { f - ^ f g } a P ^ - p a i r i n H . T h e n { f - ^ f g } i s d e p e n d e n t . P r o o f : L e t f ^ X ^ A X ^ + X ^ A X ^ . T h e n x ^ , . . . , x ^ i s a b a s i s o f U ( f ) a n d T h e o r e m 11.16, f g h a s a r e p r e s e n t a t i o n f = x-^Au + V A W w h e r e = . S i n c e , a r e 2 - d i m e n s i o n a l s u b s p a c e s of t h e 3 - d i m e n s i o n a l s u b s p a c e , t h e n d i m D _> 1 . B y p r o p e r t y 23. 4.c ( I n t r o d u c t i o n ) , we can without'loss of .generality assume x^ e o > and hence f g = x-^Au + x^Aw' . Now l e t u = S b.x, 1=2 w S d . X , ; i= 2 , 4 ' ; 1 1 f 2 = V ^ V J + V ^ J ^ V i b.,d \u00C2\u00B1 \u00E2\u0082\u00AC F . Then We now show that there e x i s t s a nonzero \ \u00E2\u0082\u00AC F such that + f g has rank at most one. This then implies that {f-^fg} cannot be independent i n H . For \ e F , z = \f\u00C2\u00B1 + f 2 = x 1 A ( x x 2 + b 2 x 2 + t > 5 x 5 + b 4x^) + xy.A(\xk + d 2 x 2 + d ^ ) . The c o n d i t i o n that the vectors x^, ( \ x 2 + b 2 x 2 + bye^ + b^x^) , x^ , \x^ + d 2 x 2 + d^x^ be independent i s equivalent to the c o n d i t i o n that the determinant r ( \ , f 1 3 f 2 ) = 1 0 0 0 0 \+b2 b 5 b 4 0 0 1 0 0 d 2 0 X+d^ be nonzero Now r ( \ , f 1 , f 2 ) = x 2 + x(d 4 + b 2 ) + ( b 2 d 4 - d 2 b 4 ) = g(x) . Since u,w' are independent, (t>2d^ - dgb^) 4 \u00C2\u00B0 \u00E2\u0080\u00A2 Thus r ( x , f 1 3 f 2 ) i s a n o n - t r i v i a l polynomial g(x) i n x a n d 3 a non- zero value of X i n F such that g(x) = o , Aand therefore.. r(x,f-j_,f 2) '= o . For t h i s value of X , R(z) <_ 1 . Hence f f ^ f g } are dependent i n H . C o r o l l a r y 11.21 Let dim U = 4 , H e R 2(U) . Then dim H = 1 . 2 4 . The f o l l o w i n g e x a m p l e shows t h a t i f F i s n o t a l g e b r a i c a l l y c l o s e d a n d d i m U = 4 , t h e n t h e r e e x i s t s a n H \u00E2\u0082\u00AC R|(U) o f d i m e n s i o n 2 . E x a m p l e 1 1 . 2 2 U = < x ^ , . . . , x ^ > ; F a R e a l s . x , y e c | ( U ) : X = X^AXg + XjAX^ y = X 1 A ( X ^ + X ^ ) + ( X ^ - X 2 ) A X ^ . F o r X e F z = \ x + y = x 1 A(xx 2 + x ^ + x i f ) + ( X X 5 + X ^ - X 2 ) A X ^ T ( X , x , y ) 1 0 0 0 0 X -1 0 0 1 x+i 0 0 1 0 1 = X ( X + 1 ) + 1 = g(x) Now g(x) = (X + 1/2) + 3/4 > o i n R e a l s . H e n c e p R ( z ) = 2 f o r a l l X e P . H e n c e H = e R 2 ( U ) , a n d d i m H = 2 . I n T h e o r e m 1 1 . 2 0 a n d E x a m p l e 1 1 . 2 2 , we see t h a t t h e n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n f o r x , y t o be i n d e p e n d e n t p i n H \u00E2\u0082\u00AC R 2 ( U ) , d i m U = 4 , i s t h a t t h e d e t e r m i n a n t P ( x , x , y ) be n o n z e r o i n F f o r a l l X . S i n c e t h i s d e t e r m i n a n t r(x . , x , y ) i s a q u a d r a t i c p o l y n o m i a l g(x) i n F , t h e c o n d i t i o n t h a t r ( X , x , y ) be n o n z e r o f o r a l l X i s e q u i v a l e n t t o t h e c o n d i t i o n t h a t g(X) be a n i r r e d u c i b l e , q u a d r a t i c p o l y n o m i a l i n F . 25-i t i s i n t e r e s t i n g to note that i f a f i e l d F has an i r r e d u c i b l e quadratic polynomial h(X) over i t , dim U = 4 } then we can f i n d two independent vectors x,y \u00C2\u00A3 II 3 H e R 2(U) > such that the corresponding determinant r(X,x 5y) i s equal to h(x) . In t h i s way,, given such a non-a l g e b r a i c a l l y closed f i e l d F s dim U = 4 } we can f i n d an H i n R 2(U) w i t h dim K =. 2 . The method used i s the f o l l o w i n g : Let dim U = 4 3 U = Oc-^ ..> . .X ; , > . Let h( X) = X -f a^X -i- a Q be i r r e d u c i b l e i n F (non - a l g e b r a i c a l l y closed). 0 1 The companion matrix of h(X) i s B = - a Q - a 1 X - I XI - B = a Q X-!-a1 1 0 0 0 0 'X 0 -1 uf \ \ X Now det (XI - B) = u K u = h( X ) T 0 0 1 0 0 a 0 X+a-, O JL, We now take t h i s determinant to be T(X,x >y) corresponding to z = Xx + y , where x^y e C|(U) ., X e F a x = x-^AXg + X ^ A X 1 ( _ y = x - L A(-x i t) + X ^ A ( a n x p + a. x^) . Then y are independent i n H \u00E2\u0082\u00AC R 2(U) , since z = X - L A ( Xx 2-x^) + X ^ A ( Xx^+a^g+a-jX^) has rank 2 f o r a l l X e F . We noxi i l l u s t r a t e by an example. r Example U = <2p...,x1|> , dim U = 4 3 F = Rationals. h(x) = \c 26. Then x = x-^Ax^ 4- xyNx^ s i n c e : y = X ^ A ( - X ^ ) + ( - 2)x^Ax 2 are independent i n H e R 2(U) For \ e F ; z = Xx + y = x 1A(X.x 2 - x^) + xy \ ( x ^ - 2 x 2 ) r ( ^ x , y ) 1 0 0 \u00E2\u0080\u00A2 0 0 X 0 - 1 0 0 1 0 0 -2 0 X = X 2 - 2 4= o i n F Hence dim = 2 . In the r e s t of t h i s t h e s i s , as i n a l l except the d i s c u s s i o n immediately preceding, we assume F i s a l g e b r a i c a l l y closed. We now show that i f { f - ^ f 2 ) i s a P^-pair , and are independent vectors i n H e R 2(U) , then k = 5 or 6 . We have already seen that k cannot be 4 . We must show that k cannot be J or 8 . p Lemma 11.23 Let f 1 ? f 2 e C 2 ( l l ) be a Pg-pair; o =}= a , ( 3 \u00E2\u0082\u00AC F , and z = a f 1 + p>f2 . Then R(z) = 4 . Proof: This follows immediately from Theorem 1.13. Lemma 11.24 Let { f ^ f g } be a P^-pair i n C 2(U) , a,p 6 F , both nonzero , z = a f ^ + f3 f 2 . Then R(z) = 3 . Proof: Let U ( f 1 ) fl U ( f 2 ) = . By Theorem I I . 16 , we can f i n d a b a s i s {x-^ ...,x^ } of U ( f 1 ) such that f ^ = x^Ax 2 + x^Ax^ , and a basis {x^}x^,x^)Xj} of U ( f 2 ) such 27. that f 2 = x\u00C2\u00B1Ax5 + XgAx 7 . Now since dim [ U ( f 1 ) + U ( f g ) ] = 7 dim [U(f-j_) 0 U ( F 2 ) ] = 1 , and {x-^Xg,. . . ,x } i s independent. For o + o ^ P e F , z = a f_^ + _ + QX ) 4- ax^Ax^ + pxgAx^ ; and R(z) = 3 by Theorem 1.13. We make the f o l l o w i n g d e f i n i t i o n f o r convenience. D e f i n i t i o n 11 .25 Let f , f g s Cg(U) . We define ,. \ f ( t l t t 2 ) = u ( f 1 ) n U ( f 2 ) . Theorem 11 .26 Let H \u00E2\u0082\u00AC R?(U) . Let [ L , . . . ^ , } be an independent subset of H . Then ( i ) 3 > dim W(f. , f .) > 2 f o r 1 <_ i < j <_ k ( i i ) dim ^ U(f ) < dim \u00C2\u00A3 U(f,) < dim ^ U(f.) + 2 . 1=1 1 i = i 1 ~ 1=1 1 Proof: Since f\u00C2\u00B1 e Cg(U) , dim U(f j L) = 4 , i < i < k . By Theorem 1 1 . 2 0 , Lemmas 1 1 . 2 3 , 24, dim [U(f.) + U(f .)] = 5 , 6 , 1 < i < j < k . Since W(f,,f^) = U ( f \u00C2\u00B1 ) n U(f j ) , the r e s u l t ( i ) fo l l o w s . F i n a l l y , ( i i ) follows d i r e c t l y from ( i ) . Theorem 11 .26 shows that any p a i r of independent p vectors i n H \u00E2\u0082\u00AC R 2(U) must be a P,.-pair or a Pg-pair. We s h a l l now obtain representations of such a p a i r of vectors, and these w i l l be extremely u s e f u l f o r f i n d i n g the basis of an H e R|(U) . Section 3 deals with a P^-pair of Independent vectors i n H ; Secti o n 4 deals with the Pg-pairs. We s h a l l i n f a c t show that such p a i r s can be expressed i n a form we c a l l a 28. (1,1) form, defined thus: 2 D e f i n i t i o n 11.27 For z e AU , we define z has a representation ~] V(z) = ,| {U 1 5U 2} j z = x x A x 2 + x 5Ax 4 e C 2 ( U ) and U 1 = , U 2 = j D e f i n i t i o n 11.28 {f-^fg} e c|(U) can be expressed i n ( l , l ) form i f there e x i s t f U 1 1 , U 2 i ) e V(f^) , i = 1,2 such that dim [ U 1 1 n U 1 2 ] =1 and dim [ U 2 1 n U 2 2 ] = 1 . Mote that i f {f ^ , ^ } can be expressed i n (1,1) form, then { f 1 ? f 2 ) have representations f-^ = X-j^A^ + XgAUg f 2 = x ^ A v 1 + X g A v 2 where = fi U ^ 2 and = U 2 1 D U g 2 SECTION J. The P c - P a i r s . In t h i s s e c t i o n we obtain representations f o r a P^-pair of independent vectors i n H e R 2(U) . Theorem 11.29 Let H e R 2(U) and l e t {f-^fg} be a P ^ p a i r of independent vectors i n H . Then there i s a representation: f l = y 4 A u l + u 2 A u 3 f 2 = y 5 A ^ 2 + u i A u 3 where { u ^ u ^ u ^ y ^ y - p . } i s some basis of [ g ( f 1 ) + U ( f 2 ) ] and = W ( f x , f 2 ) . 29. P r o o f : L e t U = W ( f n , f 0 ) . L e t x 1 e U(f ] _) , x\u00C2\u00B1 1 U Q a n d x 2 e U ( f g ) , x ? | U Q . T h e n b y T h e o r e m 11.16, t h e r e a r e r e p r e s e n t a t i o n s f l = x l A v l + V v 3 f 2 = x 2 A w - L + WgAw^ , w h e r e < v 1 >Vg,v ^ > = = U O . We show f i r s t t h a t v^^w^ a r e i n d e p e n d e n t . S u p p o s e o n t h e c o n t r a r y t h a t t h e y a r e d e p e n d e n t ; i . e . = , o ^ \ e F . T h e n s i n c e = < w 1 , W g , w ^ > , we h a v e v^AVgAv-^ = juw^AWgA^^ f o r some o 4= u e F ; h e n c e V ^ A ( VgAv^ - XjiWgAw^) = o, T h i s I m p l i e s VgAv^ - XuWgAw^ = w.^Az f o r some z e U q (Lemma 1.8). B u t t h e n - X u f g = - X^iXg - Z ) A W - ^ h a s r a n k o n e , c o n t r a r y t o h y p o t h e s i s . T h u s v-^w-^ a r e i n d e p e n d e n t . Now s i n c e < v 1 , w 1 > n a n d D a r e b o t h n o n z e r o , we may assume w i t h o u t l o s s o f g e n e r a l i t y t h a t b o t h V g a n d Wg a r e i n < v 1 , w 1 > . L e t Vg = av^ + b w 1 , Wg =\u00C2\u00BB c v 1 + dw^ . C l e a r l y b ^ o , c ^ o . F i n a l l y , w^ e = i m p l i e s w^ = av^ + pw^ + yv^ , w h e r e y 4= o ( s i n c e o t h e r w i s e = , w h i c h i s i m p o s s i b l e ) . Now we s e t y 4 - b Y \" 1 c \" 1 ( x 1 - a v ? ) y 5 - x g - dw^ + cp V ] _ u l \" b _ l Y C V l ' U2 = w l ' U5 = b v 3 a n d o b t a i n t h e r e p r e s e n t a t i o n s : t1 \u00C2\u00BB y ^ A ^ + UgA^ f 3 = y 5 A U 2 + u l A u 3 ' 30. C o r o l l a r y I I . 30 L e t H e R ^ U ) and l e t { f ^ f g } b e a P 5 - P a i r o f Independent v e c t o r s i n H . Then { f 1 > f 2 ^ can be e x p r e s s e d i n ( 1 , 1 ) form. P r o o f : By Theorem 1 1 . 2 9 , t h e r e a r e r e p r e s e n t a t i o n s f 2 = y ^ A u 2 + u xAu^ = ( - U - 5 ) A U 1 4- UgA(-y 5) Thus {f-j^jf 2} a r e i n ( 1 , 1 ) form. SECTION 4. The P g - p a i r s . T h i s s e c t i o n d e a l s m a i n l y w i t h a P g - p a i r o f ind e p e n d e n t v e c t o r s i n H e R 2 ( U ) . We show t h a t a P g - p a i r can be e x p r e s s e d i n ( 1 , 1 ) form (Theorem II.3 1). C o r o l l a r y II.3 2 r e v e a l s a s t r o n g \" s t r u c t u r e r e l a t i o n \" i n the P g - p a i r . Theorem II.31 L e t H \u00E2\u0082\u00AC Rg(U) . L e t {f^,f^} be a P g - p a i r o f i n d e p e n d e n t v e c t o r s i n H . Then {f^,fg} can be e x p r e s s e d i n ( l , 1) form. P r o o f : L e t c W(f tfg) . By Theorem II.16, t h e r e a r e r e p r e s e n t a t i o n s f ^ = x^Au 4- vAw , f 2 = X ^ A U ' 4- v ' A W ' , where and are 3 - d i m e n s i o n a l subspaces o f ; [ U ( f 1 ) 4- U ( f 2 ) ] = (x 1,...,Xg> . F o r a,B, e F , z = a f 1 + p f = x 1 A ( a u + pu') + avAW + pv'Aw' , a n d 5 . We now show n < v ' , w ' > 4= o . S u p p o s e o n t h e c o n t r a r y t h a t n = o . I f f o r some a,p ; .say a ' , p ' , t h e v e c t o r a ' u + p ' u ' 4 , t h e n f x 1 , a ' u + p ' u ' , v , w , v ' , w ' } i s i n d e p e n d e n t , a n d R(z) = 3 b y T h e o r e m 1.13. On t h e o t h e r o t h e r h a n d , i f a u + pu' \u00E2\u0082\u00AC f o r a l l a , p e F , t h e n d i m < x 1 , u , u ' , v , v ' , w , w ' > = 5 c o n t r a d i c t i n g t h e h y p o t h e s i s t h a t { f - ^ f 2 } i s a P g - p a i r . H e n c e n < v ' , w ' > =)= 0 \u00E2\u0080\u00A2 Now 4= f o r i f - , t h e n f o r some o ^ \ e F , f ^ = X-^Au 4- vAw , f 2 = X-^Au' + \ V A W ; a n d f-^ - \~\"^f2 = X ^ A ( U - x _ 1 u ' ) h a s r a n k o n e . T h e r e f o r e , d i m fl = 1 , a n d f f - ^ f g } a r e i n (1 ,1) f o r m . C o r o l l a r y 11 .32 L e t H e R|(U) a n d l e t { f - ^ f g } be a P g - p a i r o f i n d e p e n d e n t v e c t o r s i n H . L e t {x 1,...,Xg} be some b a s i s o f [ U ( f x ) + U ( f 2 ) ] s u c h t h a t c W ( f 1 , f 2 ) a n d f l = x i A u + v A w y f 2 = x i A u ' + v ' A w ' w h e r e , < u ' , v ' , w ' > a r e c o n t a i n e d i n . T h e n d i m D = 1 . P r o o f : The p r o o f i s c o n t a i n e d i n t h e p r o o f o f T h e o r e m I I . 3 1 . 32. We s h a l l o b t a i n a n o t h e r c o r o l l a r y t o T h e o r e m I I . 3 1 ; b u t f i r s t we n e e d t h e f o l l o w i n g D e f i n i t i o n 11 .33 { f -^ . . .,f^} i s a ( 1 , 1 ) b a s i s f o r i f < f 1 , . . . , f ] t > e R | ( U ) a n d t h e r e e x i s t { U j ^ U g l ^ e V ( f \u00C2\u00B1 ) , 1 < i < k ( s e e D e f i n i t i o n I I . 2 7 ) s u c h t h a t k k d i m f l U , = 1 a n d d i m D U = 1 . 1=1 1 1 i = l 2 1 C o r o l l a r y 11 .34 L e t H e R 2 ( U ) , l e t { f - ^ f ^ f ^ } be p a i r w i s e -Pg , i n d e p e n d e n t i n H , s u c h t h a t U ( f ^ ) o W ( f 1 , f g ) . T h e n { f - ^ f ^ f } i s a ( 1 , 1 ) b a s i s f o r < f 1 , f 2 , f ^ > . P r o o f : L e t {u-^Ug} be a b a s i s o f W ( f 1 3 f 2 ) . B y T h e o r e m I I . 31 , t h e r e a r e r e p r e s e n t a t i o n s f-^ = u 1 A v 1 + u 2Aw 1 , f 2 = u 1 A v 2 + UgAWg . S i n c e U ( f ^ ) 3 < u l S u 2 > a n d { f 2 , f ^ } i s a P g - p a i r , t h e r e i s a r e p r e s e n t a t i o n f ^ = u ^ v ^ + u 2Aw^ . B y d e f i n i t i o n , { f ^ f ^ f ^ } i s a ( 1 , 1 ) b a s i s f o r < f 1 , f 2 , f 5 > . 3 2 . CHAPTER I I I THE RANK TWO SUBSPACES WHEN U HAS DIMENSION 5 I n t h i s c h a p t e r , we w i l l show t h a t , when d i m U = 5 , t h e r a n k two s u b s p a c e s h a v e d i m e n s i o n a t m o s t t h r e e ( T h e o r e m I I I . 3 8 ) . I f t h e r a n k two s u b s p a c e h a s d i m e n s i o n t w o , t h e n i t h a s a ( l , l ) b a s i s ( D e f i n i t i o n I I . 3 3 ) . I f t h e r a n k two s u b -s p a c e h a s d i m e n s i o n t h r e e , t h e n i t h a s a b a s i s o f v e c t o r s whose r e p r e s e n t a t i o n s a r e g i v e n i n T h e o r e m I I I . 4 5 . We a l s o show t h a t i f d i m U = n a n d i f H i s a r a n k two s u b s p a c e w i t h a (1,1 ) b a s i s , t h e n d i m H <_ n - 3 ( T h e o r e m I I I . 4 o ) . We w i l l n e e d t o r e f e r t o t h e f o l l o w i n g r e s u l t ( s e e p . 14 o f [3]): R e m a r k I I I . 3 5 L e t {V^,...,Vk} b e n - d i m e n s i o n a l v e c t o r s p a c e s t h a t i n t e r s e c t p a i r w i s e i n d i m e n s i o n ( n - l ) . T h e n e i t h e r t h e r e e x i s t s a n ( n - l ) - d i m e n s i o n a l s p a c e W Q s u c h t h a t Y\u00C2\u00B1 3 W Q , 1 \u00E2\u0080\u00A2< i <_ k o r t h e r e e x i s t s a n (n+1) - d i m e n s i o n a l s p a c e W s u c h t h a t V j . c W , 1 <_ i <: k . The f o l l o w i n g two r e m a r k s w i l l be u s e f u l f o r f i n d i n g t h e maximum d i m e n s i o n o f t h e r a n k two s u b s p a c e s , a n d f o r f i n d i n g t h e i r b a s e s . 33. Remark I I I . 36 L e t dim U = 5 , H e R 2 ( l l ) . L e t { f 1 , . . . ,f^} be i n d e p e n d e n t i n H . Then {f-L>. . . , f k ] a r e p a i r w i s e - P ^ . P r o o f : S i n c e dim U(f j.) =-4 , i = 1,...,k , and dim U = 5 , t h e n dim \u00C2\u00A5 ( f i , f j ) > 3 , i ^ j . H i j a . By Lemma 1.19 ( l ) , dim W ( f \u00C2\u00B1 , f j ) = 3 , i f j , 1 <_ i , j < k . Hence {f^, . . . , f f e } a r e pairwise-P,- . Remark I I I . 3 7 L e t { f ^ = o : i = 1 ,...,r] be a system o f homogeneous q u a d r a t i c e q u a t i o n s i n i n d e t e r m i n a t e s w i t h c o e f f i c i e n t s i n an a l g e b r a i c a l l y c l o s e d f i e l d F . I f r < n , t h e n t h e r e e x i s t s a n o n - t r i v i a l s o l u t i o n f o r the system o f homogeneous e q u a t i o n s i n F . The above r e s u l t i s c o n t a i n e d i n Chapter 11 o f [ 4 ] . Theorem I I I . 38 L e t dim U = 5 > H e R ^ U ) . L e t [ f - ^ , . . . , f k ) be i n d e p e n d e n t i n H . Then k < 3 . P r o o f : L e t [ x 1 , . . . , x , - J be a b a s i s o f U . Then each f ^ , 1 <_ 1 <_ k has the f o r m * f = E a f . x . A x , ; a | . e F . l 4 and some {B^} n o t a l l z e r o . 34. Now z = \u00C2\u00A3 8,f, = \u00C2\u00A3 p ( i , i 0 ) x , A X . where 1=1 1 1 l ^ X i ^ 1 2 1 1 x2 p ( k a ( l ) , k a ( 2 ) = s g n CT * P ( k i > k 2 ^ * a i s a p e r m u t a t i o n o f {1,2} , and {k.^} a r e a r b i t r a r y i n t e g e r s 1'<_ ^ <. 5 , i = 1,2 . From (*) and the above statement, i t i s easy t o see t h a t { ^ ( i - ^ i g ) } a r e l i n e a r homogeneous f u n c t i o n s o f {^,...,6^.} . Hence the n o n - t r i v i a l e q u a t i o n s i n the sustem o f e q a t i o n s ( l ) i n Theorem 1.2 , v i z : ( w i t h r = 2, n = 5) (1) p ( l , 2 ) p ( 3 , 4 ) + p( l , 3 ) p ( 4 , 2 ) + p(l , 4 ) p ( 2 , 3 ) = o (2) p ( l , 2 ) p ( 3 , 5 ) + p ( l , 3 ) p ( 5 , 2 ) + p( l , 5 ) p ( 2 , 3 ) = o (**) (3) p ( l , 2 ) p ( 4 , 5 ) + p(l , 4 ) p ( 5 , 2 ) + p( l , 5 ) p ( 2 , 4 ) = o W P ( l , 3 ) p ( 4 , 5 ) + p(l , 4 ) p ( 5 , 3 ) + p ( l , 5 ) p ( 3 ^ ) = o (5) p(2,3)p(4,5) + p(2,4)p(5,3) + P(2,5) P(3,4) = o a r e i n f a c t q u a d r a t i c homogeneous i n the i n d e t e r m i n a t e s B-^,...,^ i n F . A l s o , by Theorem 1.2, (**) has 3 i n d e -pendent e q u a t i o n s . Hence, by Remark I I I . 3 7 , i f k > 4 , t h e n t h e r e e x i s t s a n o n - t r i v i a l s o l u t i o n f o r (**) . F o r t h e s e v a l u e s o f B 1,...,P k , n o t a l l z e r o ; . R ( z ) <_ 1 by Theorem 1.2. Hence i f { f ^ , . .. , f k } e H , the n k < 4 . We w i l l now e x h i b i t a 3 - d i m e n s i o n a l r a n k 2 subspace i n R 2 ( U ) . Example I I I . 3 9 U = . H - < f - L , f 2 , f 5 > ; . n u ( f \u00C2\u00B1 ) = . 35. f , = X)tAU-, + U^AUU 1 4 1 3 2 d 5 2 3 1 f 2 = ( x ^ + X^)AU^ + UgAU-^ 3 F o r e F ; z = - E a ^ i = u-j^ Aa-j^ x^ + u 2 A ( a 1 u ^ +\u00E2\u0080\u00A2 a 2 x ^ +. a^u.^) + u^A(a^x^ + a^x^ - a2u]_) \u00E2\u0080\u00A2 I f a n y one o f [a\u00C2\u00B1) i s z e r o , R ( z ) = 2. I f + o , 1 < 1 < ^ \u00C2\u00BB \u00C2\u00B0S z = u i A a i x 4 + ( u 2 + a u 3 ^ A ^ a l u 3 + ' a 2 x 5 + a 3 u i ' + u y \ ( a y t ^ - ( a 2 + a ^ ) ^ ) = ( u l + H 7 U 3 ) A ( a l X 4 - H ^ a 2 + a 3 ) u l - } 1 v 3 * < u2 + \u00C2\u00A7S ) A ( a l u3 + a 2 X 5 + a 3 U l } d a n d R ( z ) = 2 . R e s u l t s o b t a i n e d l a t e r i n t h i s p a p e r w i l l j u s t i f y o u r I n t e r e s t i n a p a r t i c u l a r k i n d o f r a n k two s u b s p a c e . T h i s i s t h e r a n k two s u b s p a c e w i t h a ( 1 , 1 ) b a s i s ( s e e D e f i n i t i o n II.3 3 ) . We now o b t a i n t h e maximum d i m e n s i o n o f a r a n k two s u b s p a c e w i t h s u c h a b a s i s when d i m U = n . We g i v e ' a l s o a n e x a m p l e o f s u c h a b a s i s . T h e o r e m III.40 L e t d i m U = n . L e t {f , . . . , f k } be a (1 ,1) b a s i s f o r < f 1 , . . . , f k > \u00E2\u0082\u00AC R|(U) . T h e n k = ( n - 3 ) . 36 , P r o o f : S u p p o s e k = n-2 . T h e n we can r e p r e s e n t f l = U l A y l + u 2 A z i f 2 = u x A y 2 + U 2 A Z 2 f n - 2 * u l A y n - 2 + V Z n - 2 > < u 1 , u 2 , y 1 , . . . , y n _ 2 , Z l , . . . , z n _ 2 > E U . Now { u ^ u ^ y ^ , . . . j y n _ 2 ^ ' m u s t b e i n d e p e n d e n t f o r , i f n o t , some l i n e a r c o m b i n a t i o n o f f - ^ , . . . , f n 2 h a s t h e f o r m u-^Av + \ V A W } o =(= XeP a n d t h i s h a s r a n k <_ o n e . T h i s i m p l i e s U = < u 1 , u 2 , y 1 , . . . j y n _ 2 > s i n c e d i m U = n . H e n c e {z^^,. . . , z n _ 2 } i s d e p e n d e n t o n {\x-L>y1,. . . , y n _ 2 3 . T h u s n - 2 Z j * = j ^ i a J y J + P j u l ' a j * P j e F } 1 i J 1 n~ 2 I f =1= o , w r i t e f j = u 1 A ( y J - p j U 2 ) + u 2 A ( n E 2 a j y j ) j 1 < j < n - 2 . H e n c e , w i t h o u t l o s s o f g e n e r a l i t y , we c a n assume [ z 1 , . . . , z n _ 2 3 i s d e p e n d e n t o n { y - ^ , . . . >yn_2^ \u00E2\u0080\u00A2 U s i n g a s i m i l a r a r g u m e n t t o t h e one a b o v e , we c a n s a y f y l ' ' ' * * yn - 2 ^ i s d e P ' e n d e n 1 i o n lz\u00C2\u00B1> \u00E2\u0080\u00A2, \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 >zn-2^\" \u00E2\u0080\u00A2 H e n c e < y l ' \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 ' y n _ 2 > = < z 1 , . . . , z n _ 2 > \u00E2\u0080\u00A2 T h e r e f o r e f o r some \u00E2\u0082\u00AC P , n o t a l l z e r o , n - 2 n - 2 , E a . y , = \ L a . z . = y f o r some o f X e F , a n d i - l 1 1 1=1 1 1 n - 2 1 z = \u00C2\u00A3 a , f , = u . Ay 4- u Q A \ y h a s r a n k a t m o s t o n e . H e n c e 1=1 1 1 1 d 3 7 . k i s at most (n-3) \u00E2\u0080\u00A2 We now e x h i b i t (n-3) such vectors {f^} i n p H e R 2 ( u ) ' L e t U = < ui> u2* x3> . . . , y f 1 = U-j^Ax^ + UgAX^ f 2 = U-^AX^ + UgAX^ fn-4 ^ U l A X n - 2 + V X n - l f n - 3 = U l A X n - l + U 2 A x n For a i \u00E2\u0082\u00AC F , not a l l zero , n - 3 z = E fs i = l 1 /n-3 \ /n-3 = V ^ V i + a ) + U 2 A l 1 ? 1 a i X i + 3 and R(z) = 2 by Theorem 1,15- Thus any l i n e a r combination z of {f-^, . . . ,f n_^} has rank 2 . This implies that \" ^ l * ' ' * ***n-3 a r e independent and generate a rank two subspace. C o r o l l a r y III.41 Let dim U = 5 , H e R|(U) , with dim H > 1 , and l e t { f \u00C2\u00B1 , b e a (1,1) basis f o r H . Then k = 2 . We have seen so f a r that i f dim U = 5 , and H e R 2(U) , then H has dimension at most 3 . I f dim H = 2 , then H has a ( l , l ) ba s i s ( C o r o l l a r y I I . 3 0 ) . We see by Example III.3 9 that there do e x i s t 3-dimensional subspaces H \u00E2\u0082\u00AC R 2(U) . By C o r o l l a r y III.41, such subspaces cannot have 38. (1,1) b a s e s . We s h a l l p r o c e e d t o o b t a i n t h e b a s e s o f H when H h a s d i m e n s i o n t h r e e . I f { f 1 , f 2 , f ^ } i s a b a s i s o f H , t h e n b y R e m a r k I I I . 3 6 , { f - j ^ f g , ^ } a r e p a i r w i s e - P , - . T h u s d i m W ( f i , f j ) = 3 , i + j , 1 < i , j < 3 . Now d i m U ( f \u00C2\u00B1 ) = 4 ; i = 1,2,3-H e n c e b y R e m a r k I I I . 3 5 , e i t h e r U(f ^) o U , d i m U o = 3 , i = 1,2,3 o r U ( f i ) c w , d i m W = 5 , i = 1,2,3 \u00E2\u0080\u00A2 We n o t e now t h a t s i n c e d i m U = 5 , d i m W ( f ^ , f g ) = 3 a n d d i m U(f^) = 4 , t h e n d i m [ W ( f 1 , f g ) n U ( f ^ ) ] >_ 2 . Thus d i m f? U(f.) > 2 . B u t s i n c e { f , , f p , f } are , p a i r w i s e - P , i - l 1 y . 5 t h e n d i m hi U ( f . ) =2,3 . We s h a l l p r o v e two t h e o r e m s w h i c h i = l 1 g i v e r e p r e s e n t a t i o n s f o r { f 1 , f g , f J i n e a c h o f t h e s e c a s e s ; b u t f i r s t i t w i l l be c o n v e n i e n t f o r u s t o h a v e t h e f o l l o w i n g d e f i n i t i o n s . D e f i n i t i o n I I I . 4 2 F o r s u b s e t s S , T o f U , [ S ; T ] = \ . I n t h e c a s e w h e r e S = { x ^ , . . . , x } a n d T = { x g + 1 , . . . , x k } , x 1 c U , 1 \u00E2\u0080\u00A2< i <_ k , we s h a l l u s e t h e f o l l o w i n g c o n v e n t i o n ? [ S ; T ] = [ x 1 , . . . , x s ; x s + 1 , . . . , 0 ^ ] \u00E2\u0080\u00A2 We n o t e t h a t i n t h i s c a s e , i f y e [ S ; T ] , t h e n y = 2 a . x , , a , e F , 1 <: i <. k , w h e r e a t l e a s t one o f i = l 1 1 1 a-, > \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 i s n o n z e r o . 39. D e f i n i t i o n I I I . 4 5 . F o r s u b s e t s S , T o f U , SAT = {xAy : x e S a n d y e T} . I n t h e c a s e w h e r e S i s t h e s i n g l e t o n {x} , we s h a l l w r i t e SAT = xAT . S i m i l a r l y f o r T . A l s o , i f S i s t h e s p a c e < x ^ , . . . > x k > , t h e n , t o k e e p t h e n o t a t i o n c o n s i s t e n t , we s h a l l r e g a r d S a s a s e t a n d w r i t e SAT as [x^,...,x f c]AT . S i m i l a r l y f o r T . T h e o r e m I I I . 4 4 L e t d i m U = 5 . L e t { f - ^ f 2 , f } be a b a s i s f o r H e R g ( U ) , s u c h t h a t U ( f i ) z> U Q , d i m U Q = 3 , i = 1 , 2 , 3 \u00E2\u0080\u00A2 T h e n U h a s a b a s i s { u - ^ u ^ u ,x^,Xj-} , U\" o = < u 1 , u 2 , u ^ > s u c h t h a t { f - ^ f 2 , f ^ } h a v e r e p r e s e n t a t i o n s : -f l = X^AU^ + U^U - J f 2 = X^AU 2 + U-, AU^, 1 3 f 3 = - yAu^ + U 2 A U 1 y y \u00E2\u0082\u00AC [ x ^ ; x ^ , u ^ , u 2 ] n [ x ^ ; x ^ , u ^ , u 2 ] P r o o f : S i n c e { f - ^ f g } i s a P ^ - p a i r , t h e n b y T h e o r e m 1 1 . 2 9 , U h a s a b a s i s { w 1 , w 2 , w ^ , y i l _ , y ^ } U 0 = W ( f - L , f 2 ) = , s u c h t h a t f 1 = y^Aw 1 + WgAW^ , f 2 = y 5 A ^ 2 + wxAw^ , Now U ( f , ) 3 U , d i m U ( f J =4 a n d d i m U = 5 . p o \u00E2\u0080\u00A2 p H e n c e t h e r e e x i s t s y ' e ^ ( f ^ ) s u c h t h a t y ' 4 u 0 > y ' \u00C2\u00AB [ y . V y 5 ; w i ^ w 2 j W 3 - ' \" B u t ^ f l j f 2 ' f 3 ^ a r e P a l r w i s e - p 5 \u00E2\u0080\u00A2 H e n c e 40. y' G iy^i y y w-^w^w^My^; y^w^v^w ] . By Theorem 11.16, f h a s a re p r e s e n t a t i o n = y'Au + vAw , = U . j- J o We show now that u 1 . Suppose on the contrary that u e ; i . e . , u = aw 1 + bwg , a,b e F . Since , are 2-dimensional subspaces of the 3-dimensional subspace U Q , then dim n _> 1 . Hence we can represent f ^ = y'A(aw 1 + bw2) + (cw1 + dw2)Aw' . Then f ^ = ( a y ' - c w ' ) A W 1 + (by' - dw')Aw2 , making { f 1 , f 2 , f ^ } a (1,1) basis f o r H . This c o n t r a d i c t s Theorem I I I . 4 0 , and the hypothesis that dim H = 3 . Hence u ^ , and u e [wy, w^,w2] . Without l o s s of ge n e r a l i t y , we can say u = w^ + c-^ w^ + CpW^ , c^ e F . We have now: f l = y ^ A w i + w2Aw^ = ( y ^ - C 1 W 2 ) A W 1 + W 2 A U , f 2 = y 5 A w 2 + W X A W ^ = (y 5 - G 2 W 1 ) A W 2 + WjAu , f = y'Au + V A W . Since , and are 2-dimensional subspaces o f U o , they i n t e r s e c t pairwise i n dimension at l e a s t one. Now u 4 . Hence we may suppose v \u00E2\u0082\u00AC [w2; u] , w e [w^ u] . Let v = aw 2 + bu , w = cw1 + du . Then f = y'Au + (aw 2 + bu)A(cw 1 + du) = (y' + adw - bcw 1)Au + Y^2A'W1 ' o 4= Y e F . P 2 - l -1 Let or = y 3 w2 = a \xQ , w\u00C2\u00B1 = a u\u00C2\u00B1 , u = au^ . Then f 1 = (y^ - c 1w 2)Aa\" 1u 1 + u 2Au^ , f 2 = (y 5 - C 2W 1)ACT 1U 2 + ^ A U ^ , 41. f ^ = j\" Aan^ + u - 2 A u i We h a v e t h e r e s u l t o n s e t t i n g x ^ = a - 1 ( y ^ -c- jWg) , x 5 = a _ 1 ( y 5 - , y = a y \" , a n d n o t i n g t h a t y \u00E2\u0082\u00AC [ x ^ j x 5 , u 1 , u 2 ] f l ( x 5 ; x ^ u - ^ U g ] . T h e o r e m I I I . 4 5 L e t d i m U = 5 \u00E2\u0080\u00A2 L e t ' Cf^fg,!\" } be a b a s i s f o r H e R p ( U ) - s u c h t h a t d i m fl U ( f ) = 2 . T h e n U h a s a 1=1 1 b a s i s < u 1 } u 2 , u ^ x ^ x 5 > s u c h t h a t { f ^ f ^ f ^} h a v e r e p r e s e n t a t i o n s g i v e n b y e i t h e r ( l ) o r ( 2 ) b e l o w : -3 (1) ^ 0 ^ U ( f \u00C2\u00B1 ) = j u e < u 1 > u 2 > , W ( f l 5 f 2 ) = < u x , u 2 , u 5 > . f ^ = x ^ A U - [ _ + u 2 A u 3 f g = Xj-AUg + U^AU^ = uAy + u^Ay' y , y ' e [ x ^ x ^ u ^ u ^ u ^ ] . 3 ( 2 ) _0 U ( f 1 ) = < u , u ' > , U \u00E2\u0082\u00AC < u - L , U g > , u ' \u00C2\u00A3 < u 1 , U g , u ^ > . = Xj^AU^ + UgAU^ f g = X^AUg + U - j ^ A U ^ f ? = yuAu' + yAy' , y , y ' e [ x ^ x ^ u ^ u ^ u ^ ] \u00E2\u0080\u00A2 O + Y 6 F \u00E2\u0080\u00A2 P r o o f : S i n c e { f - ^ f g ) i s a P ^ p a i r , b y T h e o r e m 1 1 . 2 9 , U h a s a b a s i s { u ^ U g j U ^ y ^ y g ) s u c h t h a t { f - ^ f g } h a v e r e p r e s e n t a t i o n s 42. f 2 = y 5 A U 2 + U 1 A U 3 ; W ( f 1 ? f 2 ) = . By hypothesis, dim f[ U(f ) = 2 . Let = /? U(f, ) . \" i = l 1=1 1 T h i s . i n t e r s e c t i o n i s contained i n . 1 d J Since are 2-dimensional subspaces of the 3-dimenslonal space then dim n _> l . Hence, by property 4.c (Introduction) we can assume u e . Since U(f^) => , then by Theorem 11.17 f ^ has a r e p r e s e n t a t i o n e i t h e r uAv' + U ' A W ' or Y U A U ' + vAw f o r some o f Y e F \u00E2\u0080\u00A2 Suppose f ^ = U A Y ' + U ' A w t . I f = then i t follows from Lemma II . 1 8 that f ^ = u ^ v ' + UgAw* . Then { f 1 , f 2 , f ^ } i s a ( l , l ) ba s i s f o r H , c o n t r a d i c t i n g C o r o l l a r y I I I . 4 l . Hence 4\u00C2\u00BB . Therefore, since u e , u' e [u^; \x1}u.2] . How we can without l o s s of g e n e r a l i t y assume u e [u, j u_] , since i t i s easy to see that a s i m i l a r case holds 1 2 i f u \u00E2\u0082\u00AC [ u 2 ; U.J . Thus, we can assume u' e [u^; u 2] . Let u' = au^ + b u 2 ; a ^ Q-. I f b = o , then f 5 = U A V ' + uy\aw'; hence {t\u00C2\u00B1; i =1,2 ,3} have represent-ations as i n (1). I f b \u00C2\u00B1 o , we rewrite f\u00C2\u00B1 = y ^ A ^ + a - 1 u 2 A ( a u 5 + bu 2) ; f g = y ^ U g + i ^ A u ^ = (y^ - a~ 1bu 1)A u 2 + a * 1 u 1 A ( a u 5 + bug) . Now f ? = uAv' + ( a u ? + bUg)Aw' . Writing u x f o r a - 1 ^ , u g f o r a - 1 u p ; x^ f o r a y 4 , 43. x 5 f o r a ( y 5 - a ~ 1 b u 1 ) , and u^ f o r ( a u ^ + b u 2 ) , we have (1) . I t i s easy t o see t h a t v', w' e [x^,x,_; u ^ u ^ u ^ ] . I n t h e a l t e r n a t i v e c a s e , f = YuAu' + vAw f o r some o i Y e F \u00E2\u0080\u00A2 T h i s i s J u s t case ( 2 ) , s i n c e dim n U ( f ) = 2 , 1=1 1 and hence v,w e [ x h , x _ ; u_,u_,u_] , 4 5 1 2 3 Prom the p r e c e d i n g two theorems, i t would seem t h a t any H e Rg(U) such t h a t dim H = 3 and dim U = 5 would have a b a s i s w h i c h i s one o f t h r e e t y p e s . I t i s i n t e r e s t i n g to n o t e t h a t H i n f a c t has a b a s i s which i s one o f the two t y p e s i n the second theorem ( i . e . I I I . 4 5 ) . To s u b s t a n t i a t e t h i s c l a i m , we s h a l l show t h a t the b a s i s o f H i n Theorem III.44 can be r e p l a c e d by one o f those i n Theorem II I . 4 5 . Theorem III.46. L e t dim U = 5 , H e Rg(U) , dim H = 3 . Then H has a b a s i s {f, , f 0 , f , } such t h a t dim r? U ( f , ) = 2 . 1 d <> 1=1 1 P r o o f : By Remark I I I . 3 6 , i f {-f , f 2 , f ^ } i s a b a s i s o f H , t h e n { f i : i = 1,2,3} are p a i r w i s e - P 5 . S i n c e W ( f - L , f 2 ) i s a 3 - d i m e n s i o n a l subspace and U(f^) i s a 4 - d i m e n s i o n a l subspace of the 5 - d i m e n s i o n a l space U , t h e n d i m [ U ( f , ) 0 W ( f 1 , f j ] > 2 i . e . dim fi U ( f ) > 2 . 3 1=1 A 3 I f dim fl U ( f 1 ) = 2 , we have the r e s u l t . I t remains 1=1 1 3 t o show t h a t - i f dim D U ( f , ) = 3 > t h e n t h e r e e x i s t s a b a s i s 1=1 w i t h the r e q u i r e d p r o p e r t i e s . 44. 3 If d i m n U(f,) = 3 , t h e n b y T h e o r e m 111,44 , U h a s 1=1 1 a b a s i s { u ^ U g j U ^ x ^ x , - } s u c i l t h a t [f 1 5 f 2 , f ^ } h a v e r e p r e s e n t a t i o n s f l = X^AU^ + UgAu^ , f 2 = X R A U . + U , A U , , 5 2 1 \u00E2\u0080\u00A2 3 f 3 = yAu^ + U g A l ^ y \u00E2\u0082\u00AC [ x ^ j x ^ U - ^ U g l n [ X 5 J X ^ , U - L , U 2 ] L e t S i - f l + f 2 = ( x ^ - U ^ ) A U 1 + (x_ - U , ) A U , : v 5 3 2 u(gx) = < x 4 L e t g 2 = f 2 ; g-j = f j . T h e n n u ( g ) = , a n d h a s d i m e n s i o n 2 . H e n c e i=r 1 {g-^, g 2 , g-^ } i s a b a s i s f o r H w i t h t h e r e q u i r e d p r o p e r t i e s . The f o l l o w i n g a r e e x a m p l e s o f r a n k two s u b s p a c e s o f d i m e n s i o n 3 , when d i m U = 5 , o f t h e two t y p e s g i v e n i n T h e o r e m III.45. N o t e t h a t E x a m p l e III.39 i s an e x a m p l e o f H e Rg(U) , d i m U = 5 ; d i m H = 3 > where t h e b a s i s { f r f g , f 5 } i s 3 U ( f \u00C2\u00B1 ) 3 U Q j - d i m U o = 3 ; i = 1,2,3. E x a m p l e III. 47 U = < u 1,Ug , u ^ , x ^ , x 5 > ; f ? U ( f i ) = f l = X^AU^ + UgAU^ f 2 5 2 + U N AU, 1 3 f 3 = UgAx^ + 3 5 3 F o r a \u00C2\u00B1 e F ; z = Z a j ^ f i 2 A ( a i u 5 - a 2 x 5 + a ? x 4 ) +..u 1A(a 1x 4 + agU^) = u . 4- a,u,Ax,-3 3 5 45. If = o ; z = U g A ( - a 2 x ^ + -a^x^) + a g U ^ A u ^ + a ^ u ^ A x ; -= u 2 A ( - a 2 x 5 + a 5 x 4 ) + u 5 A ( a 5 x 5 - a ^ ) and R ( z ) = 2 . If a 2 _ = o ; z = x ^ A ( a 1 u 1 - a ^ u 2 ) + U ^ A ( - ^ U g 4- a ^ x ^ ) and R ( z ) =2 . If a i 4 \u00C2\u00B0 > i = 1,2,3; t h e n z - a 3 u 2 A ( ^ i - ^ x 5 + x 4 ) + o ^ u ^ + ^ u^) + 0 ^ X 5 a2 a l a2 = ( a ^ u 2 + a 1 u 1 ) A ( x 4 + a2/a\u00C2\u00B1 u ? ) + c ^ i y N f \u00E2\u0080\u0094 x 5 + ( \u00E2\u0080\u0094 - \u00E2\u0080\u0094 ) u ? ] 3 3 5 a , a 2 \ L e t Y' = - a 2 / a l ) ' I F Y' = o ; z = ( a ^ U g H - ^ ^ ) A ( x 4 + ^ - u 3 ) 3 \u00E2\u0080\u00A2 1 + ( _ a2 U2 + a3 u3) A x5 a n d R ( z ) = 2 . a2 I f Y ' + o ; z = ( a ^ u 2 + a 1 u 1 ) A ( x 4 + \u00E2\u0080\u0094 u ? ) + ( a3 U2 - \u00C2\u00A7 U 3 ) A ( - ^ X 5 + Y U 3 } and . R ( z ) = 2 . 3 Example III.48 U = . W(f\u00C2\u00B1) = < u g ; u 1 + > . f x = X 4 A U X :+ UgAUj f 2 = X 5 A U g + U - J A U J , f = U 2 A ( U X + U ^ ) + X 4 A X 5 46. z = Ea^f^ = x ^ A ( a 1 u 1 + a^x^) + (a^Ug + a 2 u 1 + a^u 2)Au^ + U 2 A ( - a 2 x ^ + a y ^ ) I f 0 ^ = 0 ; z = X , - A ( C I 2 U 2 - ayc^) + ( a 2 u 1 + a ^ u 2 ) A ( u 1 + u^) and R ( z ) = 2 . I f a 2 = o , z = UgA(a 1 u ^ + + a^u^) + x ^ A ( a 1 u 1 + aye,-) and R ( z ) = 2 . I f a\u00C2\u00B1 + o ; i = 1 , 2 , 3 ; a. a., a . z - ( a 3 x 4 - a 2 u 2 ) A ( - l U i + x 5 ) + ( + a 2 u a ) A ( ^ + - A ) ^ + ( a 1 u 2 + cXgU-^ + a^Ug)Au^ - (,a^4 - a 2u 2)A(a 1/a 5 ^ + x^) + [ ( a x + a 5 ) u 2 + a 2u 1]A[u 3 - + i ) u 2 ] ; and R ( z ) = 2 . CHAPTER IV THE RANK TWO SUBSPACES WHEN U HAS DIMENSION 6 I n t h i s c h a p t e r , we show t h a t i f dim U - 6 , E \u00E2\u0082\u00AC R 2 ( U ) 3 t h e n dim H < 3 (Theorem IV.72) . I f dim H = 2 i t has a (1,1) b a s i s . I f dim H = 3 , l e t f f - ^ f ^ f ^ } be i t s b a s i s . Theorem 11.26 shows t h a t dim [ ^ U ( f . ) ] = 5,6 . 1 = 1 1 The f i r s t case i s c o n t a i n e d i n Ch a p t e r I I I . In t h e second c a s e , we show t h a t H has a b a s i s o f p a i r w i s e ~ P g v e c t o r s (Theorem I V . 6 l ) . There are 3 p o s s i b i l i t i e s f o r such a b a s i s (Theorem IV.62) . R e p r e s e n t a t i o n s f o r t h e s e b a s i s v e c t o r s a r e g i v e n i n remark IV.53 and Theorems TV. 57, 59-Our f i r s t major aim i s t o o b t a i n the r e s u l t t h a t i f f ^ , f 0 , f -3, a r e independent v e c t o r s i n a rank two subspace o f d i m e n s i o n a t l e a s t t h r e e , and dim[ i U ( f . ) ] = 6 , t h e n 1=1 < f 1 , f g , f ^ > c o n t a i n s a b a s i s o f p a i r w i s e - P g v e c t o r s . I f i n a d d i t i o n { f 1 , f 2 } i s a P g - p a i r , then t h i s p a i r can be extended t o a b a s i s o f p a i r w i s e - P g v e c t o r s f o r < f 1 , f 2 , f j > . T h i s r e s u l t i s c o n t a i n e d i n Theorem I V . 6 l . To a c h i e v e - t h i s a i m , we c o n s i d e r a l l p o s s i b l e t r i p l e s o f v e c t o r s { f ^ f 2 , f } s a t i s f y i n g the above c o n d i t i o n s . Theorem I I . 26 t e l l s us t h a t each p a i r { t ^ t ^ , 1 < i < J < J must be e i t h e r a P^- o r a P g - p a i r . Thus e i t h e r { f - ^ f ^ f ^ ) i s p a i r w i s e - P ^ o r a t l e a s t one p a i r i s a P g - p a i r . F o r the case when { f n , f^,f^} i s p a i r w i s e - P ^ 3 we e x h i b i t two bases f o r i one o f w h i c h i s a (1,1) b a s i s . However, t h i s case i s n o t o f g r e a t importance as we can show t h a t fy> c o n t a i n s a t l e a s t one \u00E2\u0080\u00A2 P g - p a i r . Thus t h i s case i s r e d u c e d t o the second c a s e . I n t r e a t i n g the l a t t e r c a s e , the i n t e r s e c t i o n [ fj U ( f . ) ] , p l a y s a prominent r o l e . I t has d i m e n s i o n a t i\u00C2\u00BBl 1 most two_, s i n c e we have a P g - p a i r say { f ^ f } . I n o r d e r t o show t h a t [ f j , f 2 ) can be extended t o a p a i r w i s e - P g b a s i s { f ^ f ^ g - j } f o r Kt^f^, f^> , we have found it-n e c e s s a r y t o c o n s i d e r the t h r e e cases? dim[ 3 U ( f . ) ] = o ,1,2 . 1=1 Theorem IV. 49. L e t H e R g ( U ) , and l e t { f - ^ f ^ f ^ } be p a i r w i s e - P , - and independent i n H such t h a t dim | U ( f \u00C2\u00B1 ) = 6 . Then [ 5 U ( f \u00C2\u00B1 ) ] has a b a s i s ( u ^ u ^ u ^ x ^ x ^ X g ) such t h a t f , f0, have r e p r e s e n t a t i o n s f3 ^ x 6 a U + v A U 3 ^ w h e r e = , u 4 3 u 4 . P r o o f ; S i n c e {f^f^ f^} a r e p a i r w i s e - P ^ , f T j ( f \u00C2\u00B1 ) : i = 1,2 i n t e r s e c t p a i r w i s e i n d i m e n s i o n 3 . Now dim U( f \u00C2\u00B1 ) = 4 , i = 1 , 2 , 3 . H e n c e i t f o l l o w s f r o m r e m a r k 111.35 a n d t h e h y p o t h e s i s t h a t d i m ^ U( f.) = 6 t h a t U( f . ) U , d i m U i - 1 , 2 , 3 \u00E2\u0080\u00A2 B y t h e o r e m 11 .29 , t h e r e e x i s t r e p r e s e n t a t i o n s f l = y 4 A U i + U 2 A V 3 > f 2 = y 5 A U 2 + u i A V 3 > where { u^,Ug , v-j , y 4 , y ^ } i s i n d e p e n d e n t . B y T h e o r e m I I . 1 6 , we h a v e f-j = y^AW-L + WgAW^ , w h e r e = v^> = T]Q a n d | U ( f i ) = . S i n c e d i m n > 1 , we c a n WLG assume Wg e . S u p p o s e , o n t h e c o n t r a r y : t h a t w^ 4 ; i . e . , = U Q . L e t Z = -(f^+fg+f-j) . T h e n Z - u l A ( y 4 - v ; 5 - a w 5 ) 4- U g A ( y 5 - v 5 - b w 5 ) + w l A y 6 a n d R( Z) = 3 b y T h e o r e m 1.13, c o n t r a d i c t i n g t h e h y p o t h e s i s t h a t < f l 3 f g , f 5 > s R g ( U ) . Hence w 1 e . P r o m t h e p r o o f o f T h e o r e m 1 1 . 2 9 , we s e e t h a t w\u00C2\u00B1 4 a n d w1 4 . S i n c e = < u 1 ? u 2 > a n d U Q = w2,w^> = < u 1 , u 2 J ) w - j > = \"Cu-pUg, v^ > , t h e n w^ e [ v ^ ; u-^u.g] . Thus w3 = X(v^+u) , f o r some o 4 X \u00E2\u0082\u00AC F a n d u e < u i > u 2 > \u00E2\u0080\u00A2 R e w r i t i n g : - -f l = y j j A U x + U 2 A ( V 3 + U ) , f 2 = ^ 5 A U 2 + U I A ( V 3 + U ) , \u00C2\u00A3 3 - y 6 A W l + X W 2 A ( v 3 + ^ ) \u00E2\u0080\u00A2 We s e t x 4 = y^ , X5 = y 5 3 u ^ = + u , u = a n d v = X w 2 , a n d we h a v e t h e r e q u i r e d f o r m s f o r f ^ , f 2 , . C o r o l l a r y 17.50 L e t H \u00E2\u0082\u00AC Rg(U) a n d l e t { f ^ f ^ f ^ } he p a i r w i s e - P p . a n d i n d e p e n d e n t i n H such t h a t d i m [ | U ( f , ) ] = 6 5 I X Then ff-pfg, ?j) is a (1,1) basis for Kf^f^f^ \u00E2\u0082\u00AC R 2(U) P r o o f : By T h e o r e m IV.49, [J U ( f . ) J h a s a b a s i s 1 X {u 1,u 2 Ju^ , x 4 , X p , X g } such t h a t f - ^ f ^ f ^ h a v e r e p r e s e n t a t i o n s f j_ \u00E2\u0080\u00A2= X 4 A u 2 _ + u o A u - j > f 2 - X AUg + U L A U 5 , \u00C2\u00AB X g A t t + v A ^ - J , w h e r e = < u 1 , u 2 > u 4 <^> a n d 4 . How u = a ^ + b u 2 f o r some a , b b o t h n o n z e r o i n F ; v = cu-^ + d u 2 ; c,d e F . Thus f ^ = X g A ( a u 1 + b u 2 ) + (cu-j^+dUg) A u ^ . On r e a r r a n g e m e n t we h a v e : 51. ? 2 = ( - \u00E2\u0080\u00A2 a 3 ) A u 1 + U 2 A ( - X 5 ) , f 3 = ( & ^ 6 - c u 3 ) A u 1 + u 2A(du 5-bx 6) , a n d h e n c e {f^i',.,1%} i s a ( 1 , 1 ) basis for \u00E2\u0082\u00AC R | ( U ) Theorem 17.49 a n d i t s c o r o l l a r y give t h e form of a b a s i s o f a r a n k two subspace s a t i s f y i n g the conditions of t h e t h e o r e m . We now e x h i b i t a n example of a rank two subspace o f t h e a b o v e k i n d . E x a m p l e I T . 5 1 U o = < U p U 2 ) u ^ > f1 = x ] | A u 1 + U 5 A ( U X + U 2 ) f 2 = x ^ A u 2 + u^ A(u 1+u 2) ? 3 = X . 6 A ( U 1 + 3 U 2 ) + U 5 A ( ^ 1 + U 2 ) , o + B e F , 3 + 1 . F o r a . e F , not a l l z e r o , Z = -' 1 a \u00C2\u00B1 f i = u l A[a 1x l j+a 5x 6+(a 1+a 2+a- 3)u 3] i = l + U 2 A [ a 2 x 5 + a 56x 6+( a ^ a ^ a ^ ) u^ ] j a n d R(Z) = 2 . Hence { f p f g , ^ } generate a rank two subspace o f d i m e n s i o n three. We s h a l l now show t h a t , i f { f ^ f ^ f ^ - } i s a b a s i s o f p a i r w i s e - P , - v e c t o r s f o r H \u00E2\u0082\u00AC R 0 ( U ) a n d d i m \u00C2\u00A3 U ( I \ ) = 6 t h e n H h a s a b a s i s o f p a i r w i s e - P g v e c t o r s . We n e e d t h e 52. f o l l o w i n g . Lemma IV. 52 Let ( f ^ f ^ } be a (1,1) basis f o r < f 1 ? f 2 , f 3 > \u00E2\u0082\u00AC R 2(U) s a t i s f y i n g : ., <1) dim \ U(f j ) = 6 (2) {f-^fg} i s a Pg-pair . Then 3 g } e f y } {f-p f 2 , g^} are pairwise-Pg and form a (1,1) basis f o r ^ ^ f ^ f > . Proof: Since { f ^ f g ] i s a Pg-pair, \ U(f \u00C2\u00B1) has a basis (u -^Ug^x-j, . . . , X g ] ^ {f-^^fg] have representations : -f x = u l A x 5 + U 2 A X ^ , f g = tu^x^ + ^ 2A X 5 (Theorem II.31) . Since {f^, f , f^} i s a ( l , l ) b a s i s , i t follows f r o B . Theorem I I . 18 that f ^ has a representation u-^y + u 2Ay' . Since U(f j) c [ I J (f 1)+U( f p) ] , and by Lemma 11.16, we can assume {y^y'} e . Let y = u + ax-j + bx^ + cx^ + dxg , y' - u' + a'x^ + b'x^ + c ' X p . + d'xg , where {u,u'} e Since both polynomials above are n o n - t r i v i a l , we can choose X - , , X p 6 F such that a+X 1 b c+X 2 d 4 o and a ' b ' + X l e' d'+X 2 4 o Then = X 1f ]_ + X \u00C2\u00A3 f 2 + f ^ = u l A [ a+X 1)x 3+bx 4+( X 2+c )x^+dxg ] + U g A t u ' + a ' x ^ X-j+b' ) x 4 + c v x 5 + ( x a+d' )xg] i s s u c h t h a t V(g^) n >yih> = o and tl( g^) n = o . Hence [ f ^ f 2,g^} a r e p a i r w i s e - P g and the lemma i s p r o v e d . Remark IV.55 L e t { f ^ f ^ f ^ } be a ( 1 , 1 ) b a s i s o f p a i r w i s e - P g v e c t o r s f o r c < u 2 , x ^ .. . ,xg> , n u ( f \u00C2\u00B1 ) \u00C2\u00BB o , i = 1,2 . P r o o f : The r e s u l t f o l l o w s d i r e c t l y f r o m t h e d e f i n i t i o n o f a (1 , 1 ) b a s i s and the f a c t t h a t { f \u00C2\u00B1 : i = 1,2,3} are p a i r w i s e - P g . 54. Theorem I T . 5^ L e t H e R 2 (TJ) . L e t { f ^ f ^ f ^ } be p a i r w i s e - P 5 , ind e p e n d e n t v e c t o r s i n H such t h a t dim \ U ( f . ) = 6 . Then i = l 1 f2> f 3 > h a s a C1*1) 'basis o f p a i r w i s e - P g v e c t o r s . P r o o f : By C o r o l l a r y IT.50 and i t s p r o o f , { f ^ f ^ f ^ } have r e p r e s e n t a t i o n s f ! = X 4 A U X + u 2 A U ? ; f 2 = ( - u 5 ) A u 1 + U 2 A ( - X 5 ) , f 3 = ( a x g - c u 5 ) A u 1 + u 2 A ( d u 5 - b x 6 ) , a + o , b + o , where J| - . L e t S l - f x 4-= ( x 4 + a x g - c u 5 ) A U 1 + m^u^+du^-bxg) . Then {g-^f 2 ) i s a P g - p a i r , \u00C2\u00A5(f 2 , g.^ ) \u00C2\u00AB= , and 2>f^ i s a C 1 ^ b a s i s f o r < f 1 J f 2 > f 3 > * T n e r e s u l t f o l l o w s by Lemma TV.52. C o r o l l a r y I T . 55 L e t { f ^ f ^ f ^ } be a ( 1 , 1 ) b a s i s f o r < f , , f p , f , > e R p ( U ) ; dim ? U ( f \u00C2\u00B1 ) = 6 . Then t h e r e e x i s t s a ( 1 , 1 ) b a s i s of p a i r w i s e - P g v e c t o r s f o r < f 1 , f 2 , f 5 > . P r o o f : Suppose { f ^ f g , ^ } a r e p a i r w i s e - P 5 . Then Theorem IV. 54 g i v e s the r e s u l t . O t h e r w i s e a t l e a s t two of { f ^ f ^ f ^ } , say {f f } f o r m a P g - p a i r . Lemma IT.5 2 g i v e s the r e s u l t . 5 5 . Remark 17.56 Let H e R 2(U) . Let { f ^ f ^ f } be independent i n H such that ( i ) dim \ u(f,) = 6 1 1 ( i i ) 3 U(f.) = o . i = l 1 Then f f - p f g ^ f ^ } are pairwise-Pg . Proof: I f any p a i r , say f f ^ f g ] i s a P^-pair, then U(f^) n TJ(fg) has dimension three and hence must i n t e r s e c t U(f^) . Therefore no p a i r i s a P^-pair, and { f 1 , f g , f ^ } must be pairwise-Pg . Theorem 1 7 . 5 7 Let H e R 2(U) , dim H > 3 \u00E2\u0080\u00A2 Let { f ^ f ^ f ^ } be independent i n H such that ( i ) dim \ U ( f \u00C2\u00B1 ) = 6 . i=*l ( i i ) 3 U ( f \u00C2\u00B1 ) = o . i = l Then ( f ^ f g , f-^ } are pairwise-Pg and f o r any basis {u,,u } o f W(f,,.fJ , [ \ U ( f \u00C2\u00B1 ) ] has a basis J. d X d 1=1 {VL^U^XJ, . . .,xg] such that { f - ^ f g / f ^ } have representations: f x = u l A x 3 + U g A X ^ , f g = U l A X 5 + Ug AXg , 56. F5 = X5 A W I + X 4 A W 2 , < w i . w 2 > = < x 5, x6 > > r Proof: By Remark IT. 56, { f - ^ f ^ f ^ } are pairwise-Pg . Since ff ^ , f 2 ] i s a Pg-pair, [ f ^ , f } have representations f x = U 1 A X - 5 + U 2 A X 4 , f 2 = U 1 A X 5 + u 2 A X g ; ^ ^ U ( f ^ ) = . We now show we can choose x^,x^ so that f l = u i A X 3 + u 2 a X 4 . and f ^ \u00C2\u00AB x ^ A u + x^Aw j i . e . , without changing u-^,u2 . Since [f-^, f 2 , f^} are pairwise-Pg and ( i i ) holds, U(f^) = , where {y^y^ \u00E2\u0082\u00AC [ x ^ , x 4 J U 1 , U 2 ] and [z^3z^} \u00E2\u0082\u00AC [ x ^ , X g j u 1 , u 2 3 \u00E2\u0080\u00A2 Hence there e x i s t {x-j, Yl2} s u c n that = and x^ e [ x ^ ; u 1 , u 2 ] , x 4 \u00E2\u0082\u00AC [ x 4 ; u 1 , u 2 ] * Let x^ - ax^ + bu^ + c u 2 a 4 0 \u00E2\u0080\u00A2 Without l o s s of g e n e r a l i t y , we s h a l l take a = 1 . Then f ^ = U ^ A X ^ + U 2 A X ^ = u ^ x ^ + U 2 A ( X 1 ( _ + C U - L ) By Theorem I I . 16, f ^ = X ^ A W 1 + v AWg , where = . Since { f i ^ \u00C2\u00A35} l s a Pg-pair, by C o r o l l a r y 11.34, dim n = 1 . Let t h i s i n t e r s e c t i o n be . Then x^ e [x^+cu-^Ug] and we have: - f x = u ^ x ^ + U 2 A X ^ , f ^ = x ^ A w x + X J J A W 2 where = . Thus {w1,w2} e [ z 1 , z 2 ; x ^ ] . 57. In a s i m i l a r fashion, without a l t e r i n g u^ ox- u^ we can choose f x ' x'} so that o \u00C2\u00B0 < x ^ x 6 y = < z i . z 2 > > x 5 6 ^ x 5 ^ u i ^ u 2 j 1 > x6 e [ X 6 3 U 1^ U 2^ and f 2 = u i _ A X 5 + ^ 2 A X 6 5 f 3 = x ' 5 A v i + x ^ v 2 > v r h e r e Thus { v l 5 v 2 } e [ x - ^ x^ j x g ] . From above, we also have - 3 = X ^ A W x + x^ Aw 2 3 and [w pw 2| e [ x ^ x ^ x ^ ] . With respect to the independent set X ' A X ' , 3 _< i < j < 6 3 the c o e f f i c i e n t of X ^ A X ^ i s c l e a r l y zero i n the second expression obtained above f o r f ^ 3 and the c o e f f i c i e n t of X ' A X / - i s zero i n the f i r s t . I t follows that neither term D o appears i n . Therefore f ^ = X ^ A W - ^ -1 X ^ A W 2 3 = < x ^ X g > 3 and f ^ = x < 5 A vi + X g A v 2 5 < v i ^ v 2 > = \u00E2\u0080\u00A2 Writing x ^ f o r x^ , x^ f o r x^ ' 3 x_ f o r x^ and X g f o r X g 5 we have the r e s u l t . Lemma IV, 56 Let H e R 2 ( u ) \u00E2\u0080\u00A2 Let { i \ , f 2, f ~} be independent 3 i n Ii s a t i s f y i n g ( i ) dim E U ( f i ) = 6 ( i i ) {f-^fg} i s a 3 Pg-pair ( i i i ) dim_fi U ( f \u00C2\u00B1 ) = I . Then j e -) { f ^ f ' ^ g ^ } i s a basis of pairwise-Pg vectors f o r < f 1 , f ^ f ^ > and u ( f5) n WCf ^ f g ) - u(g^) n W t f ^ f g ) . 58. P r o o f : S i n c e f f - ^ f ^ i s a P g - p a i r , t h e y h a v e r e p r e s e n t a t i o n s f l = u i A X 3 + u 2 A X 4 \u00C2\u00BB f 2 = u i A X 5 + U 2 A X 6 1 3 \u00E2\u0080\u00A2 | U ( f 1 ) * < u 1 , u 2 > x ^ J ) . . . , X g > . L e t = _ n U ( f i ) j t h e n u \u00E2\u0082\u00AC < u 1 ; , u 2 > . I t f o l l o w s f r o m T h e o r e m II . 1 8 t h a t we c a n assume Uj. = u . T h u s U( f ^) < u 1 > . B y Lemma I I . 16, f 3 = u l A W + W ' A V > w h e r e c < u 2 , x ^ 3 . . . , X g > . I f f f 1 , f 2 , f - j } a r e p a i r w i s e - P g we h a v e t h e r e s u l t . C a s e 1 S u p p o s e { f . , , f ^ } i s a P g - p a i r a n d f f 0 , f ^ } i s a P ^ - p a l r . B y C o r o l l a r y I I . 3 4 a n d P r o p e r t y 4 . C ( I n t r o d u c t i o n ) w ' e [ x ^ U g ] , s i n c e u 2 | U ( f ^ ) . Now f.^ e u ^ x ^ + U g A t x^jUg] . H e n c e w i t h o u t l o s s o f g e n e r a l i t y , w ' = x^ a n d f 3 = u i A W + x 4 A v ' c < u 2 , x ^ , . . . . x g > , L e t W ( f 2 > f ^ ) = - O i ^ y y ' > . T h e n y , y ' e [ x ^ X g j U g ] . T h e r e f o r e f 3 \" U 1 A W + X 4 A V ' w \u00C2\u00A3 [ x 5 , x g ; u 2 , x 4 ] ; v ' e [ x^XgjUg] . L e t v ' = a x ^ + b X g + G U g . C h o o s e Y ^ 0 4 y + C ^ 0 . T h e n = f^ + = u - ^ w + y x ^ ) + X ^ A ( V - Y i i g ) = U - ^ z + X ^ A Z ' ; z e [ x ^ ; x 5 , X g , U g , x ^ ] 0 [ x 5 , X g j u g, x 5 , x 4 ] ; z\u00C2\u00ABe [ u 2 ; x , - , x 6 ] fl [ x 5 , x 6 ; u 2 ] . T h e n { f - p g j ] i s a P g - p a i r a n d f f 2 , g 5 ] i s a P g - p a i r . T h e n f ^ ^ S j } a r e t ' h e r e q u i r e d v e c t o r s , u ( g ? ) n v(?1,?2) = <^x> \u00E2\u0080\u00A2 Case 2 Suppose { f - ^ f ^ } , { f ^ f ^ } are P - p a i r s . Then dim W ( f 1 , f 3 ) \u00C2\u00AB 3 ; dim W ( f 2 > f 5 ) = 3 , am by ( i i i ) , dim \u00C2\u00A5 ( f 1 , f 3 ) n \u00C2\u00A5 ( f 2 , f ^ ) = 1 . But t h i s itT5jli.es dim ^ ( f - ^ f ^ ) + W(f 2 , f ) ] \u00C2\u00AB 5 , which exceeds dim U ( f ^ ) , Hence t h i s case i s not p o s s i b l e . In order to e x h i b i t d e s i r a b l e bases //hen the conditions of Theorem IV.5 8 hold, we need fchs f o l l o w i n g lemma. Lemma, I f f e Cg(U) and f \u00E2\u0082\u00AC x - ^ x ^ x ^ x ^ ] + [ x ^ ; X 2 ] A [ X ^ ; X 2 ] where U ( f ) \u00C2\u00BB , then f \u00E2\u0082\u00AC X - ^ A C X . - J + [x^ j X ^ ^ X g ] A[X-jjX^,Xg3 , Proof: Since i s a 2-dimensional subspace of U ( f ) , then, by Theorem II.17, e i t h e r f has a representation X - ^ A V ' + X 2 A w \" \u00C2\u00B0r f has a representation f - Y X-J_A XO + V A W f o r some n o n z e r o y i n P . I f f has the f i r s t form, then the c o e f f i c i e n t o f X ^ A X ^ is zero i n f , c o n t r a d i c t i n g the assumption that f \u00E2\u0082\u00AC x i A [ X 2 , X ^ , X 4 ] + [x ^ X p j A f x ^ j x , , ] , Hence f = yx 1 A xp + VAW , o 4 Y \u00E2\u0082\u00AC P , and = , Since the c o e f f i c i e n t of x,, A X , I L nonzero i n f , i t follows e a s i l y that V A W S [ x j ^ x ^ x ^ J A t x ^ x - ^ X g ] , Theorem I V . 59 Let H e Rg(U) with dim H > 3 * Let { f l 5 f ^ f ] be pairwise-Pg and independent i n H s a t i s f y i n g ( i ) dim \ U ( f t ) = 6 ( i i ) = S U ( f . ) . any re c t o r u 0 such x 1=1 1 t h a t < u x > u 2 > - W ( f x , f 2 ) , i i _ V ( f \u00C2\u00B1 ) } has a has i s fu.^, u 2 , x^_,..., } s u c h t h a t f 2 \u00C2\u00BB U ] _ A X 5 + U 2 A X 6 ' f ^ = u-, AS ' + X ^ A X 6 , where y e <~x2,Xy . \u00E2\u0080\u00A2., xg> , y 4 < u 1 3 x 3 5 x 5 > , y 4 U ( f \u00C2\u00B1 ) , i = 1, F u r t h e r m o r e , t h e r e e x i s t s g^ \u00E2\u0082\u00AC <.t'^ , f p_, f-<> s u c h t h a t < f x , f 2 , g 3 > = < f x , f 2 , f 3 > and f 2_ = U - ^ A X - J + u p A x i j . f o ~ A x ^ + U 2 A X 6 S 5 = t L l A u 2 + V A W , v e [ x ^ j u ^ U p ] , W \u00E2\u0082\u00AC [ X g J U - ^ U p ] = V ' A W ' + Y X ^ A X 5 , o 4 Y S F i v ' \u00E2\u0082\u00AC [ u - ^ x ^ X g ] , w' e [ u 2 ; x 4 , X g ] . P r o o f ; S i n c e {f-,,f 2} i s a P g - p a i r , they have r e p r e s e n t a t i o n s f l = u i A X 3 + u c \u00C2\u00BB S i n c e {f-^f-j} > { f 2^ f 3} a r e Pg-pairs and ( i i ) h o l d s , hy C o r o l l a r y 11.34, . f ? = u-^y + zAzf , z e [ x ^ j i i p j , _ z' \u00E2\u0082\u00AC [xgjttg] . With o u t l o s s o f g e n e r a l i t y , z = , z ' = Y 2 X 6 a J i a f 3 = u l A y + Yx^AXg , o + y \u00E2\u0082\u00AC P . L e t a 2 = Y . Then f 1 = U ^ A X ^ + c f ^ A C G C ^ , = + c f 1u 2Aoacg , f ^ = u^Ay + ax^Aaxg . Hence, w i t h o u t l o s s o f g e n e r a l i t y , we ca n t a k e y = 1 , and f ^ = u]_Ay + x^AXg , as r e q u i r e d . I f y \u00E2\u0082\u00AC , some l i n e a r c o m b i n a t i o n o f {f-^, f 2 , f^} has rank a t most one. Hence y | and y 4 u ( f \u00C2\u00B1 ) , i = 1 , 2 , s i n c e {f ^ , f 2 , f ^ } a r e p a i r w i s e - P g . We can t a k e a s u i t a b l e l i n e a r c o m b i n a t i o n o f [ f l f f 2 ) f j } t o o b t a i n g^ = U 1 A Z Q + V A V ' , z Q e , v \u00E2\u0082\u00AC [ x ^ ; u 2 ] , v' e [ x 6 ; u 2 ] . S i n c e = , u 2 e U(g- j) . Hence, by t h e above p r o p o s i t i o n , s 3 = Y u i A U 2 + V A W > \u00C2\u00B0 4 Y e P , v e [ x ^ u - ^ U g ] and w s [ x 6 ; u 1 , u 2 ] . S i n c e y \" ^ \u00E2\u0082\u00AC < f f 2 , f 5> and f f \u00C2\u00B1 , f 2 , Y _ 1 g 5 ) i s i n d e p e n d e n t , we can ta k e y = 1 ; hence g^ = u - L A U 2 + V A W * Now U( g^) \u00C2\u00AB . By Theorem 1 1 . 1 7 , e i t h e r g^ = X ^ A V + XgAW o r g^ \u00C2\u00AB Yx^AXg + V'AW' . The f i r s t r e p r e s e n t -a t i o n i s n o t p o s s i b l e s i n c e t h e c o e f f i c i e n t o f U ^ A U 2 i n i t i s z e r o , and t h i s i s n o t so. Hence g^ = yx^AXg + V ' A W ' , v',w' e [ u 1 , u 2 j x 4 , x g ] . S i n c e t h e c o e f f i c i e n t o f u-^u^ i s nonz e r o i n g^ , i t f o l l o w s e a s i l y t h a t we can ta k e v 7 s [ u ^ x ^ X g ] and w' e [ u 2 j x 4 , x g ] . 62. Lemma 17 .60 L e t H e Rg(U) . L e t { f - ^ f ^ f } be independent i n H su c h t h a t ( i ) dim ? U ( i \ ) = 6 ( i i ) [ f l a f 2 ] i s a P 6 - p a i r ( i i i ) dim U ( f . ) = 2 . Then t h e r e e x i s t s a g , \u00E2\u0082\u00AC 1=1 ^ f2> f 3 > s u c n t h a t { ^ f ^ g ^ } i s a b a s i s o f p a i r w i s e - P g v e c t o r s f o r < f 1 , f 2 _ > f ^ > . P r o o f ; S i n c e { f 1 > f 2 3 l s a P g - p a i r , { f x , f 2 } have r e p r e s e n t -a t i o n s f x = u-^x-^ + UgAX^ , f 2 = u l A x 5 + U 2 A X g , | U ( f \u00C2\u00B1 ) = < u i * u 2 > x 3 * * * \u00E2\u0080\u00A2 \u00C2\u00BBx6> * B y C111) * U ( f ^ ) r> . I f e i t h e r { f - ^ f ^ } o r f f 2 , f ^ } i s a P g - p a i r , t h e n f-j = u ^ A v + U 2 A W and {f-^, f g , f^} i s a (1 , l ) b a s i s f o r < f 1 , f 2 , f y > . The r e s u l t f o l l o w s from Lemma IV.52. Suppose {f^, f^} and {fg,f^} a r e b o t h P ^ - p a i r s . By Theorem 11.17, f-j i s e i t h e r u l A v + U g A w o r f ^ = u l A u 2 + V ' A * ' * I n t h e f i r s t c a se, { f 1 , f 2 , f-^} i s a (1,1) b a s i s and t h e r e s u l t f o l l o w s by Lemma IT.52. I n t h e second c a s e , we c a n assume v ' e I J (f-^ , w' e U ( f 2 ) . Thus v ' \u00E2\u0082\u00AC [ X 3 , X 4 J U 1 , U 2 ] , w' e [ x ^ X g j u ^ U g ] . Without l o s s o f g e n e r a l i t y , v ' = x^ + a ^ + b . j U 2 + and w ' = d x 5 + a 2u ]_ + b g u 2 + C g X g , d , a 1 , b 1 , c i \u00E2\u0082\u00AC P . L e t Z \u00C2\u00AB | f \u00C2\u00B1 = u l A ( x 5 + x 5 + u 2 ) + U 2 A ( X 4 + X 6 ) + ( x 3 + a i u i + b i u 2 + c i x 2 f ) A( dx 5+a 2u 1+b 2u 2+c 2 x g ) 63. Consider! 1 0 0 1 0 1 0 0 a l b l a 2 b 2 0 1 0 0 1 0 0 0 1 0 0 1 0 c l 0 0 0 0 1 0 0 0 = - c^d + c 2 . I f t h i s determinant i s nonzero, then R(Z) = 3 . I f c 2 - c 1 d = o , then R(z) _< 2 . Hence v ' = x-z + a , u , + b , u 0 + c n x , . , and w' = x c + a_u_ + b^u. + c , x , -5 1 1 1 2 1 4 ' 5 2 1 2 2 1 6 Now f 1 = u i A ( x 3 + c i x 4 ^ + ( u 2 \" c l u l ^ A X 4 \" ui A X 3 + U 2 A X 4 > f 2 = u 1A(x^+c 1Xg) + ( u 2 ~ c i u i ) A X 6 = u i A X 5 + u 2AXg . Hence without l o s s of g e n e r a l i t y , we can assume v' = x^ + a^u^ + b ^ u 2 , w' = + a 2 u 1 + b 2 u 2 , and f ^ = u i A U 2 + ( x ^ + a ^ u i + b i u 2 ) A ( x 5 + a 2 u 1 + b 2 u 2 ) . Let g ? = f ^ - ^ + f 2 \u00C2\u00AB= ( x y-a-^+b^ U g ) A (x^+b 2 u 2 + ( a 2 + l ) u \u00C2\u00B1 ) + u 2 A ( _ u i ~ x 4 \" b i u i ) + u l A ( X 5 + B 2 U 2 ' 1 + u 2A(xg+b 2u 1) = [ x 5 + b 1 u 2 + ( a 1 + l ) u 1 ] A [ x 5 + b 2 u 2 + ( a 2 + l ) u 1 ] + u 2 A [ x g + ( b 2 - l - b 1 ) u 1 - x 4 ] . Then { f - ^ f ^ g ^ } are pairwise-Pg and < f 1 , f 2 , g 3 > \u00C2\u00AB < f 1 , f 2 , f 3 > . Theorem IV\u00C2\u00AB6l Let H e R 2(U) with dim H > 3 . Let (\u00C2\u00A3ltf2,fj} be independent elements of H f o r which 64. d i i a [ | U ^ ) ] = 6 . T h e n c o n t a i n s a b a s i s o f p a i r w i s e - P g v e c t o r s . I f , i n a d d i t i o n , { f ^ f g } i s a P g - p a i r , t h e n t h i s p a i r c a n be e x t e n d e d t o a b a s i s o f p a i r w i s e - P g v e c t o r s f o r < f 1 5 f 2 , f^> . P r o o f ; The f i r s t r e s u l t i s i m m e d i a t e f r o m Remark I Y . 56, Lemmas I Y . 5 8 , 6 0 . I f { f - ^ f g } i s a P g - p a i r , t h e n -d i m U ( f , ) = o 1 2 . The s e c o n d r e s u l t f o l l o w s f r o m 1=1 1 T h e o r e m I Y . 5 7 a n d Lemmas 1 7 , 5 8 , 6 0 . T h e o r e m IT.6 2 L e t H e R J ( U ) . L e t n\, i \" 2 , f^) be i n d e p e n d e n t , p a i r w i s e - P ^ i n H s u c h t h a t d i m ^ U ( f . ) = 6 . T h e n e i t h e r o 1 1 ( a ) f f - ^ f ^ f ^ } i s a ( 1 , 1 ) b a s i s f o r o r ( b ) d i m f\ U ( f . ) = o , l . 1=1 1 P r o o f : d i m f\ U ( f . ) < 2 . I f d i m 3 U ( f . ) = 2 , t h e n 1=1 1 i = l U ( f ^ ) =3 W ( f 1 , f 2 ) a n d b y C o r o l l a r y 11 .34, f f ^ f ^ f ^ } . i s a ( 1 , 1 ) b a s i s f o r < f 1 , f g , f ^ > . The r e s u l t f o l l o w s . N o t e : F o r r e p r e s e n t a t i o n s o f f f - ^ f g , ^ } see Remark I V . 53 f o r ( a ) ; T h e o r e m s I V , 5 7 , I V . 5 9 f o r ( b ) . The f o l l o w i n g a r e t h r e e e x a m p l e s o f f f - ^ f g , f y t h a t a r e p a i r w i s e - P g i n H e R 2 ( U ) , \u00C2\u00B1 U ( f \u00C2\u00B1 ) has d i m e n s i o n 6, 65. e a c h e x a m p l e i l l u s t r a t i n g one o f t h e t h r e e c a s e s i n T h e o r e m I V . 6 2 . E x a m p l e s I V . 6 5 E x a m p l e I Y . 6 5 . I d i m 3 U ( f . ) = 1 . 1=1 1 1 U ( f \u00C2\u00B1 ) = . f 1 = U l A X 3 + U 2 A X 4 , f g = U-^AX^ + UgA xg > f3 = U 1 A ( U 2 + X 3 + X 5 ^ + X 4 A X 6 \u00E2\u0080\u00A2 T h e s e v e c t o r s g e n e r a t e a r a n k 2 s u b s p a c e o f d i m e n s i o n 3 s i n c e i f a 1 \u00E2\u0082\u00AC F , 4 o , t h e n Z = \ a 1 f i = u 1 A [ a J + a 3 ) x 3 + ( a g + a ^ ^ + a y i g ] a 2 + ( \u00E2\u0080\u0094 u 2 + x ^ ) A ( a 5 x 6 - a 1 u 2 ) ; 3 a n d R ( Z ) = 2 ; a n d i f = o V Z = a 1 f 1 + ctgfg a n d R ( Z ) = 2 . a 1 3 c t g + o) E x a m p l e I V . 6 5 . 2 f f ^ f g , ^ } a ( 1 , 1 ) b a s i s f o r < f 1 , f g , f 5 > I U ( f . ) = < u 1 , u 2 , x 5 , . . . , x 6 > ; F = C o m p l e x n o s . f l = u i * x 2 + U 2 A X 4 ' f 2 = u l A x < - + u 2 A x g , f = u l A ( x 4 + 2 x 6 ) + U 2 A ( x 3 + 3 X 5 ) \u00E2\u0080\u00A2 66. F o r a i \u00E2\u0082\u00AC F n o t a l l z e r o = ? a.f. 1 1 = u 1 A ( a 1 x 3 + a 2 x 5 + a 3 x 4 + 2 a 3 x 6 ^ + U 2 A ^ a l x 4 + a 2 x 6 f a 3 x 3 + 5 a 3 x 5 ^ a n d R ( Z ) =2 . E x a m p l e I V . 6 3 . 3 ^ U ( f . ) = o 1=1 J u ( fi) = < u 1 , u 2 , x 3 AX^ + UgAX^ . , X g > \u00E2\u0080\u00A21 u i A\"5 f g = U - ^ X + U g A X g = X ^ A X 5 + X ^ A X g F o r a . e F , n o t a l l z e r o , Z = y a . f . = u . A ( a n x - ,+a.x_) + 1 * ' \u00C2\u00A3 1 1 1 v 1 3 2 5 + U g A ( a 1 x 4 + a 2 x 6 ) + a y c ^ A x ^ + a ^ x ^ A X g . I f a n y o f {0^} i s z e r o , R ( Z ) = 2 . I f a. | o , a. Z \u00C2\u00AB ( u r ^ ) A ( a 1 x 3 + a 2 x 5 ) a + ( u 2 + a T x 4 ) A ( a l x 4 + a 2 x 6 ) > 2 a n d R ( Z ) = 2 We c a n now show t h a t i f d i m U = 6 a n d H e R g ( U ) > 66. then dim H _< 3 . We already know from Chapters II and I I I that i f f f 1_, \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 } f k ] are independent i n H , and k dim r U ( f \u00C2\u00B1 ) < 5 , then k 3 . We s h a l l have the desired r e s u l t i f we now show that when {f^, . . ., f f c} are independent i n H , and . dim \ U ( f \u00C2\u00B1 ) = 6 , then k. < 3 . In f a c t , we show that any four independent vectors f f ^ . . \u00C2\u00BB, f ^ ] i n Cg(U) such that dim | U(f.) = 6 cannot generate a rank two subspace. To do t h i s , we take two 3 -dimensional rank two subspaces and , and show t h e i r sum i s not a rank two subspace. By Theorems 17 .61,62, there e x i s t pairwise-Pg bases f g ^ g g , g^l , f g ^ g ^ g j + l f o r , r e s p e c t i v e l y ; and dim 0 U(g,) = o 1 or 2 and 1 d * 1=1,2,3 dim 0 U(g,) = o , l or 2 . We must show 1 R 2(U) 1=1,2,4 1 d f o r a l l s i x p o s s i b l e cases. We note that the fol l o w i n g r e s u l t s are true f o r any dimension n of U , n > 6 , unless otherwise s p e c i f i e d . Lemma IY. 64 Let H e Rg.(U) . Let f f - ^ f g , f-j] he independent, pairwise-P^ i n H s a t i s f y i n g ( i ) dim ? U(f.) = 6 D i = l x ( i i ) h ( f j = 0 . i = l 1 67. I f f 4 e C g ( U ) , i n d e p e n d e n t o f f f ^ f ^ f } s a t i s f y i n g ( a ) d i m | U( f , ) = 6 ( b ) f f , , f 0 , f , . i a r e 1=1 1 1 d 4 p a i r w i s e - P f i ( c ) d i m n U ( f . ) = 1 , t h e n < f 1 , . . . , f k > 4 R o ( U ) 1=1,2,4 1 1 ^ d P r o o f : B y T h e o r e m I V .57, [ i U ( f , ) ] h a s a b a s i s 1=1 f u 1 j u 2 * x 3 * * \u00E2\u0080\u00A2 * } X 6 ^ s ^ c h t h a t f 1 = u i A X 3 + U 2 A X 4 * f 2 = U l A X 5 4- U g A X g , f 3 = X 5 A z + X 6 A Z ' * < Z , Z '> = \u00E2\u0080\u00A2 L e t - . = i n 2 j 4 U ( f . ) . T h e n u e . I3 2 By T h e o r e m IV.57, we c a n t a k e = u . S i n c e d i m \u00C2\u00A3 U ( f . ) = 6 a n d d i m fl U( f . ) = 1 1=1,2,4 1 i = l,2,4 1 t h e r e e x i s t s g^ e < f - L , f g , f^> s u c h t h a t = < f 1 , f g , f J + > a n d g^ = V ' A W 7 + Y ^ A X g , o ^ y e F , v ' e [ u ^ x , x g ] , w ' s [ U g j x ^ , X g ] . W i t h o u t l o s s o f g e n e r a l i t y , we w i l l t a k e Y = 1 . Now f-j = X ^ A Z + X g A Z ' , w h e r e = . H e n c e f v ' , w ' , x 4 , X g , x ^ , x ^ } i s i n d e p e n d e n t . I n f a c t { v ' , w ' , X g , x , - , z, z ' } i s i n d e p e n d e n t s i n c e < z , z ' > = . A l s o f o r some o | a 6 F , {x^+az '^z} i s i n d e p e n d e n t . F o r s u c h a n a , Z = g^ - a f = v ' A w ' H-.OZAX^ + ( x 4 4 - a z ' ) A X g h a s r a n k 3 . H e n c e < f 1 , . . . , f ^ > k R g ( U ) 68. Lemma I V . 6 5 L e t H \u00E2\u0082\u00AC R g ( U ) . L e t f f ^ f ^ f } be i n d e p e n d e n t , p a i r w i s e - P g i n H s a t i s f y i n g ( i ) d i m ^ U ( f . ) = 6 ( i i ) d i m 3 u ( f . ) = 1 . 1=1 1=1 1 I f f^ s C g ( U ) , i n d e p e n d e n t o f ( f - ^ f g , f^} s u c h t h a t ( a ) ( f , , f f k } a r e p a i r w i s e - P A ( b ) d i m n U ( f . ) = 1 a n d ( c ) d i m \u00C2\u00A3 U ( f . ) = 6 , t h e n < f n , . , . , f k > 4 R o ( U ) \u00E2\u0080\u00A2 i = l 1 x ^ 2 P r o o f : B y T h e o r e m I V . 5 9 , < f i > f 2 j f 3 > h a s a b a s i s ( fiJ f25 g3^ s u c h t h a t f1 = u l A x 5 + U g A X ^ , f 2 = U-^X + U g A X g , = u l A U g + v A w 3 w h e r e C u-^Ug^ x ^ , . . . , x g > = [ ? U ( f \u00C2\u00B1 ) ] , v \u00E2\u0082\u00AC [ x 4 , u 1 5 u 2 ] a n d W e [xgjU-^Ug] . L e t = f) U( f . ) . T h e n u e . E i t h e r 1 = 1 , 2 , 4 1 1 d = o r {u,u- L } i s . i n d e p e n d e n t . In t h e l a t t e r c a s e , we c a n t a k e u g = u ( T h e o r e m IV.59)= C a s e 1. u = . B y T h e o r e m I V . 59, we c a n r e p l a c e i f 1 , f 2 > f 4 3 b y a n o t h e r b a s i s { f ^ f g ^ } s u c h t h a t g^ = u ^ U g + V ' A W ' , v ' e [ x ^ j u ^ U g ] , w ' e [ x 6 , u 1 , u 2 ] . Thus u(g 4) = < u 1 , U g , x 4 , X g > 69. = U(g^) . H e n c e f g ^ g ^ } i s a P ^ - p a i r . By T h e o r e m 11.20, ^}^> i s n o t a r a n k 2 s u b s p a c e a n d h e n c e < f 1 , . . . , f ^ > <\u00C2\u00A3 Rg(U) . C a s e 2 . u = u ^ . F r o m t h e p r o o f o f T h e o r e m I V .59, we see t h a t we c a n r e p l a c e f f ^ f ^ f ^ } b y a b a s i s f f x > f\"2> g7^} s u c h t h a t g\ = u 2 A U i + yAZ , y \u00E2\u0082\u00AC [ x ^ u - ^ U g ] a n d z e [ x ^ u - ^ u 2 ] . T h e n Z = g^ + yg'^ = ( 1 - Y ) U 1 A U 2 + V A W + yyAZ h a s r a n k 3 f o r ( 1 - Y ) + o , a n d o 4 Y s F . H e n c e , w h i c h c o n t a i n s < g - j , g ' 4 > , i s n o t a r a n k 2 s u b s p a c e . We h a v e t h e r e s u l t . Lemma T V .66 L e t f f ^ , f ? , f . j } be a '(1,1) b a s i s o f p a i r w i s e - P g v e c t o r s f o r < f 1 , f c , f - , > \u00E2\u0082\u00AC R 2 ( U ) , s u c h t h a t d i m | U ( f , ) = 6 . x d. y 2 1 I f f^ e C g ( U ) i s i n d e p e n d e n t o f f f ^ f g , f^} s a t i s f y i n g ( i ) f f , , f , f h } a r e p a i r w i s e - P g ( i i ) d i m \ U ( f i ) = 6 1 ( i i i ) d i m 0 U ( f . ) = 1 , t h e n < f f h > 4 R p ( U ) . 1=1,2,4 1 P r o o f : F r o m Remark I V .53, 1 U ( f . ) h a s a b a s i s 1 x f u x , u 2 , x 3 , . . .,Xg] s u c h t h a t f f - ^ f g , f^} h a v e r e p r e s e n t a t i o n s f l = u i A X 3 + U 2 A X 4 \u00C2\u00BB f2 = U 1 A X 5 + U 2 A X 6 > f 3 = U 1 A Z + U 2 A Z ' 1 w h e r e < z , z ' > c < u 2 , x ^ , . . , X g > , z 4 ^ - ^ x ^ x y ; z' \ a n d < z , z ' > n U ( f i ) = o , i = 1,2. L e t 7 0 . n U ( f . ) = . I t f o l l o w s e a s i l y f r o m Lemma I I . 1 8 t h a t 1=1,2,4 1 t h a t we c a n assume w i t h o u t l o s s o f g e n e r a l i t y u = u ^ . F r o m Lemma I V . 59 a n d i t s p r o o f , f^ = u ^ y + x^AXg , y \u00E2\u0082\u00AC < u 2 , x ? , . . . , X g > y 4 < u x,x 3,x 5> , y 4 U ( f \u00C2\u00B1 ) , i = 1,2 . A s u i t a b l e l i n e a r c o m b i n a t i o n -of f ^ , f 2 , g i v e s f ' 3 = u ^ v + U 2 A V ' , w h e r e v \u00E2\u0082\u00AC < u 2,x^,Xg> a n d v ' 4 < u 2 , x 4 , X g > S i m i l a r l y , a s u i t a b l e l i n e a r c o m b i n a t i o n o f f , f 2 , f ^ g i v e s f 4 = U ^ A W + W ' A W \" , w h e r e w e < U g , x ^ , X g > a n d w ' e [ x ^ ; u 2 ] , w\" \u00E2\u0082\u00AC [ x g ; u 2 J . C h o o s e o r ^ , . . . i n F a l l n o n z e r o s u c h t h a t [a-^Xj+agX^+OjV+a^w, v ' } i s i n d e p e n d e n t . L e t = f i > ^ 'g = f Z = | a i f ' i ~ u 1A ( a 1x 3 + a 2x^ + a 3 v + a i ) _ w ) + a^UgAv' + W A W * , w h e r e w \u00E2\u0082\u00AC [ x 4 ; x 6 , u 2 ] , a n d w* e [xgjx^Ug] . H e n c e R(z) = 3 , a n d < f x , . . . , f ^ > 4 R^ U) * Lemma I V .67 L e t { f 1 , f 2 , f ^ } be a ( l , l ) b a s i s o f p a i r w i s e - P g v e c t o r s f o r < f 1 , f 2 , f ^ > 6 R 2 ( U ) , s u c h t h a t d i m J U ( f 1 ) = 6 . I f f^ \u00E2\u0082\u00AC c|(U) , i n d e p e n d e n t o f f f - ^ f ^ f ^ } s u c h t h a t ( i ) d i m | U ( f \u00C2\u00B1 ) = 6 ( i i ) f f - ^ f g , ^ } a r e p a i r w i s e - P g ( i i i ) n U ( f . ) = o , t h e n < f * . . . , f 4 > 4 R\u00C2\u00A7(U) * 1=1,2,4 7 1 . , P r o o f : B y R e m a r k X V . 53, ? U ( f \u00C2\u00B1 ) h a s a b a s i s f u ^ u ^ x - ^ . . . , x g } s u c h t h a t [ t ^ t p f j } h a v e r e p r e s e n t a t i o n s f\u00C2\u00B1 = u - ^ x ^ + U g A x 4 , f 2 = u l A x 5 + U g A x g , = u l A z + U g A z ' , w h e r e < z , z ' > c < U g , x 3 , . . . 3 x 6 > , < z f z ' > n U ( f i ) = o , 1 = 1 , 2 ; z 4 < u 1 , x 3 , x 5 > , a n d z ' 4 < U g,x 4 , X g > . P r o m T h e o r e m I V .57, we s e e t h a t we c a n assume f^ = x ^ A v + x ^ A v ' , < v , v ' > = = x,_ Aw .+ XgAw' ; = < x ^ , x 4 > . We s h a l l w r i t e f4 = x 3 A ( a 1 x 5 + a g X g ) + x ^ A ( b ^ y b g X g ) , a i , b \u00C2\u00B1 e F . We s h a l l c o n s i d e r t h e f o l l o w i n g c a s e s : C a s e 1 . a 2 , b ^ = o j a^ =f 0 , b g 4 \u00C2\u00B0 \u00E2\u0080\u00A2 W i t h o u t l o s s o f g e n e r a l i t y , a^ = 1 . fjj. = X 3 A X 5 + b 2 x 4 A x 6 . A s u i t a b l e l i n e a r c o m b i n a t i o n o f f p f 2 > f 3 \u00C2\u00A7 i v e s f ' 5 = u l A y + U g A y ' , w h e r e y e [ x ^ x ^ U g ] , and y ' 4 < u 2, x4, x5> \u00E2\u0080\u00A2 W i t h o u t l o s s o f g e n e r a l i t y , y = x 4 + a x g + b u 2 , a , b e F . L e t Z = + f g + aX'^ + \u00C2\u00AB ( u 1 + x 5 ) A ( x 3 + x 5 ) + U g A ( x 4 + x 6 + a y ' - a b ^ ) + ( a ^ - b g X g ) A ( x 4 + a x 6 ) . T h e n R ( Z ) = 3 f o r some o 4 a s F s i n c e y ' 4 < U g, x 4, x6> \u00E2\u0080\u00A2 72. C a s e 2 - a i ^ b 2 = \u00C2\u00B0 > a 2 4 \u00C2\u00B0 , \ + o . W i t h o u t l o s s o f g e n e r a l i t y , a g = 1 . f^ = X ^ A X g + b ^ A X j - . As i n c a s e 1 , f ' ? = u ^ y + u ^ y ' , w h e r e y = x^ + a x g + b u 2 a n d y ' 4 < u 2 , x 4 , X g > . L e t Z \u00C2\u00BB f 1 + f 2 + a f + f^ = ( u ^ X g ) A ( x 3 + x 5 + a x , + + a a x 6 ) + U 2 A ( x ^ + X g + a y '-abu-j^) + ( x 6 + b 1 x 4 ) A ( x 5 + a x l f ) . T h e n R ( Z ) = 3 f o r some n o n z e r o a \u00E2\u0082\u00AC F s i n c e y ' e < u 2 , x ^ , X g > . C a s e 5 - S u p p o s e one o f a'\u00C2\u00B1>&2}lol}^'2 i s z e r o 811(1 * n e ^es^ n o n - z e r o , s a y b^ = o ; w i t h o u t l o s s o f g e n e r a l i t y , a^ = 1 fk = x 5 A ( x 5 + a 2 x 6 ) + x 4 A h 2 x 6 = X 3 A X ' 5 + x ^ b g X g . Now f 2 = u - ^ x ' p . + ( u 2 - a 2 u 1 ) A X g . A s i n t h e a b o v e 2 c a s e s , we w i l l t a k e f = u - ^ y + u ^ y ' , y s < x 1 ) _ , X g > , y ' 4 < u 2 , x J + , X g > . L e t Z = f 1 + f 2 + a f ' 5 + f 4 = u - ^ x ^ + x ' ^ ) + u 2 A ( x 4 + a y / ) + ( u 2 - a 2 u 1 ) A X g + a u x A y + X ^ A X ' 5 + x ^ A b p X g = ( u 1 + x 3 ) A ( x 3 + x ' 3 ) + u 2 A ( x 4 + a y ' + x 6 ) + a u l A y + ( h 2 x l f - a 2 u 1 ) A x 6 = ( u 1 + x 5 ) A ( x 5 + x / 5 ) + u 2 A(x 4+ay'+x 6) + ay Ay ; y e [^;x6,u1] ; y e [ x g j x ^ , ^ ] since y \u00E2\u0082\u00AC . Then R(Z) =3 for some o 4 a e P since y' 4 < u 2 > x 4 > x g > . Case 4: Suppose a i ? b i are a l l nonzero, i = 1,2 . Without loss of generality, a^ = 1 . f 4 = x 5 A ( x 5 + a 2 x 6 ) + ^ ( b ^ + b p X g ) \u00C2\u00BB ( x 5 + h 1 x 4 ) A ( x 5 + a 2 x 6 ) + X 4 A Y X 6 ; Y = V b l a 2 ' Now f1 = u l A(x 3+b- Lx 4) + (u 2 - b 1 u 1 ) A X 4 , f 2 = u.1/\(x^-b1x^) + (u 2-b 1u 1)AXg , f^ = u 1A(z+b 1z / ) + ( u 2 - b 1 u 1 ) A z ' . Hence f\u00C2\u00B1 - u^x'^ + U ' 2 A X 4 , f g = xi^x/^ + u'^X g , f 5 = u l A \" i + U ' 2 A 1 and f 4 = X ' 5 A ( X ' 5 + ( a g - b - ^ X g ) + Y X 4 A X 6 . If a g - b^ = o , we have case 1 . If a 2 - b ^ 4 \u00C2\u00B0 , w e have case 3. Combining the four cases, we have the result; v iz., < f 1 , . . . . , f 4 > 4 R ^ U ) ' 74. Lemma I V . 6 8 . L e t H e R 2 ( U ) . L e t f f p f 2, f } be i n d e p e n d e n t i n H s u c h t h a t ( i ) f\ U ( f,) = o ( i i ) d i m ?U(f.) = 6 . 1=1 1=1 1 p I f f 4 \u00E2\u0082\u00AC C 2 ( U ) , i n d e p e n d e n t o f f f - ^ f ^ f 3 , s u c h t h a t ( a ) d i m | U( f ) = 6 (b) n U ( f , ) = o , t h e n i = l 1 1=1,2,4\" v < f 1 ? . . . , f 4 > 4 R2(U) . P r o o f : B y R e m a r k I T . 56, f f - ^ f ^ f } a r e p a i r w i s e - P g . Thus [ ^ U ( f i ) ] h a s a b a s i s f x 1 , . . . , X g ] s u c h t h a t f x , f 2 h a v e r e p r e s e n t a t i o n s f\u00C2\u00B1 = x ^ x ^ + x ^ x ^ , f g = x l A x 5 + x 2 A X g , a n d U ( f ^ ) = < x ^ , . . . , x g > ( T h e o r e m I V . 5 7 ) \u00E2\u0080\u00A2 A l s o U( f ^ ) n < x - L , x 2 > = o b y h y p o t h e s i s ( b ) . We w i l l show f i r s t t h a t we c a n a l w a y s f i n d a l i n e a r c o m b i n a t i o n f o f f-L a n d f 2 s u c h t h a t W ( f ^ , f ^ ) n U ( f ) + o ; i . e . , U ( f ? ) fl U( f 4 ) n U ( f ) 4 o > S i n c e W ( f ^ , f 4 ) c U ( f ^ ) a n d d i m W ( f ? , f 4 ) >2, we c a n f i n d x = | a . x . a n d y = fr B \u00C2\u00B1 x 1 , a \u00C2\u00B1 , p. s F' , 1=3 1 1=3 i n d e p e n d e n t i n W ^ , ^ ) \u00E2\u0080\u00A2 Now a f x + b f g = x ^ l a x ^ + b x ^ ) + x 2 A ( a x 4 + b x g ) . C o n s i d e r t h e v e c t o r v = X-jX + X g y -[ u 1 ( a x 3 + b x 5 ) + U 2 ( a x 4 + b x g ) j , w h e r e X-^ X 2 , a , b i U l , u 2 a r e i n d e t e r m i n a t e s i n F . T h e n t h e c o e f f i c i e n t s o f 75-x 3> \u00E2\u0080\u00A2 * \u00E2\u0080\u00A2 > x6 a r e \u00E2\u0080\u00A2 Xla3 + X ^ -+ X 2 3 4 - u 2 a ha5 + x 2 P 5 -ha6 + X2P6 \" u 2 b (**) The d e t e r m i n a n t o f t h e c o e f f i c i e n t s o f X 2, U ; L , U 2 i s \u00C2\u00B03 &3 - a o o -a a5 P5 -b o a6 o -b = a 2 ( 3 6 a 5 - P 5 a 6 ) ab(P 3a 6-P 6a 5+a 43 5-a 5P 4) + b 2(p 1 |a 5-a J +3 3) + w h i c h i s a homogeneous p o l y n o m i a l o f degree 2 i n a and b . S i n c e x and y a r e independent, at l e a s t one o f the c o e f f i c i e n t s o f a' ab and b i n the above p o l y n o m i a l i s n o n z e r o . Hence t h e r e e x i s t n o n - t r i v i a l a,b i n P making (**) a l l z e r o ; i . e . , the v e c t o r v i s z e r o . Now U 2 cannot b o t h be z e r o , f o r i f t h e y were, t h e n + X, f o r n o n t r i v i a l X x, \ 2 , and n o n t r i v i a l feu] , fp^} \u00E2\u0080\u00A2 c o n t r a d i c t i n g the independence of x and y . Hence there e x i s t \ 1 } X 2,u 1,u 2,a,b such that ( X\u00C2\u00B1, X 2) 4 (0, 0 ) , (u,u 2) + (\u00C2\u00B0,\u00C2\u00B0) \u00C2\u00BB and (a,b) 4 (0,0) , and the vector v i s zero. Thus W( f^) n u ( f ) 4 o . We now have - (say), and < f 1 , f 2 , f ? > = . Now U(f) ~> but U ( f \u00C2\u00B1 ) n = o , i = 3,4 . Hence dim[W(f 3,f 4)^U( f ) ] = 1 or 2 . In the f i r s t case, we can assume f f , f - ^ , f 4 ) are pairwise-Pg (lemma IV.5 8 ) , and the r e s u l t follows by Lemmas IV.64,65,67 . In the second case, has a pairwise-Pg, basis f{g,g3,gij.} (Lemma IV . 6 0 ) , and from the proof of Lemma IV.60 , U(g) n U(g^) n U(g 4) 4 0 \u00E2\u0080\u00A2 I f the dimension of t h i s i n t e r s e c t i o n i s one, we have the r e s u l t by a s i m i l a r proof to the one above. I f i t i s two, C o r o l l a r y 11.34 shows {g,g 3,g 4} i s a (1,1) basis f o r < f , f ? , f 4 > . The r e s u l t follows by Theorem 111.40 and Lemmas IV.66,67 . Theorem I V . 69 Let H \u00E2\u0082\u00AC R g ( U ) . Let f f ^ f ^ f ^ } be independent i n H such that dim \ U ( f 1 ) = 6 . I f 77. f ] \ e C 2(U) , independent of ff f f } such that \u00E2\u0080\u00A21' 2> 3' S 1 dim p U ( f i ) = 6 , then < f ^ . . .,f 4> 4 R 2(U) Proof : By Theorem I V . 6 l , we can assume f f ^ f ^ f ^ } are pairwise-Pg Again, by Theorem I V . 6 l , we can fur t h e r assume f f ^ f g , f^} are pairwise-Pg . Now dim r\ U(f, ) = 0,1,2, and 1=1 1 dim o U( f, ) = o,l,2. In the case where dim ? U( f. ) = 2 1=1,2,4 1 1=1 1 dim ^ U(f.) = 2 we have dim ft U(f.) = 2 . By 1=1,2,4 1 1=1 1 C o r o l l a r y 11.34, f f 1 , f 2 , f^} i s a (1,1) basis f o r < f 1 , f 2 , f 5 > , f f 1 , f 2 , f 4 ] i s a (1,1) basis f o r < f 1 , f 2 , f i f > . Hence f f 1 , . . . , f ^ } i s a (1,1) basis f o r c o n t r a d i c t i n g Theorem 111.40. The other cases are dealt with i n Lemmas IV.64 _> 68. We have the desired r e s u l t . Theorem IV. 70. Let H \u00E2\u0082\u00AC R 2(U) . Let f f ] _ , f 2 , f-^ } he independ-ent i n H ; dim \ U(f j_) = 5 . I f f 4 e C 2(U) , f4 4 f2,fy> > a n d d i m T- u ^ f i ^ = 6 ; t h e n 4 R^(u) . Proof: dim U(f- L) + U ( f 2 ) = 5 \u00E2\u0080\u00A2 Hence dim \u00C2\u00A3 U(f.) = 6 . 1 2 4 7SV S i m i l a r l y d i m j U ( f . ) = 6 . B y T h e o r e m 1 7 . 6 9 , < f 1 , . . . , f 4 > 1 R2(U) . T h e o r e m I V . 71 L e t H e R g ( U ) . L e t f f . ^ . . . . f f e ] be i n d e p e n d -e n t i n H such t h a t d i m r U ( f . ) = 6 . T h e n k < 3 . F o r 1=1 1 . = k = 3 , < f 1 , f g , f ^ > h a s a b a s i s o f p a i r w i s e - P g v e c t o r s . P r o o f : T h i s i s i m m e d i a t e f r o m ' T h e o r e m s I V . 6 9 , 7 0 a n d 6 l . T h e o r e m I V . 7 2 : L e t H e R g ( U ) . I f d i m U = 6 , t h e n d i m H < 3 . P r o o f : L e t f f ^ , . . . , f k ] be I n d e p e n d e n t i n H . Now d i m I U ( f . ) = 4 , 5 , 6 . I n t h e f i r s t c a s e ; k = 1 ( C o r o l l a r y I I . 2 1 ) . 1 1 I n t h e s e c o n d c a s e ; k _< 3 ( T h e o r e m I I I , 38). I n t h e t h i r d c a s e , k < 3 ( T h e o r e m I V . 7 1 ) . 79-CHAPTER V THE RANK TWO SUBSPACES WHEN U HAS DIMENSION J I n t h i s c h a p t e r , we show t h a t i f H e R | ( U ) , a n d d i m U = 7 , t h e n d i m H < 4 . I f d i m H = 4 , i t h a s a (1,1) b a s i s . B o t h t h e s e r e s u l t s a r e c o n t a i n e d i n T h e o r e m V . 8 5 . F o r a n e x a m p l e o f s u c h a b a s i s , see T h e o r e m 111.40. I f t h e d i m H = 2 , i t h a s a (1,1 ) b a s i s . I f d i m H = 3 , a n d H 3 h a s a b a s i s [ f - ^ f g , ^ ] , t h e n d i m TU(f\u00C2\u00B1) = 5,6,7 . I n f i r s t e a s e , T h e o r e m I I I . 4 5 g i v e s r e p r e s e n t a t i o n s f o r { f - L , f 2 , f 3 } . I n t h e s e c o n d c a s e , { f - ^ f ^ f ^ } a r e p a i r w i s e - P g ( T h e o r e m I V . 61 ) a n d T h e o r e m s I V . 62,. '57, 59 a n d R e m a r k I V . 53 g i v e r e p r e s e n t a t i o n s f o r { f - ^ f g , ^ } , I n t h e t h i r d c a s e , T h e o r e m s V . 74, 75 g i v e r e p r e s e n t a t i o n s f o r { f - ^ f g , ^ } . Theorem- V . 73 L e t H \u00E2\u0082\u00AC R g ( U ) w i t h d i m H > 3 . L e t 3 { f 1 , f g , f ^ } be i n d e p e n d e n t i n H s u c h t h a t d l m [ s u ( f ^ ) ] = 7 . T h e n { f - ^ f g , ^ } c o n t a i n s a P g - p a i r , s a y { f - ^ f g } , w h i c h c a n be e x t e n d e d t o a p a i r w i s e - P g b a s i s [ f 1 , f g , g ^ } o f < f 1 , f g , f ^ > . F u r t h e r m o r e , e i t h e r { f - ^ f g ^ } i s a ( l , l ) - b a s i s f o r < f 1 , f g , f ? > o r d i m [ u ( f 1 ) n u ( f 2 ) n u ( g 3 ) i = 1 . P r o o f : We show f i r s t t h a t { f - ^ f g , ^ } c o n t a i n s a t l e a s t two P . - p a i r s . S i n c e d i m [ l u ( f . ) ] < ? d i m ! U ( f ) - 2 d i m W ( f . , f \u00E2\u0080\u00A2 6 1 1 \"1=1 1 l < i < j < 3 3 d i m U ( f . ) , we h a v e (*) x\u00C2\u00B12 + X g ? + x - 5 < x w h e r e 1=1 80 x i j = d i m W ^ f I * f ^ ' a n d x = d i m ^ \u00E2\u0080\u00A2 B y T h e o r e m 1 1 . 2 6 , 2 \u00C2\u00A3 x i j 1 ^ , 1 < i < j < .3 a n d o < x < 3 . A l s o , p u t t i n g y = x 1 2 + X g ^ + x , we h a v e 6 < y \u00C2\u00A3 9 . d e a r l y x c a n n o t be z e r o . I f x = 1 , we h a v e y = 6 a n d h e n c e e a c h x ^ j = 2 . I f x = 2 , we h a v e y <_ J so t h a t a t l e a s t two o f t h e x ' s X J a r e 2 . I f x = 3 s t h e n e a c h x . = 5 a n d h e n c e y m u s t be 9 \u00E2\u0080\u00A2 B u t b y e q u a t i o n (*) a b o v e , y <_ 8 . We h a v e a c o n t r a d i c t i o n a n d h e n c e t h i s c a s e i s n o t p o s s i b l e . T h e r e f o r e d i m ?i U ( f . ) = 1 , 2 , . I n t h e f i r s t c a s e , [ f , , f _ , f _ } a r e 1=1 1 1 2 3 p a i r w i s e - P g . I n the s e c o n d c a s e , { f - ^ f ^ f ^ ] c o n t a i n s a t l e a s t two P g - p a i r s . 3 We now c o n s i d e r t h e c a s e when d i m [ n U ( f . ) ] = 2 . 1=1 1 3 L e t { f 1 ^ 2 } , {?2>fj} b e t h e P g - p a i r s a n d PI U ( f \u00C2\u00B1 ) = B y T h e o r e m s I I . 1 8 , 3 1 , t h e r e e x i s t s a b a s i s { u ^ u ^ x ^ , . . . , x y } 3 f o r [ \u00C2\u00A3. U ( f . ) ] s u c h t h a t f = U , A X , + u . A x , , f 2 = u i A x 5 + U 2 A X 6 ' a n d f 3 = u l A w + U 2 A W ' w i t h u ( f 3 ^ 3 < x 7 > H e n c e { f ^ f g , : ^ ) i s a U* 1) b a s l s f o r < f 1 ^ 2 , f 3 > * I t now r e m a i n s t o show t h a t t h e r e e x i s t s g^ e < f 1 , f 2 , f ^ > s u c h t h a t f f 1 ? f 2 , g - j } i s a (1 ,1) b a s i s o f p a i r w i s e - P g v e c t o r s f o r < f 1 , f 2 , f > . I n t h e above r e p r e s e n t a t i o n o f f ^ , a t l e a s t one o f { w , w ' } s a y w e [Xj, u 2 , x ^ , . . . , X g ] , T h u s some l i n e a r c o m b i n a t i o n aw + Bw' e < u 2 , x ^ , \u00E2\u0080\u009E . . , X g > , w i t h B =)= \u00C2\u00B0 \u00E2\u0080\u00A2 W i t h o u t l o s s o f g e n e r a l i t y , we s h a l l t a k e 6 = 1 . T h e n 8 i . T h e n f3 = - au 2)Aw + u gA(aw + w ' ) f\u00C2\u00B1 = ( u x - a u 2 ) A x 3 + u 2 A ( a x ? + x^) f2 = ^ u l \" a u 2 ) A x 5 + u 2 A ( a x 5 + x g ) \u00E2\u0080\u00A2 R e n a m i n g ( i ^ - a u 2 ) = u j , w = x j > aw + w' = v , ax^ + x^ = x^ , ax,_ 4- X g = X g , we h a v e f l = u i A x 3 + u 2 A x i f 2 = U ^ A X 5 + UgAXg f ^ = U^Ax| + UgAv , V \u00E2\u0082\u00AC < U g , X , X ^ , X , X g > . I f v 4 U ( f \u00C2\u00B1 ) , i = 1,2, t h e n { f - ^ f ^ f } a r e p a i r w i s e - P g . I f , s a y , v e U ( f l ) , i . e . ( f ^ j - s a p 5 _ p a l r , t h e n l e t g^ = fg + f = U ^ A ( X 5 4- X ^ ) 4- U 2 A ( X g 4- v) . T h e n { f - ^ f g j g , } f o r m a (1,1) b a s i s o f p a i r w i s e - P g v e c t o r s f o r < f 1 , f g , f ^ > . The t h e o r e m i s p r o v e d . We now g i v e r e p r e s e n t a t i o n s f o r t h e 2 k i n d s o f p a i r w i s e - P g b a s e s i n T h e o r e m V.73. T h e o r e m V . 74. L e t H e R ^ ( U ) . L e t { f - ^ f g , ^ } be p a i r w i s e -Pg , i n d e p e n d e n t i n .H s a t i s f y i n g ( i ) { f ^ f g , ^ } i s a (1,1) b a s i s f o r < f x , f g , f 3 > ; ( i i ) d i m \ U ( f , ) = 7 * 1=1 T h e n [ \u00C2\u00A3 U(f,)] h a s a b a s i s [ u - ^ u ^ x . . . , x } s u c h t h a t 82 { f - j ^ f g j f } h a v e r e p r e s e n t a t i o n s : f l = u l A x 3 + . U 2 A X 4 > f ^ = u^Ax-, + UgAv , w h e r e v \u00E2\u0082\u00AC , v 4 a n d v 4 U ( f i ) , i = 1,2 . P r o o f : The p r o o f i s c o n t a i n e d i n t h e p r o o f o f T h e o r e m V . 7 3 . T h e o r e m V . 75 L e t H e R | ( U ) \u00E2\u0080\u009E L e t { f - ^ f ^ f ^ } be i n d e p e n d e n t i n H s a t i s f y i n g ( i ) d i m [ 1 U ( f . ) ] = 7 ; ( i i ) d i m n U ( f , ) = 1 . 1=1 1=1 T h e n f o r any b a s i s o f \u00C2\u00A5 ( f ^ , f g ) such t h a t < u l > = ~j\ u ( f i ) , t h e r e i s a b a s i s { u ^ u ^ x ^ , . . . , x - J o f 3 [ \u00C2\u00A3-U(f.)] s u c h t h a t 1=1 1 f l = U,AX 7 1 3 + U 2 A X j + , f 2 = U , A x r 1 5 + U 2 A X g , f 3 = U l A x 7 + X^AXg F u r t h e r m o r e , a n y t h r e e v e c t o r s { f ^ f ^ f } w i t h ' r e p r e s e n t a t i o n s (*) g e n e r a t e a r a n k two s u b s p a c e o f d i m e n s i o n t h r e e . P r o o f : B y T h e o r e m V . 7 3 , i f 1 > f 2 } f ^ a r e P a i r w i s e - p 6 \u00E2\u0080\u00A2 T h u s , 8 3 . 3 b y T h e o r e m s 1 1 . 1 6 , 1 1 . 3 1 , f o r = n U ( f ) , a n d a n y v e c t o r 1=1 x u 2 f o r w h i c h < u 1 , u 2 > = W f f - ^ f ) , t h e r e i s a b a s i s 3 {u-j^u^x-^,... ,Xj) f o r [ E U ( f i ) ] s u c h t h a t f l = U 1 A X 3 + u 2 A x 4 > f g = U X A X 5 + U 2AXg , f^ = u^Au' + V A W , w h e r e c a n d U ( f - 5 ) 3 < x 7 > . S i n c e { f - ^ f } , { f g , f } a r e , P g - p a i r s , b y C o r o l l a r y 1 1 . 3 2 , i n t e r s e c t s e a c h o f < U g , x ^ > a n d < U g,Xg> i n d i m e n s i o n 1 . I f u 0 i s i n e i t h e r i n t e r s e c t i o n ; t h e n d i m fl U ( f . ) >_ 2 , d 1=1 c o n t r a d i c t i n g h y p o t h e s i s ( i i ) . H e n c e V A W e [x^; U g ] A [ x g ; U g ] . Now UgAx^ e UgAfx^; U g ] a n d UgAXg \u00E2\u0082\u00AC u 2A[Xg; U g ] \u00E2\u0080\u00A2 Therefore there e x i s t x' \u00E2\u0082\u00AC [x^ ; and Xg \u00E2\u0082\u00AC [x6; Ug] such t h a t f l = U, Ax, 1 3 + UgAxj^ , f 2 = U, Ax,_ 1 5 + UgAXg , f 3 = U^Au' + yx^AXg , f o r some o |= y e F . 2 L e t a = Y Then f l = u., Ax-, 1 3 + a UgAcxx^ , f 2 = U-. A x c 1 D + a_ 1UgACiXg , f 3 = u xAu' + bx^AaXg Since U ( f 5 ) 3 < x 7 > , u e [x-,; x ^ , X g ] . Renaming a _ 1 u 2 84. a x ^ b y x ^ , a x g b y X g a n d u ' b y x ? , we h a v e t h e d e s i r e d r e p r e s e n t a t i o n s (*) , w i t h i U ( f . ) = 1=1 1 ' We now c o n s i d e r z = 2 aj_\u00C2\u00A3j_ > e F , n o t a l l z e r o . T h e n z = U ^ A ( a ] _ x ^ + ctgX^ + a ^ x 7 ) + u 2 A (a-^x^ + a ^ X g ) + a y c ^ A X g I f a2 = o , z = u-^A ( a - ^ X j + a 2 x ^ + a y e - , ) + ( a 1 u 2 - a y C g ) A X 4 a n d R ( z ) = 2 . I f ctg 4= o , a-, z = u 1 A ( a 1 x 5 + a g x 5 + a-f-j) + ( u g + ^ x ^ A ^ x ^ + c ^ X g ) a n d R ( z ) = 2 . T h u s e v e r y n o n - t r i v i a l l i n e a r c o m b i n a t i o n o f ( f ^ , f 2 , f ^ } i s a r a n k 2 v e c t o r , a n d h e n c e { f ^ , f 2 , f } a r e i n d e p e n d e n t a n d g e n e r a t e a r a n k 2 s u b s p a c e . We w i l l u s e t h e f o l l o w i n g lemmas t o show t h a t i f ? ft { f , , . . . , f , } i s a b a s i s f o r H \u00E2\u0082\u00AC R (U) a n d d i m ' 2 U ( f . ) = J , t h e n H h a s a ( 1 , 1 ) b a s i s ' , t h r e e o f whose members a r e p a i r w i s e -Pg ( T h e o r e m V . 8 l ) . T h e r e a r e t h r e e c a s e s , a c c o r d i n g as d i m J 5 U ( f ) = 5,6, o r 7 . We p r o v e t h e r e s u l t f o r t h e t h i r d 1=1 c a s e a n d u s e i t t o p r o v e t h e o t h e r s . 85. Lemma V . 7 6 . L e t H e R 2 ( U ) w i t h d i m H > 4 . L e t 4 { f x , . . . , f J be i n d e p e n d e n t i n H s u c h t h a t d i m 2 U ( f ) = 7 , i = l x 3 a n d l e t { f , , f 2 , f } be p a i r w i s e - P ^ s u c h t h a t d i m 2 U ( f . ) = 7 ;? o 1=1 1 T h e n two o f { f - ^ f ^ f } , s a y { f - ^ f g } , a r e s u c h t h a t d i m 2 U ( f . ) = 7 . 1=1,2,4 1 P r o o f ; S i n c e { f - ^ f 2 , f } a r e p a i r w i s e - P g , d l m [ U ( f ) + U ( f p ) ] = 6 . T h e n d i m 2 U ( f . ) = 6 , 7 . I n . i = l , 2 , 4 1 t h e f i r s t c a s e , d i m v, U ( f . ) = 7 . I n t h e s e c o n d c a s e , 1=1,2,4 1 we h a v e t h e r e s u l t . Lemma V . 7 7 . L e t { f 1 , f 2 , f , } be a ( l , l ) b a s i s f o r p 3 < f , , f 0 , f ^ > e R o ( U ) , s u c h t h a t d i m 2 U ( f , ) = 7 . I f 1 2 3 2 1=1 1 f 4 e C 2 ( U ) , i n d e p e n d e n t o f { f ^ f ^ f ^ } , s u c h t h a t 4 ( i ) d i m 2 U ( f \u00C2\u00B1 ) = 7 \u00E2\u0080\u00A2 ( i i ) < f x , . . . , f 4 > e R 2 ( U ) , t h e n < f 1 , . . . , f J | _ > h a s a ( l , l ) b a s i s , t h r e e o f whose members a r e p a i r w i s e - P g . P r o o f : By T h e o r e m V.73> we c a n assume { f - ^ f 2 , f ] a r e p a i r w i s e Pg . By T h e o r e m V . 7 4 , { f - ^ f ^ f ^ } h a v e r e p r e s e n t a t i o n s f x = U XAX 5 + u 2Ax^ , f 2 = U 1 A X 5 + U 2 A X 6 ' = u-^x^ + , v 6 < u 2 , x ? , . . . , X g > a n d v 1 < u 2,x^,Xg> . 86. B y Lemma V . 7 6 a n d T h e o r e m V.7 3 , we c a n assume d i m E U ( f ) = 7 a n d { f y ' , f 0 , f , , } a r e p a i r w i s e - P ^ 1=1,2,4 D F o r d i m E U ( f , ) = 7 , T h e o r e m \u00C2\u00A5 . 7 3 i m p l i e s 1=1,2,4 1 e i t h e r {f^fg,?^} i s a ( x ^ ) b a s i s f o r < f 1 , f 2 , f ^ > o r d i m n U ( f , ) = 1 . 1=1,2,4 1 I n t h e f i r s t c a s e , i t i s e a s y t o see t h a t { f ^ . . . , f ^} i s a ( 1 , 1 ) b a s i s f o r < f 1 , . . . , f ^ > , a n d t h e r e s u l t f o l l o w s . I n t h e s e c o n d c a s e , l e t - = n U ( f ^ ) . S i n c e i. \u00E2\u0080\u0094=1 ^ 2 $ ^ \u00C2\u00A5 ( f n , f 0 ) s D U ( f . ) , t h e n u \u00E2\u0082\u00AC < u , , u 0 > . B y T h e o r e m I I . 1 8 , 1 d 1=1 ,2 ,4 1 1 2 we c a n w i t h o u t l o s s o f g e n e r a l i t y assume = u . F r o m T h e o r e m V . 7 5 a n d i t s p r o o f , f 4 = U^J^AW + yx^Axg f o r some o 4= Y e F a n d w e [x ?, x^,Xg]. L e t z = f^ + f g + f- j + a f ^ = u 1 A ( x ^ + x,~ + x ? + aw) + UgAv + (Ug + a Y x 4 ) A ( x 4 + x g) . S i n c e v | , f o r some o a e F s u c h t h a t x ? + aw e [x ?; x^,Xg] , R ( z ) = 3 . H e n c e t h i s c a s e i s n o t p o s s i b l e i f < f 1 , . . . , f 4 > 6 R g ( U ) . . T h e r e s u l t f o l l o w s . Lemma V . 7 8 . L e t H e R g ( U ) . L e t { f ^ f ^ f } be i n d e p e n d e n t i n H s u c h t h a t d i m 1 U ( f \u00C2\u00B1 ) = 7 \u00E2\u0080\u00A2 I f f 4 e c | ( U ) , 1=1 4 i n d e p e n d e n t o f { * W f 3 } s u c h t h a t ( l ) d i m 5 U ^ f i ) = 7 ' 87. ( i i ) < f 1 , . . . , f 4 > e R|(U) , t h e n < f x , . . . , f 4 > h a s a ( 1 , 1 ) b a s i s , t h r e e o f w h o s e members a r e p a i r w i s e - P g . P r o o f : B y T h e o r e m V . 7 3 , we c a n assume { f - ^ f ^ f ^ } a r e p a i r w i s e -P g . M o r e o v e r , e i t h e r { f - ^ f ^ f } i s a ( l , l ) b a s i s f o r o r d i m n U ( f , ) = 1 . The f i r s t c a s e i s c o n t a i n e d 1=1 1 i n Lemma V . 7 7 - We n e e d c o n s i d e r t h e n o n l y t h e s e c o n d c a s e . B y Lemma V . 7 6 , we c a n assume d i m E U ( f . ) = 7 , 1 = 1 , 2 , 4 1 a n d b y T h e o r e m V . 7 3 , { f ^ f ^ f 4 } c a n be t a k e n t o be p a i r w i s e -P g . A s a b o v e , e i t h e r { f - ^ f ^ f ^ } i s a ( 1 , 1 ) b a s i s f o r o r d i m H U ( f ) = 1 . A g a i n , t h e f i r s t c a s e 1 d 4 i = l , 2 , 4 1 i s c o n t a i n e d i n T h e o r e m V . 7 7 -3 B y T h e o r e m V . 7 5 , [ E U ( f . ) ] h a s a b a s i s 1=1 {u , , u 0 , x ^ , . . . , x 7 } s u c h t h a t f l = U_ AX^ 1 3 4- UgAx^ , f 2 = U, AX\u00E2\u0080\u009E 1 5 + UgAXg , f 3 = U-^AX^ + X^AXg . L e t = H U ( f , ) . T h e n u e W ( f , f ) = < u n , u _ > . 1=1,2,4 1 1 2 1 2 B y T h e o r e m V . 7 5 , we n e e d c o n s i d e r o n l y two p o s s i b i l i t i e s : e i t h e r = \" o r { u , ^ } i s i n d e p e n d e n t i . e . , = . C a s e 1. u = u n . 8 8 . A s i n t h e p r o o f o f T h e o r e m V . 7 5 , f 4 = u i A y + v A w > y \u00E2\u0082\u00AC [Xj, u 2 , x 4 , x 6 ] , v e [ x 4 ; u g ] , w e [x g; Ug] . T h u s , w i t h o u t l o s s o f g e n e r a l i t y , f ^ = u^Ay 4- ( x ^ + a ^ U g ^ b ^ X g + h 2 u 2 ) , a^,b^ e F , b^ =(= o . I f a 2 = o , ( b 1 f 5 \u00E2\u0080\u00A2- b 2 f 1 - f ^ ) = u 1 A ( b 1 x 7 - b 2 x 3 - y) h a s r a n k o n e . I f b 2 = o , ( a 2 b ] _ f 2 + b 1 f ? - f 4 ) = u 1 A ( a 2 b 1 x 5 + b ^ - y) h a s r a n k o n e . 2 H e n c e i n t h i s c a s e , . . . , f 4 > i R 2 ( u ) . C a s e 2 . u = u 2 . A s i n t h e p r o o f o f T h e o r e m V . 7 5 , f^ = UgAy + V A W , y e fx,,; u ^ x ^ x ^ > v e \] , w e [x,-j u.^ . T h e n { f 3 , f 4 } i s a P k - p a i r , k > 7 \u00E2\u0080\u00A2 B y T h e o r e m I I . 2 6 , 2 z = a f ^ + p f ^ 4 C 2 ( U ) , a , p b o t h n o n z e r o i n F . A g a i n 2 < f - L , . . . , f ^ > 4 R 2 ( U ) . The lemma i s p r o v e d . Lemma V . 79- L e t H e R g ( U ) . L e t { f ^ f ^ f ^ } be i n d e p e n d e n t i n H s u c h t h a t d i m E U ( f ) = 6 . I f f. e C 2 ( U ) i n d e p e n d e n t 1=1 1 4 d 4 o f { f , ,f_,f_} s u c h t h a t ( i ) d i m E U ( f . ) = 7 1 2 3 i = l 1 ( i i ) < f 1 , . . . , f i ) _ > e R 2 ( U ) , t h e n < f 1 , . . . , f ^ > h a s a (1,1) , b a s i s , t h r e e o f whose members a r e p a i r w i s e - P g . P r o o f : B y T h e o r e m V . 6 l , we may assume t h a t { f ^ f ^ f i j ' a r e p a i r w i s e - P g . . , T h e n d i m [ U ( f 1 ) + U ( f 2 ) ] = 6 . T h e n d i m E U ( f . ) = 7 . The r e s u l t f o l l o w s b y Lemma V . 7 8 . 1 = 1 , 2 , 4 , '\u00E2\u0080\u00A2\u00E2\u0080\u00A2 v \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 ^ Lemma V . 8 0 . L e t H e R\u00C2\u00A7(U) \u00E2\u0080\u00A2 L e t tt\u00C2\u00B1>t2>f2^ b e i n d e P e n d e n t 89. In H such that dim \u00C2\u00A3 U(f\, ) = 5 . I f f ^ e C^(U) inde-4 pendent of { f ^ f ^ f ^ } such that ( i ) dim \u00C2\u00A3 U ( f \u00C2\u00B1 ) = 7 ( i l ) e R^U) then has a (1,1) o a s i s , three of whose members are pairwise-Pg . 3 Proof; dim E U(f ) = 5 and therefore dim[U(f 0) +- U(f )] = 5 \u00E2\u0080\u0094 \u00E2\u0080\u0094 \u00E2\u0080\u0094 1 3 (Theorem 1 1 . 2 6 ) . Hence dim \u00C2\u00A3 U(f.) = 7 = dim E U(f ) . The 1=2 1 1=1 1 r e s u l t f o l l o w s immediately from Theorem V . 7 8 . Theorem V . 8 l . Let H e R p ( U ) . Let {f ,. . . , f j j be independent 4 P i n H such that dim E V(\u00C2\u00B1\u00C2\u00B1) = 7 , I f e Rg (U) , then i t has a (1,1) b a s i s , three of whose members are pairwise-P 6 \u00E2\u0080\u00A2 Proof; By Theorem 1 1 . 2 6 , dim \u00C2\u00A3 U ( f \ . ) = 5*6,7 \u00E2\u0080\u00A2 The r e s u l t 1 1 follows immediately from Lemmas V.78, 79 , 80 . P Lemma V . 8 2 . Let H e R^(U) \u00E2\u0080\u00A2 Let [ f f . , } be independent 2 4 1 4 i n H such that ( i ) dim \u00C2\u00A3 lUf\u00C2\u00B1) =7 , ( i l ) { ^ , . . . ^ 4 } i s a (1,1) b a s i s f o r , and ( i l l ) { f ^ f ^ f } are pairwise-Pg . Then two of { f ^ f g , ^ } , say { f ^ f g ) , are such that dim E U ( f i ) = 7 . i, \u00E2\u0084\u00A21 2 $ ^ Proof: I f dim 1 U(f, ) = 6 , then dim E U ( f . ) = 7 . 1 1 1=1,2,4 x Otherwise dim \ U(f,) = 7 . By Theorem I I I . 4 0 , 1 1 9 0 . 4 d i m S ' U ( f 1 ) = 6 , 7 \u00E2\u0080\u00A2 I n the f i r s t c a s e , d i m \u00C2\u00A3 U ( f . ) = 7 2 1=1,3,4 1 I n t h e s e c o n d c a s e , we h a v e t h e r e s u l t . The p r e c e d i n g r e s u l t w i l l be u s e f u l i n t h e f o l l o w i n g lemma, a n d i n t h e n e x t c h a p t e r . We s h a l l now show t h a t i f d i m U = 7 , H e R | ( U ) , t h e n d i m H < 4 . I f d i m H = 4 , i t h a s a ( 1 , 1 ) b a s i s , three of whose members a r e p a i r w i s e - P g . To show t h e f i r s t r e s u l t , we p r o v e t h a t d i m H + 5 . T h e o r e m V . 8 3 L e t H s R ? ( U ) a n d l e t {f, ,. . . ,f k ) be d ^ x 4 i n d e p e n d e n t i n H s u c h t h a t d i m \u00C2\u00A3 U ( f . ) = 7 . I f 1=1 1 f,_ e C 2 ( U ) , i n d e p e n d e n t o f {f ^ , . . . , f ^ } s u c h t h a t d i m \u00C2\u00A3 U ( f ) = 7 . t h e n 4 R ^ ( U ) . P r o o f : B y T h e o r e m V . 8 l , we may assume {f-^,. . . , f ^ } i s a (1 , 1 ) b a s i s f o r < f - ^ , . . . , f 4 > e R 2 ( U ) , a n d { f - ^ f g , ^ } a r e p a i r w i s e -P g . B y Lemma V.82, we may f u r t h e r assume d i m \u00C2\u00A3 U ( f ^ ) = 7. 5 H e n c e d i m \u00C2\u00A3 U ( f . ) = 7 , a n d [ f ,. . . , f } i s a ( l , l ) b a s i s 1=2 5 f o r < f 2 , . . . , f 3 > b y T h e o r e m V . 8 1 . S i n c e [ f g , f ^ } i s a P g -p a i r , d i m W ( f g , f 3 ) = 2 . L e t < u 1 , U g > = W ( f g , f 5 ) . T h e n f ^ h a s a r e p r e s e n t a t i o n u^Av^ 4- UgAw^ 1 <_ i _< 5 , s i n c e 3 U ( f \u00C2\u00B1 ) = < u 1 , U g > , a n d { f . ^ . . . , ^ } , ( f g , . . .tfJ are (1 ,1) b a s e s f o r t h e i r r e s p e c t i v e s u b s p a c e s . H e n c e { f ^ , . . . , f ^ } i s a ( 1 , 1 ) b a s i s f o r c o n t r a d i c t i n g T h e o r e m I I I . 4 0 . T h e r e f o r e < f x , . . . , f ^ > 4 ' 91. T h e o r e m V . 8 4 L e t H e R | ( U ) . L e t { f i s . . . , f k ) be k i n d e p e n d e n t i n H s u c h t h a t d i m L U(f. ) = J . 1=1 1 T h e n k < 4 . I f k = 4 , < f 1 , . \u00E2\u0080\u009E \u00E2\u0080\u009E , f 1 | > h a s a (1 , 1 ) b a s i s t h r e e o f whose members a r e p a i r w i s e - P g . P r o o f : B y T h e o r e m 1 1 . 2 6 , d i m ^ U ^ ) = 5 , 6 , 7 . I f k - 1 w , d i m 2 U ( f ) = 5 , t h e n k - 1 < 3 b y T h e o r e m I I I . 3 8 , a n d h e n c e 1=1 k - 1 k < 4 . I f d i m 2 U(f, ) = 6 , t h e n k - 1 < 3 b y T h e o r e m 1=1 ~~ k - 1 I I I . 3 8 , a n d h e n c e k < 4 . I f d i m 2 U ( f , ) = 6 , k - 1 < 3 1=1 ~ b y T h e o r e m I V . 7 1 , a n d a g a i n k < 4 . F i n a l l y , f o r k - 1 k d i m 2 U ( f ) = 7 a n d d i m 2 U(f.) = 7 , T h e o r e m V . 8 3 shows 1=1 1 1=1 1 k c a n n o t b e g r e a t e r t h a n 4 . H e n c e k <_ 4 . The s e c o n d r e s u l t i s c o n t a i n e d i n T h e o r e m V \u00E2\u0080\u009E 8 l . T h e o r e m V . 85 L e t H e R|(U) , dim U = 7 . T h e n d i m H < 4 . When d i m H = 4 , H h a s a ( 1 , 1 ) b a s i s , t h r e e o f whose members a r e p a i r w i s e - P g . P r o o f : L e t f f - ^ . . . , f f c } be i n d e p e n d e n t i n H . B y T h e o r e m 1 1 . 2 6 , d i m 2 U(f,) = 5 , 6 , 7 i f k > 2 . The r e s u l t 1=1 . . . f o l l o w s b y T h e o r e m s I I I . 3 8 , I V . 7 1 , a n d V . 8 4 . 9 2 . CHAPTER V I THE M A I N RESULTS I n t h i s c h a p t e r , we show t h a t ' i f H e R 2 ( U ) , d i m U = n a n d n > 6, t h e n d i m H < n - 3 - F u r t h e r m o r e , i f d i m H > 4, i t h a s a ( l , l ) b a s i s . T h e s e r e s u l t s a r e c o n t a i n e d i n T h e o r e m V I . 1 0 0 . L e t H = < f ^ , . . . , f k > \u00E2\u0080\u00A2 I n C h a p t e r s I I I , I V , V , we c o n s i d e r e d t h e c a s e s d i m \u00C2\u00A3 U ( f . ) = 5 , 6, J. I n t h i s c h a p t e r , 1 Ic we w i l l c o n s i d e r t h e c a s e d i m 2 U ( f i ) = 8 , show t h a t k <_ 5 , a n d t h a t ( f 3 , . . . , f k ) i s a ( 1 , 1 ) b a s i s f o r < f 1 , . . . , f k > , k <_ 5 ( T h e o r e m V I . 9 2 ) . We s h a l l t h e n u s e t h e r e s u l t s f r o m t h e p r e c e d i n g c h a p t e r s a n d a b o v e t o o b t a i n t h e m a i n r e s u l t s s t a t e d i n T h e o r e m V I . 1 0 0 . T h e o r e m V I . 8 6 . L e t H e R ^ U ) . L e t { f ^ f g , ^ } be i n d e p e n d e n t i n H s u c h t h a t d i m | U ( f \u00C2\u00B1 ) = 8 , T h e n { f ^ f ^ f ^ i s a ( l , l ) 2 b a s i s o f p a i r w i s e - P g v e c t o r s f o r < f ^ f 2 , f 3 > e R 2 ( U ) . P r o o f ; By T h e o r e m 1 1 , 2 6 , d i m [ U ( f 1 ) + U ( f 2 ) ] = 6 , a n d h e n c e ^ f 1 , f 2 ) i s a P g - p a i r . S i m i l a r l y i t ^ t ^ ) , { f - ^ f ^ a r e P g - p a i r s . H e n c e [ f ^ f ^ f ^ } a r e p a i r w i s e - P g . S i n c e d i m | U ( f 1 ) = 8 a n d ( f 1 , f 2 , f 3 ) a r e p a i r w i s e - P g , t h e n d i m 3 U ( f ) = 2 . T h u s U(f-) => W f f ^ f g ) a n d h e n c e b y C o r o l l a r y 1 1 - 3 4 , ( f 1 , f 2 , f 3 ) f o r m a ( 1 , 1 ) b a s i s f o r < f 1 , f 2 , f 3 > e R 2 ( U ) . 93. T h e o r e m V I . 8 7 . L e t H e R 2(U). L e t [ f ^ f ^ f ) be i n d e p e n d e n t i n H s u c h t h a t d i m i u ( f \u00C2\u00B1 ) = 8. T h e n i U(f\u00C2\u00B1) h a s a b a s i s ( u 1 , u 2 , x 3 , . . . , X Q ] s u c h t h a t ( f - ^ f ^ f } have r e p r e s e n t a t i o n s f l = u l A x 3 + u 2 A x 4 ' f 2 - u i A x 5 + V*6' f 3 = u l A x 7 + u 2 A x 8 -P r o o f : B y T h e o r e m V I .86, [ f - ^ f g , ^ } , i s a ( l , l ) b a s i s o f p a i r w i s e - P g v e c t o r s f o r < f 1 , f 2 , f 3 > . Hence ( f ^ f ^ f ^ h a v e r e p r e s e n t a t i o n s f1 = ^ A x ^ + UgAx^, f 2 = u-^x,. + u g A x g , f 2 = u 1 A x 7 + u 2A x8 ' a n d t u i ' u 2 j X 3 > . . . ,Xg] a r e i n d e p e n d e n t s i n c e d i m i U ( f \u00C2\u00B1 ) = 8. T h e o r e m V I . 8 8 . L e t H e R 2 ( U ) . L e t { f ^ f ^ f ^ be i n d e -p e n d e n t i n H s u c h t h a t d i m | U ( f \u00C2\u00B1 ) = 8 . I f f ^ e C 2 ( U ) , i n d e p e n d e n t o f { f . ^ f 2 , f ^3 s u c h t h a t < f ^ . . . , f ^ > e R 2(U), t h e n { f 1 , . . . , f 1 ^ } i s a ( 1 , 1 ) b a s i s f o r e R 2(U), a n d { f ^ , f 2 , f ^ ) a r e p a i r w i s e - P g . P r o o f ; B y T h e o r e m V I .86, [ f ^ f ^ f ^ ] i s a ( 1 , 1 ) b a s i s o f p a i r w i s e - P g v e c t o r s f o r < f 1 , f 2 , f 3 > . B y T h e o r e m V I . 8 7 , ( f ^ , f 2 , f ^ } h a v e r e p r e s e n t a t i o n s f ^ = u-^Ax^ + UgAx^, f 2 = u.jAx + u g A x g , f 3 = u 1Ax^, + u 2 A X g j I U(f\u00C2\u00B1) = . B y T h e o r e m 11 . 2 6 , 3 > d i m \u00C2\u00A5 ( f ^ , f 1 ) > 2; i = 1 , 2 , 3 -H e n c e d i m J ^ U ^ ) = d i m [ U ( f ^ ) n < u - L , u 2 > ] > 1, f o r i f t h i s were n o t s o a n d t h e i n t e r s e c t i o n were e m p t y , t h e n d i m U(f^) = 6, w h i c h c o n t r a d i c t s t h e f a c t t h a t d i m U(f^) = 4 . L e t c $ TJ(f. ) . s o t h a t u e . B y Lemma I I . 1 8 , e a c h o f 1=1 1 ' X d. ( f 1 , f 2 , f 3 ) c a n be r e p r e s e n t e d i n t h e f o r m f\u00C2\u00B1 = u A v \u00C2\u00B1 + u\u00C2\u00BBAv j ; i = 1 , 2 , 3 ; < u , u ' > = - I t i s e a s y t o s e e ( u 1 , u 2 , v 1 , v j _ , . . . , v 3 , v ^ ) a r e i n d e p e n d e n t . Hence WLG, we c a n t a k e u = u ^ . 94. 4 Suppose dim n U(f \u00C2\u00B1) =1; i . e . , U ( f i f ) n has d i m e n s i o n 1. S i n c e u-L e U ( f ^ ) , t h e n u 2 \u00C2\u00A3 U ( f 4 ) . S i n c e dim U ( f 4 ) =\u00E2\u0080\u00A2 4 , and Uff^) r> < U j>, then dim W(f^,f ) = 2 , i = 1, 2 , 3 , and thus (f , . . . , f }^ a r e p a i r w i s e - P g . T h e r e f o r e , U ( f 4) c < u 1 , u 2 , x 3 , . . . ,xg>; and by Theorem I I . 1 6 , f i j . = u ] _ A v + wAw', c . By C o r o l l a r y 11-32, dim n = 1, dim n = 1, (*) dim n = 1. I f u 2 ^ , (*) i m p l i e s dim = 3 , which i s a c o n t r a d i c t i o n . Hence u 2 e and t h e r e f o r e .$ U ( f ^ ) = . Now dim U ( f ^ ) = 4 and U ( f 4) r> . Hence f ^ forms a P g - p a i r w i t h a t l e a s t one o f ( f ^ f ^ f ^ } , s a y f 1 . Hence f ^ has a r e p r e s e n t a t i o n u^Av^ + u2Av^_ (Theorem I I . 3 1 ) . T h e r e f o r e ( f ^ , . . . , f ^ ) i s a (1 ,1) b a s i s f o r < f 1 , . . . , f 4 > \u00E2\u0080\u00A2 Theorem V I , 8 9 . L e t H e R 2(TJ). L e t ( f 1 , . . . ,f^} be i n d e -4 pendent i n H suc h t h a t dim S TJ(f 1 ) = 8 . Then has a (1 , 1 ) b a s i s ( g - ^ - . - j g ^ ) such t h a t dim | U ( g i ) = 8 . P r o o f : By Theorem 11 .26, dim | U ( f j L ) = 6, 7, 8. By C o r o l l a r i e s I V . 6 l , V - 7 3 , Theorem VI.8 6 , we can assume [ f . , f p , f _ } 3 4 a r e p a i r w i s e - P g . I f dim J U ( f \u00C2\u00B1 ) = 6; dim Jyjff^) = 8 . I f dim i U ( f . ) = 7, then, by Theorem V . 7 3 , t h e r e a r e two c a s e s ; 1 e i t h e r ( i ) ( f - L , f 2 , f 3 } i s a (.1,1) b a s i s o r ( i i ) dim 3 U f f ^ = 1. Case ( i ) By Theorem V . 7 4 , ( f - ^ f ^ , ^ } have r e p r e s e n t a t i o n s : f x = u ] A x 3 + u 2 A x 4 , f 2 = u ] A x 5 + u 2 A x 6 ; f 3 = u 1 A x 7 + u 2Av, 95. 3 2 U ( f 1 ) = < u 1 , u 2 , x 3 > . . . , x y ; v e < u l i , u 2 , x 3 , . . . , x g > , a n d v \u00C2\u00A3 U ( f 1 ) , i = 1, 2., s i n c e { f ^ f ^ f ^ } a r e p a i r w i s e - P g . L e t E U ( f 1 ) = n2,x^,...,XQ> a n d U ( f ^ ) = < x g > . L e t W* = < u p u 2 ) x 3 , . ..,x ?>. T h e n d i m U ( f ^ ) H \u00C2\u00A5* = 3; a n d U ( f ^ ) H < u 1 , u 2 > T 0, f o r i f n o t , b y T h e o r e m 11.26, d i m W ( f 4 , f \u00C2\u00B1 ) > 2, 1 = 1 , 2 , 3, a n d t h i s i m p l i e s d i m U(f l l _) fl \u00C2\u00A5* > 3-S u p p o s e U ( f 4 ) 3 < u 1 , u 2 > . T h e n U ( f 4 ) = < x g , u 1 , u 2 , y > ; y e W * . I f y e [ x ^ j U - ^ u ^ x ^ x ^ x ^ x ^ X g ] , t h e n d i m i = 1 E 2 4 U ( f i ) = 8. O t h e r w i s e , y e \u00C2\u00A5 * * = < u 1 , u 2 , x ^ , . . . > x g > , a n d d i m U ( f ^ ) fl W** = 3. I t f o l l o w s t h e n t h a t f ^ f o r m s a P p . - p a i r w i t h a t m o s t one o f { f ^ , f 2 , f ^ } , s a y f ^ . Hence [ f 2 , f 3 , f ^ } a r e p a i r w i s e - P g a n d f ^ = u^Aw + u 2Aw'; ( w , w ' } e [ x g ^ u ^ U g ] . C l e a r l y f ^ 4= u-^Aw + Y U 2 A X ^ , 0 4= Y e F , f o r o t h e r w i s e ( f ^ - y~xf^) h a s r a n k o n e . S i m i l a r l y f ^ 4 yu^Ax^ + u 2 A w ' , 0 4= Y e P . L e t g = 5 f ^ + f ^ = U ^ A ( 6 X 3 + w) + UgA ( 5 x 4 + w ' ) . F o r some 6 e F, [ u ^ , u 2 , x,_,Xg, v , x 7 , 6 x ^ + w, 6x^ + w ' } i s i n d e p e n d e n t . F o r s u c h a 6 , ( f 2 , f 3 , g ) a r e p a i r w i s e - P g a n d d i m [ X J ( f 2 ) + U ( f 3 ) + U ( g ) ] = 8. S u p p o s e d i m U ( f 4 ) 0 = 1. B y T h e o r e m I I . 1 8 , i t f o l l o w s e a s i l y t h a t we c a n WLG t a k e U ( f ^ ) r> . T h e o r e m 11.26, t h e a b o v e a s u m p t i o n , a n d t h e f a c t t h a t d i m [ U ( f i ) _ ) H W*] = 3 i m p l y U ( f ^ ) 3 < y , y ' > ; y e [ x ^ x ^ j u ^ U g ] ; y ' e [ x ^ ^ x g ; u^Ug] a n d a y + jSy 1 e [ v j u^Ug] f o r some a , 0 e P . F o r some 0 4= Y e P j g = f]_ + Yf* 3 = t ^ A ( x 3 + Y X 7 ) + U g A ^ + y v ) , a n d U ( f 4 ) n U(g) = m a k i n g [ f ^ g j a P ? - p a i r . T h i s c o n t r a d i c t s T h e o r e m I I . 2 6 a n d h e n c e t h i s c a s e does n o t o c c u r . 96. C a s e ( i i ) . B y T h e o r e m V . 7 5 , { f - , f g,^} h a v e r e p r e s e n t a t i o n s : f l = u l A x 3 + u 2 A x V f2 = u i A x 5 + u | A x 6 J f 3 = u l A x 7 + x 4 A x 6 ; i U ( f \u00C2\u00B1 ) = < u 1 , u 2 , x 3 , . . .,x7>. L e t T, U ( f \u00C2\u00B1 ) = KvL^u^Xy . . . ,x g>, U(f^) 3 - L e t W* = . T h e n d i m U(f4)0W*= 3- S i n c e [f . ^ f 2 , f ^} a r e p a i r w i s e - P g , b y t h e p r e c e d i n g f a c t a n d T h e o r e m 11.26, i t f o l l o w s t h a t f^ 'forms a P ^ - p a i r w i t h a t m o s t one o f ( f 1 , f 2 , f 3 ) , s a y f . ^ H e n c e [ f 2 , f 3 , f ^} a r e p a i r w i s e - P g . S u p p o s e U ( f ^ ) < u 1 > . T h e o r e m 11.26 i m p l i e s d i m \u00C2\u00A5 ( f i f , f \u00C2\u00B1 ) > 2, i = 1, 2, 3- T o g e t h e r w i t h t h e f a c t t h a t d i m U ( f 1 | ) 0 W* = 3 , t h i s i m p l i e s U ( f ^ ) o < u , v , w > ; u e [ U 2 J U ] ; v e [ x ^ ; u ^ 3 , w e [ x g j u ^ ] . B u t t h e n [ f ^ f ^ } i s a P g - p a i r , w h i c h c a n n o t be w r i t t e n i n ( l , l ) f o r m , c o n t r a d i c t i n g T h e o r e m 11.31. H e n c e U ( f 4 ) -D < U ^ > , a n d U ( f ^ ) = < u ^ , X g , z , z ' >, c < u 2 , x 3 , \u00E2\u0080\u0094 , x ^ > . Now d i m n U(f.j_) = 1, i = 2, 3 a n d d i m fi U ( f 1 ) >. 1. T h u s , i f < z , z ' > fl < u ] [ , u 2 > ={= 0, t h e n => a n d t h e r e f o r e U ( f ^) o < u 2 > . S i n c e [ f 2 , f ^} i s a P g - p a i r , t h e n f ^ has a r e p r e s e n t a t i o n u^Aw + u 2 A w ' . T h e n ( f 1 , f 2 , f i f } i s a (1,1) b a s i s f o r < f 1 , f 2 , f i f > . As i n t h e p r o o f o f T h e o r e m V . 7 3 , we c a n assume ( f x , f 2 , f ^ 1 a r e p a i r w i s e - P g . We h a v e t h e f i r s t c a s e a g a i n . O t h e r w i s e < z , z ' > fi < u ^ , u 2 > = 0 , a n d WLG, z e [x^x^jU-^Ug] . I f z e [ x i ] _ ; u ] L ] , t h e n as a b o v e , ( f ^ f ^ f ^ ) i s a (1,1) b a s i s f o r < f 1 , f 3 , f J ^ > . We h a v e Case ( i ) a g a i n . O t h e r w i s e , z \u00C2\u00A3 U ( f \u00C2\u00B1 ) , i = 2, 3- T h e n < z , z ' > fi < u 1 , X g > T 0 ' a n d z ' e [ x g j ^ J . A g a i n , as a b o v e , f f ^ f ^ f ^ } i s a (1,1) b a s i s f o r a n d we h a v e Case ( i ) a g a i n . F i n a l l y , s i n c e we have shown t h a t < f 1 , . . . , f J + > h a s a b a s i s w h i c h we s h a l l s i m p l y c a l l (g1,...,g>4) s u c h t h a t 9 7 . 3 dim 2 U(g 1) = 8 , i t follows from Theorem VI.88 that {g \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 . ,g^} i s a (1,1) basis. Theorem V I . 90. Let H e \u00C2\u00A3 ( U ) . Let [f, , . . . , f j be inde-d 1 D pendent i n H such that dim t U(f ) = 8 . Then < f f > 1 i 5 has a (1,1) basis {g : 1=1,... , 5 } such that dim 2 U(g ) = 8 . \u00E2\u0080\u00A2 1 1 Proof: B y Theorems I I . 2 6 , IV.71, 4 dim E U(f, ) = 7 , 8 . 1 1 B y Theorems I V . 8 l , VI89, has a ( l , l ) basis, three of whose members are pairwise-Pg. We can assume that (f ^ , . . . , f 4 } are i n fact these basis members, and that [f ,f ,f_) are 4 5 pairwise-Pg. If dim t U(f^) = 8 , we can assume by Theorem V I . 8 9 that dim E U(f.) = 8 . The result follows by Theorem V I . 8 8 . 1 3-4 If dim E U(f 1) = 7 , we can assume that by Theorem V . 8 2 , that dim \u00C2\u00B1 = 1 \ ^{?\u00C2\u00B1) = 7- Then d i m i = 1 ^ _U(f 1) = 8 and the result follows from Theorems V I . 8 9 , 8 8 . Theorem VI. 91. Let H e R ^ U ) \u00E2\u0080\u00A2 Let {t^...,? } be inde-pendent i n H such that dim | U(f \u00C2\u00B1) = 8 . If fg e c|(U), independent of [ f 1 , . . . , f }, such that dim | U(f 1) = 8 , then k Re-proof: B y Theorem V I . 9 0 , we can assume {f ^ . . . ,f,_} i s a (1,1) basis for such that dim | U(f j L) = 8 . B y Theorem V I . 8 8 , {f 1,...,fg} i s a (1,1) basis for . B y Theorem III.40, \u00C2\u00A3 R 2(U). 98. Theorem \u00C2\u00A51.92. Let H e Rg(TJ) with dim H > 5. Let if-[_,\u00E2\u0080\u00A2\u00E2\u0080\u00A2 \u00E2\u0080\u00A2, f k) b \u00C2\u00A9 independent i n H such that d i m ^ U ^ ) = 8 . Then k < 5, and has a (1,1) basis {g 1 : 1 < i < k) such that dim j Ll 1U(g 1) = 8 . Proof: By Theorem 11.26, k-1 dim \u00C2\u00A3 U(f,) = 6, 7, 8 . 1=1 1 k-1 I f d i m ^ U ( f 1 ) = 6, then by Theorem I V . 7 1 , k - 1 <_ 3, and thus k <_ K. I f dim i| 1lT(f ) = 7, then by Theorem V.84-, k - 1 <_ 4 , and k < 5 . k-1 , k I f d i m 1 E 1 U ( f 1 ) = 8 and d i m 1 | 1 U ( f i ) = 8 , then Theorem VI.91 shows k <_ 5 . Hence k <_ 5. By Theorem 11.26, k > 3- The r e s u l t f o l l o w s by Theorems VI.8 8 , 8 9 , 90. Lemma VI.9 3 . Let H e R | ( U ) . Let [ f ^ f ^ f ^ } be inde-pendent i n H such that dim 1 U ( f \u00C2\u00B1 ) = 7 - I f f ^ e C 2(U), independent of [ f ^ f ^ f . ^ ] such that ( i ) e Rg(U) and ( i i ) dim 2 IT(f i) = 9 , then has a ( l , l ) basis {g \u00C2\u00B1,.\u00E2\u0080\u00A2\u00E2\u0080\u00A2,g 4) such that dim ^ U(g 1) = 8 . Proof: By C o r o l l a r y V . 7 3 , we can assume [ f - ^ f ^ f ^ } are pairwise-Pg. Then d i m 1 = 1 E 2 2^U(f 1) = 8 . The r e s u l t follows from Theorem VI.8 8 . Lemma VI.94. Let H e R | ( U ) . Let { f 1 , . . . , f i f ) be independent 9 9 . 4 0 i n H s u c h t h a t d i m J U ( f \u00C2\u00B1 ) = 7- I f f\"5 \u00E2\u0082\u00AC C 2 ( U ) , i n d e p e n -d e n t o f { f 1 , . . . , f 4 ) s u c h t h a t ( i ) < ? ^ . . . , f > e R 2 ( U ) a n d ( i i ) d i n | U ( f \u00C2\u00B1 ) = 9 , t h e n < f 1 , . . . , f > h a s a (1 ,1) b a s i s i&i,-'-such t h a t d i m ^ U ( g 1 ) = 8 . P r o o f : B y T h e o r e m V . 8 l , we c a n assume f f 1 , \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 . , f ^} i s a (1 , 1 ) b a s i s f o r s u c h t h a t f f ^ f ^ f } a r e p a i r w i s e - P ^ - Hence d i m E U ( f . ) = 8 . The r e s u l t f o l l o w s 6 1=1,2,5 i y b y T h e o r e m V I . 8 8 . T h e o r e m V I . 95- L e t H e R ^ U ) . L e t {f ]_, . \u00E2\u0080\u00A2 . , f k ) be i n d e -k . . _ p e n d e n t i n H s u c h t h a t d i m E ^ U ^ f ^ ) = n , n _> 6 . T h e n k <_ n -F o r n = 7, k = 4 , < f x , . . . , f ^ > has a ( l , l ) b a s i s . F o r n > 8 , < f 1 , . . . , f k > h a s a (1 ,1) b a s i s . P r o o f : F o r n = 6 , 7, 8 , t h e r e s u l t i s c o n t a i n e d i n T h e o r e m s V . 7 1 , V . 8 4 , V I . 9 2 \u00E2\u0080\u00A2 F o r n > 9 , d i m \u00C2\u00A3 U ( f , ) _> 7, 8 b y T h e o r e m 1 1 . 2 6 . 1 The r e s u l t f o l l o w s b y i n d u c t i o n f r o m T h e o r e m s V I . 9 3 , 9 4 , 9 2 , 8 8 , a n d I I I . 4 0 . The f o l l o w i n g i s a n e x a m p l e of a (1 ,1) b a s i s f o r H e R 2 ( U ) , d i m H = n - 3 , d i m U = n . E x a m p l e V I . 9 6 . L e t [ x ^ . . . , x n ) be a b a s i s f o r U . f 1 = x x A x^ + x 2 A x ^ f 2 = X ]_ A x 4 + x 2 A x 5 f 3 = X l A x 5 + x 2 A X g 100. f ii \u00E2\u0080\u0094 X-, A x _ -j- x ^ A x - i n-4 1 n-2 2 n-1 f ~ = x, A x , + x\u00E2\u0080\u009E A x\u00E2\u0080\u009E n-3 1 n-1 2 n n-3 For a\u00C2\u00B1 e F; z = 2 a \u00C2\u00B1 f \u00C2\u00B1 = x . [1 A < a l x 3 + a 2 x 4 + \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 + a n - 3 X n - l } + x \u00C2\u00A3 A ( a 1 x I | + a 2x^ + ... + a n._3 x n) > and R(z) = 2 . I t i s p o s s i b l e f o r H e R 2(lT) to have a ( l , l ) basis of pairwise-P.- vectors when dim H = n-3; dim U = n > 7. We s h a l l show t h i s i n the f o l l o w i n g theorems and example. Theorem VI.97- Let H e R|,(U). Let [ f , . . . , f k ) be inde-pendent i n H such that ( i ) U ( f \u00C2\u00B1 ) 3 U Q , dim U o = 3 , 1 < 1 < k ; ( i i ) dim 2 U ( f . ) = m, 6 < i < dim U . Then k < m-3 and ( f . 1 , . . . , f 1 1 i s a ( 1 , 1 ) basis fo pairwise P^ vectors f o r . Proof: Since ( i ) holds, [f-^, \u00E2\u0080\u00A2.' ,^ k) are pairwise -P,.. By ( i ) and ( i i ) , k > 3 j and f o r k = 3 , dim i U ( f \u00C2\u00B1 ) = 6. By Theorem I V . 4 9 , [ f 1 , \u00E2\u0080\u00A2 . . , f k ) have representations: f 1 = x l A u l + W l A V l -f 2 = X 2 A U 2 + Wg A V g f ^ = X ^ A ( ^ + ^ 3 U 2 ) + W3 A V ^ 101. f k = X k A K + ^ k U 2 ' + W k A V k > w i t h \u00C2\u00A33* \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 n o n - z e r o a n d d i s t i n c t o n F , u\"o = < u 1 , u 2 , u ^ > ; < w 1 , v i > c U Q, 1 < i < k , v\u00C2\u00B1 e < u 1 , u 2 > , 1 < i < k , I U ( f i ) = <\i^,u2,u^,x^} . . . ,x^>. As i n t h e p r o o f o f c o r o l l a r y IV.50, e a c h f\u00C2\u00B1 c a n be e x p r e s s e d i n t h e f o r m vAu 1 + v'Au 2, 1 < i < k , a n d h e n c e f f ^ . . . , f k ) i s a ( 1 , 1 ) b a s i s f o r < f 1 , . . . , f k > . , B y T h e o r e m I I I . 4 0 , k <_ m~3 \u00E2\u0080\u00A2 T h e o r e m VI.98. L e t H e R|(U); d i m U = n>7- L e t { f , . . . , f } be i n d e p e n d e n t i n H s u c h t h a t U ( f ^) 3 U Q , d i m u 0 = 3, 1 < i <_ k . T h e n k <_ n-3 \u00E2\u0080\u00A2 F o r k = n-3, {f \u00C2\u00B1 i . . . , f n _ 3 3 . i s a (1,1) b a s i s f o r < f f ,> . k ^ ' P r o o f : d i m I u ( f . ) <_ n , a n d h e n c e b y T h e o r e m V I . 95, k < n-3 1 1 k : ) \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 When k = n-3 , d i m \u00C2\u00A3 U ^ T J) = n , 7 < n = d i m U . The r e s u l t . . \u00E2\u0080\u00A2 . 1 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 f o l l o w s b y T h e o r e m V I .97. The f o l l o w i n g i s an e x a m p l e o f a ( l , l ) b a s i s o f 2 p a i r w i s e - P j - v e c t o r s f o r H e R 2 ( U ) , d i m H = n-3, d i m U = n ; H = < f ^ , . . \u00E2\u0080\u00A2 , f n _ 3 > \u00E2\u0080\u00A2 E x a m p l e V I . 9 9 - U = < u 1 , u 2 , u 3 , x 1 , . . . , x n _ 3 > i U = < u , , u 0 , u Q > . 0 1 d' 3 f i = x i A ^ U l + ^ i ^ + u 3 A u i j 1 1 1 -^ n\"3-6, ,...,6 \u00E2\u0080\u009E n o n - z e r o a n d d i s t i n c t i n F . ^1' \" n - 3 F o r a 1 , . . . , a n _ 3 e F, n o t a l l z e r o ; . z = - S a f \u00C2\u00B1 , z = u,A ( \u00C2\u00A3 a , x , + \u00C2\u00A3 a , u ) 1 1=1 x x 1=1 ^ + U 2 A (\u00C2\u00A3g a ^ ) . 102. Now R ( z ) < 2 i m p l i e s n - 3 n - 3 n - 3 2 a 0 x \[ T, a x , + E a,u_ 1 1=1 1 1 1 1=1 1 1 i=l 1 3 f o r some X 4= 0 i n P. Since j3. 4 0. ! < _ ! < . n - 3 , \ 4 0, n - 3 1 t h e n a \u00C2\u00B1 = 0 a n d a ^ = Xa\u00C2\u00B1; 1 <_ 1 < n - 3 . I f a \u00C2\u00B1 4 0 f o r two o r more values of i , then ^6\u00C2\u00B1\" = X for two or more 1, c o n t r a d i c t i n g the d i s t i n c t n e s s o f . But a\u00C2\u00B1 4 0 f o r n - 3 e x a c t l y one i implies 0 , w h i c h i s a c o n t r a d i c t i o n . H e n c e E ( z ) = 2. T h e o r e m VI.100. L e t d i m U = n > 6. I f H e R ^ ( U ) , t h e n d i m H < n - 3 - I f d i m H > 4, H h a s a (1,1) b a s i s . P r o o f : L e t {f^, . ..,f^} be i n d e p e n d e n t i n H . T h e n k . , d i m ^ E ^ U ( f 1 ) <_ n . The r e s u l t follows b y c o r o l l a r y 11,21, a n d T h e o r e m s I I I . 3 8 , TI.9 5 . I f a r a n k two s u b s p a c e H of d i m e n s i o n k has a ( l , l ) b a s i s ( f ^, \u00E2\u0080\u00A2 . . , f k ) , t h e n (f ^ , .. . ,f^.} h a v e r e p r e s e n t a -t i o n s ? - f 1 = xAx i + yAy^, 1 \u00C2\u00A3 i < k , where i s a f i x e d 2 - d i m e n s i o n a l s u b s p a c e i n U . We n o t e t h a t a n y l i n e a r c o m b i n a t i o n z of a n y 2 b a s i s members f i , f j i s z = af i + /3f j | a,j3 e F , 1 <_ I, J <_ k. Thus z = x A ( a x i + 0Xj) + yA(ay. + )3y . ) . Hence a n y n o n - z e r o vector f i n H has a 1 J r e p r e s e n t a t i o n x A x f + yAy\u00C2\u00B1,; x f , y f e U . We c a n t h u s r e s t a t e t h e m a i n r e s u l t s ( i . e . , T h e o r e m VI.100) u s i n g t h e f o l l o w i n g d e f i n i t i o n of H . D e f i n i t i o n VI. 1 0 1 . A r a n k two s u b s p a c e H i s a f l , 1 ) - t y p e s u b s p a c e i f a n y n o n - z e r o v e c t o r f e H has a r e p r e s e n t a t i o n 103, f = x A x f + y A y f i n Cg(Cr), where i s a f i x e d 2 - d i m e n s i o n a l subspace of U . T h e o r e m . V I . 1 0 2 . L e t dim U = n > 6. I f H e R|('C), t h e n d im H < n-3- I f dim H > 4, then E i s a ( 1 , 1 ) - t y p e s u b -s p a c e . F o r the convenience of the r e a d e r , we s h a l l now g i v e a summary o f the r e s u l t s o b t a i n e d i n t h i s p a p e r . A r a n k two v e c t o r f has a r e p r e s e n t a t i o n f = x^AXg + XgAx^; where x,, . .. are independent i n IT. The 4 - d i m e n s i o n a l subspace i s w e l l - d e f i n e d i n the s e n s e t h a t i f y 1 A y 2 + 7^AY^ l s any o t h e r r e p r e s e n t a t i o n o f f , Yj, \u00E2\u0080\u00A2 \u00E2\u0080\u00A2> \u00E2\u0080\u00A2 e I T , t h e n = . We r e f e r t o t h i s 4 - d i m e n s i o n a l subspace as U ( f ) . Thus f o r the r a n k two subspaces t o be n o n - t r i v i a l , dim U >. 4. When dim U = 4, the r a n k two subspaces are of d i m e n s i o n one . Thus the i n t e r e s t i n g r a n k 2 subspaces are those when dim. U >. 5 \u00E2\u0080\u00A2 The main r e s u l t s are as f o l l o w s . L e t dim U \u00C2\u00BB n>6. Then the r a n k two subspaces have d i m e n s i o n a t most ( n-3), and t h i s maximum d i m e n s i o n i s a t t a i n e d . When a r a n k two subspace H has d i m e n s i o n h i g h e r than o r e q u a l t o 4, then e v e r y n o n - z e r o v e c t o r f e H has the form f = xAx f + y A y f , where i s a f i x e d 2 - d i m e n s i o n a l s u b e p a s e of II, and x f . , y f e U . We c a l l s u c h a subspace a (1,1)-type subspace . An example of a b a s i s f o r such a subspace i s the' f o l l o w i n g ; -104'. Let dim H = k; dim TJ = n; U = . H = ; k < 3-f i = X l A x i + 2 + X 2 A x i + 3 ; 1 ^ 1 < k \" Other r e s u l t s obtained characterize the rank two subspaces of dimension two and three. The rank two subspaces of dimension two e x i s t when dim U >_ 5. Every 2-dimensional rank two subspace i s a ( l , 1 ) -type subspace. The 3-dimensional rank two subspaces are the most v a r i e d . F i r s t , the rank two subspaces [ H ] when dim U = 5 are anomalous i n the sense that, whereas when dim U > 5, dim H < n-3j we have that when dim U = 5, there do e x i s t rank two subspaces of dimension 3\u00E2\u0080\u00A2 However, we have not found i t convenient to character-i z e the rank two suspaces by considering the underlying space IT. We have treated then i n the f o l l o w i n g way instead. Let [ f ^ , f 2 , f ^ } he a basis of the rank 2 subspace H. We considered the vector space U'f^)] . We showed that dim U ( f \u00C2\u00B1 ) ] i s e i t h e r 5, 6, 7 or 8. 3 Case 1. dim E U ( f 1 ) = 5* 1=1 Then H has a basis ( g 1 , g 2 , g 3 ) which i s one of the f o l l o w i n g two types:-105, 3 x^ A \u00E2\u0080\u00A2u1 + u 2 A l2 = x,_ A U2 + U.? A S3 = U A y + u 3 A y< \u00E2\u0080\u009E 3 1(b) [2 U ' f ^ ) } = < U , j , U p , U - , X i l , X K > | VJ 6 : ,S> c. ^ J_ ^ U * \u00E2\u0082\u00AC \"\u00E2\u0080\u00A2\u00E2\u0080\u00A2^2^3\" \u00E2\u0080\u00A2 g x = X^ A U., + U 2 A u g 2 = X,- A U 2 + U 1 A U 3 g 3 s u A u' 4- y A y ' 3 Qui a_ j . dim S ,U{f 1) = 6, fhen E has a basis (g_^g^g,} whieh i s ont of th\u00C2\u00A9 ' \"\"'4 * 3 f o l l o w i n g three types?\u00C2\u00BB 2{a,) (g^gg^gg} i s a ( l / I ) basis and go H i s a * type gribBpaoe. 3 2(b) t E t'(f t)3 \u00C2\u00AB , u g , x - , . . . , x g > . g g . u 1 A + U g A x g g 3 . X 3 A y + X 4 A y t j \u00C2\u00AB < X g , X g > , 3 2(c) [ 2 m I i ' ( f 1 ) j - < u 1 , ' a 2 , x 3 , \u00E2\u0080\u00A2. . , x g > . 1=1 106. 3(a) g 1 = A x 3 + u 2 A x^ gg = U-t A Xp. + Ug A Xg 63 = A y + x^ A xg j y e . 3 dim [ 2 T . \ > . )1 = 7. i = l A K has a basia (g^,g2,g^J which Is one of two types {&\u00E2\u0080\u00A2\u00E2\u0080\u00A2 pi2-\"g3^ i s s (~?~) M i l s ind so H i\u00C2\u00A7 s (\u00C2\u00A3,1)-3 \u00C2\u00AB1 s t32 A x 3 + U g A x^ ss A X^ + X| A Xg. If dim E tr{f ) - 8 P th#n H is a (1,1)-fcyp\u00C2\u00AB i - l 1 subspace. 1 0 7 . BIBLIOGRAPHY [1] B o u r b a k l , N., E l e m e n t s de Mathematique, A l g e b r e , Chap. I l l A l g e b r e M u l t l l l n e a r e , 1948. [2] Hodge, W.V.D. and Pedoe, D., Methods o f A l g e b r a i c Geometry, V o l . I* Cambridge 1947. [3] B e r t i n i , E . , E i n f u e h r u n g i n d i e P r o j e c t i v e G e ometrie M e h r - D i m e n s i o n a l e r Raume, V i e n n a , 1924. [4] Van Der Waerden, B. L., Modern A l g e b r a , V o l . . I I . [5] Halmos, P. R., F i n i t e D i m e n s i o n a l V e c t o r Spaces, 2nd E d i t i o n , 1958. "@en . "Thesis/Dissertation"@en . "10.14288/1.0080623"@en . "eng"@en . "Mathematics"@en . "Vancouver : University of British Columbia Library"@en . "University of British Columbia"@en . "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en . "Graduate"@en . "Characterization of subspaces of rank two grassmann vectors of order two"@en . "Text"@en . "http://hdl.handle.net/2429/41206"@en .