ממן15

ממן15 - 15 " 1...

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Unformatted text preview: 15 " 1 ,VI.19 , dim Im T < dim ker T , dim Im T < 2 - , . dim Im T + dim ker T = dim R 4 = 4 dim Im T = 0 : dim Im T v 0 - . dim Im T + dim ker T > 4 . Tv = 0 , v v R 4 , Im T = {0} , t , T (1,1,1,1) v 0 - , [ T (1,1,1,1) ] B = (1,1,1,1) , [ T ] B .1 ." , dim Im T = [ T ] B . dim Im T = 1 - , {(1,1,1,1)t } . [ T ] B = dim Im T = 1 - , [ T ] B . [ T ] B . (2, a1 , a2 , a3 )t , [ T ] B - . {(1,1,1,1)t } , , . a1 = a2 = a3 = 2 , (2, a1 , a2 , a3 )t = 2 (1,1,1,1)t , , c1 = c2 = c3 = 4 b1 = b2 = b3 = 3 1 1 = 1 1 2 2 2 2 3 3 3 3 4 4 4 4 T ( x1 , x2 , x3 , x4 ) . B = (v1 , v2 , v3 , v4 ) : B ( x1 , x2 , x3 , x4 ) : ( x1 , x2 , x3 , x4 ) v R 4 ( x1 , x2 , x3 , x4 ) = x4 v1 + ( x3 - x4 )v2 + ( x2 - x3 )v3 + ( x1 - x2 )v4 [T]B , T ( x1 , x2 , x3 , x4 ) = Tx4 v1 + T ( x3 - x4 )v2 + T ( x2 - x3 )v3 + T ( x1 - x2 )v4 = = x4Tv1 + ( x3 - x4 )Tv2 + ( x2 - x3 )Tv3 + ( x1 - x2 )Tv4 = = x4 (1,1,1,1) + ( x3 - x4 )(2, 2, 2, 2) + ( x2 - x3 )(3,3,3,3) + ( x1 - x2 )(4, 4, 4, 4) = = (4 x1 - x2 - x3 - x4 , 4 x1 - x2 - x3 - x4 , 4 x1 - x2 - x3 - x4 , 4 x1 - x2 - x3 - x4 ) t , . [ T (1,1,1,1) ] B = (1,1,1,1) - , dim Im T = 1 ,' .2 , . T (1,1,1,1) = (1,1,1,1) + (1,1,1, 0) + (1,1, 0, 0) + (1, 0, 0, 0) = (4,3, 2,1) . ImT - {(4,3, 2,1)} , Tv = 0 , v = ( x1 , x2 , x3 , x4 ) : , x1 = ker T . ker T - x2 + x3 + x4 , . 4 x1 - x2 - x3 - x4 = 0 4 x x x x + x3 + x4 v = 2 , x2 , x3 , x4 2 , x2 , 0, 0 3 , 0, x3 , 0 4 , 0, 0, x4 = + + = 4 4 4 4 = x2 ( 1 4 ,1, 0, 0) + x3 ( 1 4 , 0,1, 0) + x4 ( 1 4 , 0, 0,1) . ker T {( 1 4 ,1, 0, 0), ( 1 4 , 0,1, 0), ( 1 4 , 0, 0,1)} , ' ,) ker T - , .( dim ker T = 3 - 2 - { p ( x ), q( x)} - . q ( x) = x 3 + x 2 - x + 3 - p ( x) = x 3 - 2 x + 1 : : . 1 ( x 3 - 2 x + 1) + 2 ( x 3 + x 2 - x + 3) = 0 , 1 p( x) + 2 q( x) = 0 .1 1 + 2 = 0 2 = 0 , . : -21 - 2 = 0 1 + 3 2 = 0 . - { p ( x), q ( x)} . B = { p ( x), q ( x), x 3 , x} : R 4 [ x] { p ( x), q ( x)} , - , 4 , 1 , 2 , 3 , 4 1 + 2 + 3 = 0 2 = 0 . , -21 - 2 + 4 = 0 1 + 3 2 = 0 R : T : R 4 [ x] v M 2v 3 - 1 1 2 1 0 0 3 , , T ( p ( x)) = T (q ( x)) = 02v 3 , T ( x ) = T ( x) = 0 0 3 0 -1 0 . B T , R 4 [ x] - , VI.12 T . : q ( x) - p ( x) - KerT - .ii . T (u ( x)) = 02v 3 , . u ( x) v Ker T - , u ( x) = 1 p ( x) + 2 q ( x) + 3 x3 + 4 x , u ( x) v R 4 [ x] T . T (u ( x)) = T (1 p ( x) + 2 q ( x) + 3 x 3 + 4 x ) = 02 , 1T ( p ( x )) + 2T (q ( x)) + 3T ( x 3 ) + 4T ( x) = 02 3 3 .i , - 1 1 2 1 0 0 + = 1 02 3 + 2 02 3 + 3 4 0 0 02 3 0 -1 0 3 - 3 + 4 , 3 = 4 = 0 . 33 23 -3 3 =0 0 23 , Ker T u ( x) . u ( x) = 1 p( x) + 2 q ( x) . Ker T , q ( x) - p ( x) 1 2 1 - : W = Sp Im T - .iii -1 0 3 - 1 2 1 , A = T ( x3 ) , A = - , A v W 3 -1 0 , . A v Im T , R 4 [ x] - A . A = T ( x 3 ) . W v Im T A v S Rv 2 . 2 d 2 S Rd 2 2 2 a b . a, b, c v R b c 1 0 1 0 0 0 R , , - : S 2d2 - B = 0 0 1 1 0 0 R R , dim R 3 [ x] = 3 . dim S 2d 2 = 3 , . S 2d2 R R R - S 2d 2 - , S 2v 2 R 3 [ x] , dim S 2v 2 = dim R 3 [ x] .2 . R 3 [ x] a b R T . T ( A) = ax 2 + bx + c , A = : T : S 2v 2 v R 3 [ x ] b c , 1 , 2 a R - , A = 1 b1 b1 a 2 , B = c1 b 2 b2 R S , c2 2v 2 : a T (1 A + 2 B ) = T 1 1 b1 a + 2 a2 =T 1 1 1b1 + 2 b2 b1 2 a + 2 c1 2 b 1 a1 b2 = T b c2 11 1b1 2 a2 + b 1c1 2 2 2 b2 = 2 c2 1b1 + 2 b2 2 = (1 a1 + 2 a2 ) x + (1b1 + 2 b2 ) x + 1c1 + 2 c2 = 1c1 + 2 c2 = (1 a1 x 2 + 1b1 x + 1c1 ) + (2 a2 x 2 + 2 b2 x + 2 c2 ) = = 1 (a1 x 2 + b1 x + c1 ) + 2 ( a2 x 2 + b2 x + c2 ) = 1T ( A) + 2T ( B ) a b , ax 2 + bx + c = 0 , T ( A) = 0 , A = , Ker T = {0} b c . A = 0 , a = b = c = 0 . T ,VI.20 3 (i = 1, 2,K , n - 1) i - n v n A = [ T ] B n - ; , i + 1 - 1 : 0 1 = 0 M 0 0 0 1 M 0 L L O M L 0 0 0 M 1 0 0 0 0 0 .2 .1 A = [T]B , (k = 1, 2,K , n) k : k k T u j = u j + k , i = n - k + 1, n - k + 2,K , n T ui = 0 .( k < n ) j = 1, 2,K , n - k Tu j = u j +1 , Tun = 0 , , k = 1 . j = 1, 2,K , n - 1 (i = n - p + 1,K , n) T p ui = 0 , , k = p p . ( j = 1, 2,K , n - p ) T u j = u j + p . T , T p +1 un - p = T (T p un - p ) = T (un - p + p ) = T (un ) = 0 , p +1 p . (i = n - p + 1,K , n) T ui = T (T ui ) = T (0) = 0 : T p +1 p . ( j = 1, 2,K , n - p - 1) T u j = T (T u j ) = T (u j + p ) = u j + p +1 : T . , k = p + 1 ," n , (i = 1, 2,K , n) T ui = 0 - , k = n , , V T n , .T n = 0 n -1 , (i = 2,3,K , n) T ui = 0 - , , , k = n - 1 , B - . T . T n -1 0 , T n -1 n -1 u1 = u1+ n -1 = un u1 0 , un = 0 - n S v1 = 0 - , v1 v 0 , v1 v V , . S n -1 0 , S n = 0 0 .3 , . S , . S n-2 n -1 v1 = vn , vn v 0 , vn v V . S n -1v1 0 , vn -1 v1 = vn -1 - , vn -1 V . Svn = S ( S n -1v1 ) = S n v1 = 0 i -1 n-2 n -1 , (i = n - 1,K , 2) S v1 = vi , . Svn -1 = S ( S v1 ) = S v1 = vn i -1 i . Svi = S ( S v1 ) = S v1 = vi +1 . Svn = 0 - , Svi = vi +1 , i = 1, 2,K , n - 1 , i . B v = (v1 , v2 ,K , vn ) , . [ S ] Bv = A = [ T ] B , F TM : V v V , V = F n M , N v M nv n n n -1 ,' , . TM = 0, TM .4 0 , . TM v = Mv .' A , [ TM ] C = A - V C = (v1 , v2 ,K , vn ) . (i = 1, 2,K , n - 1) TM vi = vi +1 - , TM vn = 0 , - , V B = (u1 , u2 ,K , un ) . F n V . M = [ T ] B - V T : V v V ,' . T n = 0, T n -1 0 , M n = 0, M n -1 0 . C T ' A V C = (v1 , v2 ,K , vn ) . Tvn = 0 , Tvi = vi +1 , i = 1, 2,K , n - 1 , i , vi ) wi = N i v1 - , D = ( w1 , w2 ,K , wn ) , V . i = 1, 2,K , n ( , V D = ( w1 , w2 ,K , wn ) , , N n = 0 - ,( N n -1 n -1 0 - ) N w1 0 : n -1 n -1 n . wn = N w1 . N ( N w1 ) = 0 , , N w1 = 0 - . w2 = Nw1 ,' 4 . T (T ( w)) = v , T 2 ( w) = v - w v V . v v Im T 2 . Im T 2 v Im T , v v Im T , T (u ) = v - , T ( w) = u T , T 2 ( x) = T (T ( x )) = T (0) = 0 . T ( x) = 0 , x v ker T . ker T v ker T 2 , x v ker T 2 , . , dim Im T + dim ker T = n ,VI.19 . dimV = n , .( V T 2 ) dim Im T 2 + dim ker T 2 = n . dim Im T 2 + dim ker T 2 = dim Im T + dim ker T dim Im T 2 = dim Im T , . Im T 2 = Im T - (i) . ker T v ker T 2 , . dim ker T 2 = dim ker T , - ,- . ker T = ker T 2 , .- .2 .1 , ker T = ker T 2 - , (ii) . dim Im T 2 = dim Im T , dim ker T 2 = dim ker T . Im T 2 = Im T , Im T 2 v Im T , ,- - u v V . v Im T ker T . (ii)- (i)- , T 2 (u ) = 0 , T (T (u )) = 0 : . T (v ) = 0 , v = T (u ) . v = 0 , T (u ) = 0 , u v ker T , ker T = ker T 2 . u v ker T 2 . Im T ker T = {0} , ,V.31 , , dim(Im T ker T ) = dim{0} = 0 ,VI.19 . dim(Im T + ker T ) = dim Im T + dim ker T - . dim V = dim(Im T + ker T ) , dim V = dim Im T + dim ker T . V = Im T ker T - , V = Im T + ker T , Im T + ker T .3 V 5 :-- T - . T - . dimV = n ) n , , ,-- T -- T , . , ,( V , n > 1 . v v 0 - v v ker T , , ker T v {0} , . , - {v} - ) V () {v} [ T ] B . B = (v, v1 , v2 ,K , vn -1 ) - ,( v v 0 [ T ] B . T (v ) = 0 , v v ker T . B T , [ 0] B = 0 . B , 0 , , T (v) . [ T ] B F , Ax = 0 . A v M nv n .1 .2 . {0} - A , n ker TA ,VI.10 . TA (v ) = Av , TA v Hom F -- TA ,VI.14 . ker TA v {0} , A F n B ' , , Av- A . Av = [ TA ] B A , , ,VI.40 , , . ...
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