ממח03

ממח03 - :1 . T ( f ( x )) = xf ( x...

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Unformatted text preview: :1 . T ( f ( x )) = xf ( x ) = a2 x 3 + a1 x 2 + a0 x . f ( x) = a2 x 2 + a1 x + a0 . z xw == w z z w .1 .2 :2 1 2 1 0 2 1 2 3 = -8 . 2 3 0 = 2 3 0 = 2 -1 3 1 1 2 -1 0 . 0 .1 -4(1, -1, 0) - 5(-1,1,1) = (1, -1, -5) -4(1, 0) - 5(0,1) = (-4, - (2,1) 5) .2 :3 n n ImT = V , . w x Im T w = i T (vi ) = T i vi . w x V i =1 i =1 .1 . ." , T - n n = , T i vi T (0) = 0 . i vi = 0 - ," T ( A) - . i i =1 =1 .2 . ." A , 1 = K n = 0 , n i =1 i T (vi ) = 0 :4 S ( x, y ) = (0, y ) T ( x, y ) = ( y, 0) . ST ( x, y ) = S ( y, 0) = (0, 0) TS ( x, y ) = T (0, y ) = ( y, 0) S ( x, y ) = ( x, y ) ker S = {0} Im S = V . T ( x, y ) = (2 x, 2 y ) ker T = {0} Im T = V T S .1 .2 :5 ker S = {0} Im S = V dim Im S = n A = [ S] , B = [T ] ( A) = n . ( B ) = k k n ( BA) = min{n, k } = k = ( B ) dim Im TS = dim Im T S ( x, y ) = (0, x ) ker S = {(0, y )} . T ( x, y ) = ( x, x) TS ( x, y ) = T (0, x) = (0, 0) ker TS = V .1 .2 :6 T ( f ( x )) = f ( x + 1) = 0 a0 + a1 ( x + 1) + a2 ( x + 1) 2 + a3 ( x + 1)3 = 0 . (a0 + a1 + a2 + a3 ) + ( a1 + 2a2 + 3a3 ) x + ( a2 + 3a3 ) x 2 + (a3 ) x3 = 0 a3 = a2 = a2 = a1 = 0 f ( x) = 0 ker T = {0} T (1) = 1 T ( x) = 1 + x T ( x2 ) = x2 + 2x + 1 T ( x 3 ) = x 3 + 3 x 2 + 3x + 1 :7 a 11 a 21 a12 11 a + a22 12 a a21 =0 a22 .1 . .2 . 2a11 +a a 21 12 a12 = - a21 a12 + a21 =0 2a22 a11 = a22 = 0 0 1 ker T = Sp 1 0 - .1 2a11 . a 21 + a12 a12 + a21 1 0 0 0 1 0 = + + 2a11 0 (a12 + a21 ) 0 2a22 1 2a22 0 1 0 .2 :8 1 1 1 2 0 0 0 0 = + = = . T 2 0 0 0 0 0 0 0 0 1 0 0 0 .1 .2 :9 1 2 1 0 0 0 1 1 1 1 0 0 = = - + - . T 2 0 1 0 0 1 0 0 0 0 0 1 1 . . . A-1 ( AB) A = ( A-1 A)( BA) = BA .1 .2 :10 , B - A . . 2 1 11 10 = 2 0 0 1 1 2 1 2 2 13 1 2 = . 3 1 1 2 2 0 11 1 2 4 4 2 1 3 0 = 5 5 .1 4 7 3 2 = 1 1 3 3 5 2 1 2 .2 :11 S ( x, y ) = (2 x, 0) S (1, 0) = 2 (1, 0) T ( x, y ) = (0,3 y ) T (0,1) = 3 (0,1) . (T + S )( x + y ) = (2 x,3 y ) (2 x,3 y ) = 5 x,3 y ) ( x, y ) = (0, 0) (2 .1 Sv = 1v . Tv = 2 v ( S + T )v = (1 + 2 )v .2 :12 . . , 1 1 A= B=I 0 1 . . I ( A) = ( I ) = 2 tr ( A) = tr ( I ) = 2 tI - A = tI - I = (t - 1) 2 :13 . 2 0 = -4 0 -2 -1 = 1 0 0 -1 .1 .1 .2 =1 0 -1 0 0 0 . 0 0 -1 0 0 0 0 0 0 = 1 dim V1 = 2 0 0 -1 = 2 dim V1 = 1 0 :14 t -1 0 0 0 t -1 0 = t 4 - 1 = (t - i )(t + i )(t - 1)(t + 1) det(tI - A) = 0 0 t -1 -1 0 0 t . . R .2 .1 t t 0 -1 0 0 -1 0 t -1 0 t -1 0 0 0 . 0 0 t -1 0 t 0 -1 . 4 - 0 1 0 0 t 0 0 t - 1 dim V1 = dim V-1 = dim Vi = dim V- i = 1 .2 :15 K = {(1, 0)} . K = {(0, y ) | y R} ( K ) = {( x, 0) | x R} .1 K U = ( Sp( A) ) = Sp { (1, 2,1,1), (2, 2, 2, 2), (2,1, 2, 2)} (2, 2, 2, 2) = 2 3 (1, 2,1,1) + 2 3 (2,1, 2, 2) Sp( A) - A dim U = 2 dimSp( A) + dim ( Sp( A) ) = 4 .2 dimSp( A) = 2 . ." :16 . (u - v) (u + v ) = u u + u v - u v - v v = u - v u+v + u-v 2 2 2 2 .1 = (u + v) (u + v) + (u - v) (u - v) = u + + u . = u + u v * u v + v v u u - v -* u v + v v = = 2u u 2v v 2 u + + = ( 2 v 2 ) .2 :17 1 -2 3 + + = 1 14 14 14 1 -2 3 . - - =0 14 14 14 1 -2 3 2 +7 +4 =0 14 14 14 2 2 2 .1 U = Sp ( { (1, 0, - 1 3 ), (0,1, - 2 3 )} ) ( x, y, z ) (1, 0, - ( x, y, z ) (0,1, - . x = 13 z y = 23 z ) 0 = 2 ) 3= 0 1 3 .2 U = { ( 1 3 t , 2 3 t , t ) | t R} = Sp ( { (2, 4, 6} ) :18 (5,8, -2,11) = 1 (1,1, -1, 2) + 2 (2,5,1, -3) 1 + 22 = 5 : 1 + 52 = 8 -1 + 2 = -2 21 - 32 = 11 . ." / . (4, -1, -3, 4) - (3, -2, -2, -1) = (1,1, -1,5) U .1 .2 :19 . ( 1 2)2 + ( 1 2)2 + ( 1 2)2 + ( 1 2)2 = 1 K 1 1 3 1 1 3 t .1 3 3 1 1 3 3 . 1 3 - 2 3 - 2 3 I - 2uu t = 2 3 1 3 - 2 3 - 2 3 - 2 3 13 - ( 1 3 ) 2 + (- 2 3 ) 2 + (- 2 3 ) 2 = 1KKK 1 3 1 uu = 3 1 3 .2 ...
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This note was uploaded on 07/23/2009 for the course MATH. 04101 taught by Professor ישראלפרידמן during the Summer '06 term at The Open University.

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