חוברת מאוחד×&or

חוברת מאוחד×&or

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Unformatted text preview: a 20109 I 2007 - " : " - - 2006 . . () .1 .2 .3 .4 .5 .6 .7 11 " 12 " 01 " 13 " 14 " 02 " 15 " 16 " 17 " 03 " " " 1 3 5 11 13 15 19 21 23 25 ." " , , . . . . . . ! .( ) . , , : . , , .http://telem.openu.ac.il : . . 7 . " : .12:00-10:00 ,' ,09-7781423 . .myriamr@openu.ac.il - .09-7780631 : . , .1 . 12 : - (") - ,() .(") . . . .2 -n :1 :2 Rn :3 - I II :1 :2 E n - III IV V VI VII VIII .3 . ." " . . . . , , . ." " * . . , * , - . . , . : , . " " . , , . . - - , . . .4 ) . .( . . - . ! : . , , . . ." " .5 : . 15 .1 . 60 .2 . 60 .3 (2007 / 20109) .6 " " () (") * 2 ,1 1 20.10.2006-15.10.2006 3 ,2 27.10.2006-22.10.2006 2 3 3.11.2006-29.10.2006 3 11 " 10.11.2006 4 10.11.2006-5.11.2006 4 5 ,4 17.11.2006-12.11.2006 5 5 24.11.2006-19.11.2006 6 12 " 1.12.2006 7 ,6 1.12.2006-26.11.2006 7 01 " 8.12.2006 7 8.12.2006-3.12.2006 8 13 " 15.12.2006 8 15.12.2006-10.12.2006 ( ) 9 ." " , * . . - " " () (") * 9 ,8 10 22.12.2006-17.12.2006 ( -) 14 " 29.12.2006 9 29.12.2006-24.12.2006 11 02 " 5.1.2007 10 5.1.2007-31.12.2006 12 15 " 12.1.2007 11 12.1.2007-7.1.2007 13 16 " 19.1.2007 12 ,11 19.1.2007-14.1.2007 14 17 " 26.1.2007 03 " 29.1.2007 12 26.1.2007-21.1.2007 15 ." " , * . . .7 ; . , . . ,) . ,( , ? Microsoft Internet Office . Microsoft Word 7.0 , Explorer 6 . ? http://telem.openu.ac.il : , . ? . , : : . , ," , , , , , , . , . . . - . " " . . - , . . . , , http://telem.openu.ac.il/personal_notes : . ? . . , , , . . . "" , , ' . - . : . . .( ) , , . . , , . , . , , , , , , . . . . , . . , . ! ? , , . . , . ! . . , . , infodesk@openu.ac.il : ,09-7782222 .09-7781111 " , . ! . " . (") . ) , , , ) ( ' . .( . infodesk@openu.ac.il : ,09-7782222 : 19:00 - 8:30 : 12:30 - 8:30 : . " , : " . ) (09-7781111 ) ( ) ( URL . , ." 7- " 3 ,I , ," " . .( VII ' ) 1 - 2 2 2 3 3 3 4 4 4 3 5 - 2 8 - 6 12 -9 3 ,2 5 ,4 7 ,6 8 ,7 10 ,9 11 12 ,11 01 " 02 " 03 " 11 " 12 " 13 " 14 " 15 " 16 " 17 " . . . , . . , " - ? . ( ) . ' (! ) . .( " ) . ! " . , . " ." " " : ! " . " . " ." " . , , , . / . ) " .( - . . " " ." ,( ) " " , . , . ! .() " .(" 5 ) " " - . ,(" ") "- " " . " ." : , , . " . . ( ) . . " " . " . ." , . . , , , , ." " , , . . , ." " : " : " ) . .( : " www.openu.ac.il/sheilta . . " " . " . " . " " ) . .1 (. www.openu.ac.il/sheilta ."" .2 . "" , .3 ." " , .4 " " " " .5 ." .( ) . .6 ."" - .7 . " "" .8 ( " ) " . . . X . . , . . . . " . - , " , , ." . . . . . .("") 0 . ( ) .(0 ) .(0 ) " " .( ) ," . - " , X , , : ? . 9 . . ! ! . ( ) 1 ' . " .( ) " .2 , . , : . . . . . 60 ! . , . . 11 (") 1 - 20109 : 3,2 : 3 : 5 : 2007 : 10.11.2006 : 13.11.2006 : : . " . ( 20) 1 ax + y + z = 4 . x + by + z = 3 : x + 2by + z = 4 ? ? ? b , a . , ( 20) 2 : 2 x - 3 y - 7 z + 5t + 2 w = - 2 x - 2 y - 4 z + 3t + w = - 2 2 x - 4 z + 2t + w = 3 x - 5 y - 7 z + 6t + 2 w = - 7 x + 6 y - 18 z = 0 - 4 x + 5 z = 0 - 3 x + 6 y - 13 z = 0 - 2 x - 2 y + z = 0 .2 .1 1 ( 20) 3 ax + by = 0 : . A = a b . () c d cx + dy = 0 . ad - bc 0 . ad - bc = 0 A 0 . (i) . () ( x0 , y0 ) (ii) . S = {( x0 , y0 ) | R} () . . ( 20) 4 , . R n - u , v, w : . u + 3v - w , u + v - 3w , v + w (ii ) u - v - w , u + v - 2 w , u + w (i) . {v1 , v2 ,..., vk } . R n - v1 , v2 ,..., vk , w . {v1 + w, v2 + w,..., vk + w} . v1 , v2 ,..., vk w . ( 20) 5 . R n - a1 , a 2 , ... , a m , b , x1 a1 + ... + x m a m = b m n . R n - { a 1 ,..., a m } , x1 a1 + ... + x m a m = c c R n m n . R n - . . { a 1 ,..., a m } . { a 1 , ..., a m } x1 a1 + ... + x m a m = b , 2 12 (") 1 - 20109 : 5,4 : 3 : 5 : 2007 : 1.12.2006 : 4.12.2006 : : . " . ( 15) 1 . a , A = 0 a 1 0 0 a a 1 0 .( An n - ) n > 1 An , C - B , A = B + C : . ,(I.5 ) ( 20) 2 . n B, A . . AB = A + B A2004 = 0 B, A . det B = 0 - I - A . x x n - 1 = ( x - 1)( x n -1 + x n - 2 + + x + 1) : . n < m n m B - m n A . AB . 3 ( 25) 3 2 1 0 . . Dn = . . 0 0 1 2 1 . . . 0 0 0 1 2 . . . . . . 0 1 . . . . . . . . . . . . . . . . . . . 1 0 0 0 0 . . . 2 1 0 0 0 . . 0 1 2 : n . . n 3 Dn = 2 Dn -1 - Dn - 2 (i) . n 1 Dn D3 , D2 , D1 (ii) . a , R3 - v1 = ( a ,1,1), v2 = (1, a ,1), v3 = (1,1, a ) a , , . R3 . ( 15) 4 . B adj ( B ) = B I adj ( B ) . B = 1 - 8 -2 8 3 -2 4 1 3 . . adjA A . ( 25) 5 a1 . C = a2 a 3 b1 1 b1 1 b2 - B = b2 b 0 b3 3 1 0 a1 1 1 , A = a2 a 0 1 3 1 0 b1 a1 1 1 , b = b2 , a = a 2 b a 0 1 3 3 1 . x = 1 A x = b 0 . . B x = a 0 . C x = 1 1 . . 4 01 (") 1 -20109 : 2-5 : 2 : 19 : 2007 : 8.12.2006 : " www.openu.ac.il/sheilta : . 1 . . 2 . 1 x1 - 3 x1 2 x1 - x1 - x2 - x3 - 4 x4 + 2 x2 + x3 + 2 x4 + x2 - x3 - 2 x4 + + x3 + 2 x4 - x2 + x3 + 2 x4 = 8 = -3 = 1 = 1 = 3 . .1 . .2 2 x+ . 2 x + x- 2y - 2y + 3z = 7z = a b c 6 y - 11z = . c - b , a .1 . c - b , a .2 5 3 x1 + x1 + x1 + x2 + x3 = 1 x3 = 1 x3 = 1 x2 + x2 + . 1, - 2 .1 . .2 4 . n - k . , k > n .1 . , k n .2 5 . , - y - x . x + y , .1 . x + y , + = 1 - - .2 6 . k n A , Ax = b . . , k < n .1 . Ax = c , Ax = b k = n .2 7 , n - k - . .k n .1 . k = n , .2 6 8 . A = {(1,0,1,0 ), (0,1,1,1), (2,3,5,3), (0,0,1,0 ), (1,1,3,1)} . R 4 A .1 . A - (1,1,1,1) .2 9 . k 2 , R n - A = {a1 , a 2 , ..., a k } ." A - {a1} , a2 , a3 , ..., ak a1 .1 . R n A k > n .2 10 . n n A . A , Ak - k .1 . A = 0 , A2 = 0 .2 11 . n n A . det A = 0 , At = - A .1 . AB B , A .2 12 . n n A . A , A + A2 = I .1 . A3 + A2 + A , A .2 13 . n n A . , A .1 7 A , A - .2 . 14 0 1 2 3 1 9 1 2 . .1 2 1 0 1 3 2 1 0 f .d e a k - 4c 2 k + f g - 4 a 2 g + d = - 16 d g h - 4b 2 h + e b e h c f = 2 .2 k 15 . - 2 Bt (C -1 )2 = - 1 . C = 4, B = 1 - 5 C , B .1 2 1 0 1 . 1 1 1 1 1 1 1 2 1 1 1 1 3 1 1 1 1 ... ... ... ... 1 1 1 1 = - ( n - 2)! .2 1 1 1 1 1 ... n -1 16 . n n B - A . A = B , x R n A x = B x .1 . B x = c c R n , AB x = c c R n .2 17 .1 . , , .2 . 8 18 . n n A .1 . AB = 0 - n n B 0 A . Ax = b , b R n n n A .2 19 adj( AB) = (adjB)(adjA) , B - A .1 . adj (3 A) = 9 A , 3 3 A .2 2 9 10 13 (") 1 - 20109 : 53 ' 8 ,7,6 : 3 : 5 : 2007 : 15.12.2006 : 18.12.2006 : : . " . ( 20) 1 . z 4 = - 2i . aij A - , C A . det A = det A - .1 . At A = I A .2 .1- . i - . . ( 20) 2 k - (c, d ) ,( a, b) R 2 : R 2 . ( a, b ) + (c, d ) = ( a + c, b + d ) k ( a , b ) = ( ka , b ) ? R R 2 Q , Q ( 2 ) = a + b 2 | a , b Q . Q ( 2 ) { } . 11 ( 20) 3 : , , R . 2 2 C U = A M 22 AC = 0 .1 { } . W = {( x, y , z , t ) R 4 | 2 x - y + t = x - 3 y + z - 2t = 0} .2 , n 3 , R n n T .3 . . W , C = 2 0 , U - 1 0 . ( 20) 4 . V = U W - V - W ,U V . T W - S U - , T , S . S T . W , U B2 , B1 . dim V = n - . ? V B1 B2 . . ( 20) 5 ? R3 = {( , - 2 , ) | , R} {(3 , - , 2 ) | R} . . W = { p ( x ) R 4 [ x ] | p ( - 1) = p (0) = p (1)} - U = Sp({x - 1, x 2 + 1} . W R 4 [ x ] - W . ? R 4 [ x ] = U W . . 12 14 (") 1 - 20109 : 8,7 : 4 : 5 : 2007 : 29.12.2006 : 1.1.2007 : : . " . ( 20) 1 : R 4 [ x ] - W - U U = S p{x3 + 4 x 2 - x + 3, x3 + 5 x 2 + 5, 3 x3 + 10 x 2 + 5} . W = S p{x3 + 4 x 2 + 6, x3 + 2 x 2 - x + 5, 2 x3 + 2 x 2 - 3 x + 9} . U + W ,W ,U . . U W - ? U W . ( 20) 2 . F V v1 , v 2 , ... , v k , a, b . C = {v1 , v 2 , ... , v k } - B = {v1 , v 2 , ... , v k , b} , A = {v1 , v 2 ,... , v k , a} : . b Sp (C ) V A : . a Sp (B ) . V B . V - A , C . . . 13 ( 20) 3 V - v1 , v2 , v3 , u1 , u2 , u3 . R V B = {u1 - u2 , u1 + 3u3 , 4u2 + 5u3 } u1 , u2 , u3 U = Sp{v1 , v2 , v3 } - . . . U - .V - W ,U 2 ,U1 V . dim W dim U 2 - . U 1 W = {0} - V = U 1 U 2 - . ? W U 2 {0} . W {0} - . ( 20) 4 . n n A Ax = 0 A A 2 = 0 . . ( A) n - 2 Ax = 0 ( A 2 ) < ( A) . Ay 0 y A2 x = 0 . ( 20) 5 - W - R 4 - v1 = (1, 2, 2,1) , v2 = (0, 2,0,1) , v3 = ( - 2,0, - 4,3) .- R 4 . W B = {v1 , v2 , v3 } . W - v - a, b, c, d / . R 4 - v = ( a , b, c, d ) . B v . W - v = ( a , b, c, d ) . v1 ' = (1,0, 2,0), v2 ' = (0, 2,0,1), v3 ' = (0,0,0,3) . B ' - B - W B ' = {v '1 , v '2 , v '3 } . . . . 14 02 (") 1 -20109 : 6-8 : 2 : 18 : 2007 : 5.1.2007 : " www.openu.ac.il/sheilta : . 1 . . 2 . 1 . {a + bi | a , b Q} .1 R n n .2 . 2 : z . z2 0 . 1 + iz = 1 + z .1 .2 3 , z 4 + 2 z 3 + 2 z + 6 = 0 z C .1 . z . z , z 2 + iz = 0 z .2 15 4 . 2 cos 3 + i sin 3 - 1 + i .1 4 4 . 2 cos 4 + i sin 4 - 1 - i 3 .2 3 3 5 . - 1 - i - - 1 + i , 1 - i , 1 + i z 4 = -4 .1 . 3+i - 2 3 -i , -i z 3 = i .2 2 6 . 1 + w + w2 + + wn -1 = 0 , wn = 1 , w 1 , w C .1 . x n - 1 = ( x - 1)( x n -1 + x n - 2 + + x + 1) : x : 1 1 1 w w2 = -3 , w = cos 1 . 1 w2 w 2 2 + i sin .2 3 3 7 V = {(u , w) | u U , w W } , F W - U .1 :- (u, w) = (u, w) - (u1, w1) + (u2 , w2 ) = (u1 + u2 , w1 + w2 ) . R 5 [ x ] - W = { p ( x ) R 5 [ x ] | p ( - 1) = p (1) = 2} .2 8 . n 2 , V v1 , v 2 , ..., v n u - {v1 , , ..., v n } .1 . {v1 , v 2 , ..., v n , u} , {v1, v2 , ..., vn} u .2 . u = vn : , {v1, v2 , ..., vn -1} 16 9 . V S , T . Sp ( S ) + Sp (T ) = Sp ( S ) Sp (T ) . T S Sp (T ) Sp ( S ) .1 .2 10 . B , Sp( A) = Sp( B) - ( ) A B .1 . V - A , A + A = A .2 11 V , W , - , V - U .1 . U W = V - , 0 < k < n , k V - U - V A = (v1,..., vn ) .2 . U - A - k 12 R . M 2 2 - 1 1 1 0 0 1 0 0 , 0 1 , 1 0 , 1 1 .1 0 0 . R 4[ x] { x3 + x 2 , x3 + 1, x 2 - x + 1, x3 - 2 x 2 + 2 x - 1, 2 x 2 - 3 x + 4} .2 13 . u W u U . u V - V = U W .1 c 0 , W2 = d d - c a+b b c , d R , W1 = a , b R .2 0 -a R M 2 2 = W1 W2 17 14 . R5 3 - W - 2 - U . U W U + W R 5 .1 . R 5 = U W U W = {0} .2 15 , U W R 4 3 - W - 2 - U .1 . dim(U W ) = 1 . dim(U W ) = n - 2 , R n - n - 1 - U , W .2 16 (1, - 1, 0, 2) .1 . (2, - 1, 0, 1)t ((1, 0, 1, 1), (1, 1, 0, 1), (1, 1, 1, 0), (0, 1, 1, 1)) (v1, v2 , v3 ) (v1 + v2 , v2 + v3 , v3 + v1) .2 1 - 1 1 1 . - 1 1 1 3 2 1 1 -1 17 AB , n n B - A .1 . A . .2 18 R P ( AB ) , P( B) , P( A) - . A, B M n n . ABx = 0 , Bx = 0 , Ax = 0 . P( A) = P( B) ( A) = ( B ) .1 . P ( AB ) = P ( B ) ( B ) ( AB ) .2 18 15 (") 1 - 20109 : 9,10 : 4 : 5 : 2007 : 12.1.2007 : 15.1.2007 : : . " . ( 20) 1 dim(Im T ) < dim(ker T ) . T : R 4 R 4 B = ((1111), (111,0), (11,0,0), (1,0,0,0)) T ,,, ,, , . [T ] B 1 2 1 a1 = 1 a 2 1 a 3 3 b1 b2 b3 4 c1 c2 c3 . ( x1, x2 , x3, x4 ) R 4 T ( x1, x2 , x3 , x4 ) (1 i 3) ai , bi , ci . .ImT - kerT - . ( 15) 2 - KerT - T : R 4 [ x ] M 23 ( R ) . - ImT - q ( x ) = x3 + x 2 - x + 3 - p ( x ) = x3 - 2 x + 1 . W = S p{ - 1 2 1 } 3 -1 0 . R 4 [ x ] T : 2 2 . R 3 [ x ] . 19 ( 30) 3 . V B = (u1 , u2 ,..., un ) . dimV = n , F V :- T : V V . T (un ) = 0 - T (ui ) = ui +1 , i = 1,..., n - 1 , i . B T A . T n -1 0 , T n = 0 . S n -1 0 , S n = 0 S : V V B ' S V B ' .' A N n = 0 , M n = 0 - n n N , M . . . . . N - M , N n -1 0 , M n -1 0 - ( 20) 4 . T : V V , V . kerT ker T 2 - Im T 2 Im T : kerT = ker T 2 (ii) Im T 2 = Im T (i) . . . (ii) (i) - , . Im T kerT = V Im T kerT = {0} , (ii)- (i) . ( 15) 5 . T :V V V . [T ] B - V B , T . . A , A . 20 16 (") 1 - 20109 : 11 : 4 : 5 : 2007 : 19.1.2007 : 22.1.2007 : : . " . ( 20) 1 3 0 - 4 A - B , A = 1 1 - 2 2 0 - 3 . . P T : R 3 R 3 . ( x, y, z ) R 3 T ( x, y, z ) = (3x - 4 z, x + y - 2 z,2 x - 3z ) . T 2001 ( x, y, z ) ' . ( 20) 2 . T : R 3 R 3 . T (1,-1,0) = (a - 4, a + 6,0) - T (11,0) = (- 5,-5,0) , T (11,1) = (2,2,2) , , . T a T a . 21 ( 20) 3 . n n A . . A = 1 = 0 , A 2 = A - A I ,0 . a 0 0 1 1 a . ( 20) 4 , n n , n 0 . B = . . . 0 0 . 0 . . . . . . 0 0 1 1 . 1 1 . . 0 . . - A = . .. . . . . 1 1. . . 0 . . . . . . 1 1 . . . . 1 0 . . . . . . . . B = P -1 AP - P ( 20) 5 . A4 = 0 n A . . A , A 0 A = 0 . ( A) = 1 - trA = 0 - 5 5 A .11 ,26 : . ? A . 22 17 (") 1 - 20109 : 12 ,11 : 3 : 5 : 2007 : 26.1.2007 : 29.1.2007 : : . " . ( 20) 1 . BA - AB - . n B , A AB v = (1, 2) - 2 2 B , A . BA 1 . . ( 20) 2 . R3 - U = S p{(3,12, - 1)} . U U . . U v = ( - 10, 2,8) . ( 20) 3 . R n - w - v . n 3 , Sp{v - w, v + w} - u 0 R n - . T : R n R n . ker T = (Im T ) , Tu v = 0 v ker T u R n . . 23 ( 20) 4 . , . A = 1 2 1 1 1 2 2 1 1 . . . Pt AP - P ( 20) 5 . R n - U 2 ,U1 . R n = U1 U 2 . R n = U1 U 2 ? R n = U1 + U 2 . R n = U1 + U 2 . . 24 03 (") 1 -20109 : 12-9 : 2 : 19 : 2007 : 29.1.2007 : " www.openu.ac.il/sheilta : . 1 . . 2 . 1 . T ( f ( x)) = x f ( x ) - T : R 3 [ x ] R 4 [ x] .1 . C .2 . T ( z ) = z - T : C C 2 T (2,3,1) = (1,1,1) , T (1,2,3) = (0,1,2 ) - T : R 3 R 3 .1 . T (1,0,1) = (1,0, 2 ) - T (- 1,1,1) = (0,1) , T (1,-1,1) = (1,0 ) - T : R 3 R 2 .2 . T (1, - 1, - 5 ) = ( 2,1) - 25 3 T : V V , V A = {v1 , v 2 ,..., v n } . .-- T , V {Tv1 , Tv 2 ,..., Tv n } .1 . A , {Tv1 , Tv 2 ,..., Tv n } .2 4 . S , T Hom(V , V ) V TS = 0 , ST = 0 .1 . S = T , Im S = Im T - ker S = ker T .2 5 . S , T Hom(V , V ) . dim Im TS = dim Im T , dim ker S = 0 - V .1 . ker TS ker S .2 6 . T ( f ( x)) = f ( x + 1) - T : R 4 [ x ] R 4 [ x ] . T . [T ]B 1 0 = 0 0 1 1 0 0 1 2 1 0 .1 1 3 B = 1, x, x 2 , x 3 T .2 3 1 ( ) :- T : M R2 M R2 ,7-8 2 2 R . X M 22 , T ( X ) = X + X t 7 . ker T = Sp . ImT = Sp 0 - 1 1 1 .1 1 0 0 1 0 0 , , 0 0 1 0 0 1 .2 26 8 1 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 C = 0 0, 0 0, 1 1, 0 1 - B = 0 0 , 0 0 , 1 0 , 0 1 . M R 2 - 2 . [T ]B 1 0 = 0 0 0 1 1 0 0 1 1 0 0 0 0 1 .1 2 1 1 - 1 . T = 1 0 0 C - 1 .2 9 . n n B - A . B - A , B - A .1 . BA - AB , A .2 10 . n n B - A . B - A , B - A .1 . , B - A .2 11 . S , T : V V , R n V , T 2 - S 1 .1 . S + T 1 + 2 . S + T v , T S v .2 12 . n n B - A . B - A , B - A .1 . B - A , , B - A - .2 27 13 . - 0 - 1 .1 0 - 2 1 1 0 1 1 0 . 0 1 0 - 0 1 1 .2 0 0 1 0 0 1 2 0 1 0 14 0 0 .A= 0 1 1 0 0 0 1 0 0 0 1 0 0 0 . R A .1 . C A .2 15 . ( K ) = K , R n - K .1 . R 4 - A = {v1 , v 2 } .2 , . A , (Sp( A)) = Sp{(1,2,11), (2,2,2,2), (2,1,2,2)} 16 . u, v R n .1 . u - v - u + v u = v . u+v + u-v 2 2 =2 u ( 2 + v 2 ) u, v R n .2 17 . Sp({(1,-1,-1), (2,7,4)}) - 1 , - 2 , 14 14 3 14 .1 . U = Sp ({ (2,4,6 )}) , U = ( x, y, z ) R 3 | x + 2 y + 3 z = 0 .2 { } 28 18 {(1,1,-1,2 ), (2,5,1,-3)} .1 . 5 8 - 2 11 3 9 3 8 1 1 -1 2 Sp({(1,1,1,1), (1,2,2,-1), (3,4,4,1)}) (4,-1,-3,4 ) .2 . (3, -2, -2, -1) 19 1 2 1 . 2 1 2 1 2 1 2 1 2 1 - 2 1 - 2 1 2 1 - 2 1 - 2 1 2 1 2 1 - 2 .1 1 2 1 - 2 . P = I - 2uu t 1 1 1 . 2 , u = 3 1 29 ...
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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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