חוברת מאוחד×&or

חוברת מאוחד×&or

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Unformatted text preview: a 20109 I 2007 - " : " - - 2007 . . () .1 .2 .3 .4 .5 .6 .7 11 " 12 " 01 " 13 " 14 " 02 " 15 " 16 " 17 " 03 " " " 1 3 5 9 11 13 17 19 21 23 ." " , , . . . . . . ! .( ) . , , : . , , .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 16.3.2007-11.3.2007 3 ,2 23.3.2007-18.3.2007 2 3 30.3.2007-25.3.2007 3 11 " 6.4.2007 4 6.4.2007-1.4.2007 ( -) 4 4 13.4.2007-8.4.2007 ( -) 5 5 ,4 20.4.2007-15.4.2007 ( ) 6 5 27.4.2007-22.4.2007 ( ) 7 ( ) 12 " 4.5.2007 7 ,6 4.5.2007-29.4.2007 8 01 " 11.5.2007 7 11.5.2007-6.5.2007 ( " ) 9 ." " , * . . - " " () (") 13 " 18.5.2007 * 8 10 18.5.2007-13.5.2007 ( ) 9 ,8 25.5.2007-20.5.2007 ( -) 11 14 " 1.6.2007 9 1.6.2007-27.5.2007 12 02 " 8.6.2007 10 8.6.2007-3.6.2007 13 15 " 15.6.2007 11 15.6.2007-10.6.2007 14 16 " 22.6.2007 12 ,11 22.6.2007-17.6.2007 15 17 " 29.6.2007 03 " 29.6.2007 12 29.6.2007-24.6.2007 16 ." " , * . . .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 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 : 6.4.07 : 8.04.07 : : . " . ( 20) 1 - 3 x + . x + -2 x x + y + 4 z = -5 y + z = 2 z = -3 + y - 2z = 5 : . , a, b, c, d , e ,4 f ( x) = a + bx + cx 2 + dx 3 + ex 4 . . f (1) = 1 , f (2) = -1 , f ( -1) = 5 , f (3) = -59 , f (-2) = -29 .1 43 ' : ( 20) 2 : . c , b , x = 3b 4 x + 8 y + 7 z + 3cw + 2 y + 2 z + cw = b 2 x + 4 y + 2 z + ( c - 1) w = b : c - b ? (i) . ? (ii) . ? (iii) 1 ( 20) 3 : . 3 - k L . L , (2,4,6 ) - (- 1,-2,-3) . ,- L - L (1,-4,0 ) - (1,-2,-1) , (1,-2,1) . . (1,0,0 ) . k 3 L - . ( 20) 4 . R3 - v1 = (2,1, - 1) , v2 = ( - m, - 1,3) , v3 = ( - 3, 2, m + 1) , v4 = (1, 2,1) . R3 {v1 , v2 , v3 , v4 } m . . R3 - w = ( m + 1, m - 1,1) . x1v1 + x2 v2 + x3v3 + x4 v4 = w m x1v1 + x2 v2 + x3v3 + x4 v4 = 0 m , . . ( 20) 5 . R n - D , C D - . D C . C . R n C . D - R n D , C - . . ( n - ) . . 2 12 (") 1 20109 : 5 , 4 : 3 : 5 : 2007 : 4.5.07 : 6.5.07 : : . " . ( 20) 1 . Ar = 0 - r 1 A . AB = BA B , A . A + B - AB . (I.5 ) : ( 20) 2 . AB = I m n m B - m n A . Bx = 0 . m n . m = n - X = A , BX = I n X . . . ( 20) 3 . n n B - A . A = 0 , B 2 = I + BA - A 2 = AB . A( adjA) 0 - . n A . ad jA - 3 . . ( 15) 4 : n x+ y 1 x + y if i = j 0 xy if j = i + 1 aij = ", Dn = . 1 if j = i - 1 . 0 if j i, i + 1, i - 1 . 0 xy x+ y 1 . . . . 0 xy x+ y . . . . . . 0 . xy . . . . . . . . 0 . 0 . 0 . . . . . 0 . xy 1 x+ y .0 xy , 1 , x + y . Dn = x n +1 - y n +1 x y x- y ( 25) 5 1 0 . . . . 0 1 1 1 0 . . . 0 0 1 1 0 . . . . . . . . . . . . , n 3 . n n A = . . . . . . . . . . . . . . . . 0 . . . 0 1 1 0 0 . . . . 0 1 1 . A . . A n . ' . R n - B = {v1 , v 2 , ... , v n } . . R n - C = {v1 + v 2 , v 2 + v 3 ,... , v n + v1 } 4 01 (") 1 -20109 : 5-2 : 2 : 19 : 2007 : 11.5.07 : " www.openu.ac.il/sheilta : . 2 . . 1 . 1 x1 + x2 + x3 = 1 (. x1, x2 , x3 , ) . x1 + x2 + x3 = 0 ( + 1) x - x + x = 1 2 3 . = - 1 .1 . 1 .2 2 x1 + 2 x2 - 3x3 = . 3x1 - x2 + 2 x3 = x - 5x + 8 x = 2 3 1 . , , .1 . , , .2 5 ,(O) (M) - 5 -3 . , n- m 3 . m n (O)- .1 . m > n (M)- .2 4 . (M)- , (O)- .1 . (O)- (M)- .2 5 ,(M) - (M') .(M) . (M') - (M)- .1 . (M')- (M)- .2 6 . A Ax = b .1 . Ax = b b R 4 , 4 3 A .2 7 . R 4 - A = {(1,1,1,1) , (0,1,0,0) , (2,1,2,2) , (0,0,1,1) , (1,3,2,1)} .R4 A .1 . R 4 A - .2 8 .R3 {a, b, c} .R3 {a + b , b + c , c + a} .1 .R3 {a b , b + c , a + 2b + c} .2 6 9 , A .Rn - A = {a1 , a 2 , a3 } .1 . a3 = 1 a1 + 2 a 2 - 1 , 2 . n 3 ,Rn - A = {a1 , a 2 , a3 } .2 . A , {a1 , a 2 },{a1 , a3 },{a2 , a3 } .n n Y - X ,D ,B ,A 19-10 10 0 1 . 0 1 100 0 1 = 0 1 .1 1 2 . -3 -2 2007 1 2 = .2 -3 -2 11 . ( A + B )( A - B) = A 2 - B 2 .1 . 1 ( A + I )2 - ( A - I )2 = A .2 4 [ 12 . ( DA)t = DAt D .1 . AB = BA ( AB)t = At Bt .2 13 . A A2 - A + I = 0 .1 . A A2 - A = 0 .2 14 . A + B A ,B .1 . B A AB .2 7 15 .A . A AX = B - X B .1 . AY B , Y X AX = B - A .2 16 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 . = 0 .1 1 1 . k 1, -1, 0 1 k 1 k 2 1 -1 .2 1 17 1 2 3 1 + 1 . 2 + 2 3 + 3 1 + 21 - 3 1 2 + 2 2 - 3 2 3 2 2 2 1 2 3 1 2 = a 3 1 + 1 2 + 2 = 2a 3 + 3 1 + 1 2 + 2 3 + 3 3 3 2 3 .1 3 + 2 3 - 3 3 = 216a . 31 2 1 .2 18 . adj( A) = n -1adjA R .1 . adj( At ) = (adjA)t .2 19 A . b R n .1 . Ax = b , A = 2 - . Ax = 0 , At = - A - - n .2 8 13 (") 1 20109 : 7,6 : 3 : 5 : 2007 : 18.5.07 : 21.5.07 : : . " . ( 20) 1 . z 3 = z C - . 5 5 A . adj ( adjA) = A det A ( 25) 2 , . . . . C K = {( z1 , z2 , z3 ) C3 | (1 + i ) z1 - 2 z2 + 5iz3 = 0} .1 . R L = {(x1 , x 2 , x3 , x 4 ) R 4 | x1 x 2 } .2 . C M = A M C n A = - A .3 n . R P = {p (x ) R 4 [x ] | p (0 ) = 1} .4 . , . . . { } 9 ( 15) 3 . V u1 , u2 ,u3 . v1 = u1 + u2 + u3 , v2 = u1 - u2 + u3 , v3 = -u1 + 3u2 - u3 . ? Sp{u1 , u2 , u3 } = Sp{v1 , v2 , v3 } ( 20) 4 W = Sp{x 3 + ix , (1 - i ) x 2 - x} - U = Sp{x3 - 2i x , x 2 + 1} . C . C 4 [ x] - . W U {0} C - . ! ? = 0 C 4 [ x] = W U . ( 20) 5 . V = U1 U 2 - . V - W ,U 2 ,U1 . W = U1 (U 2 W ) , U1 W .1 V = U1 U 2 - W ,U 2 ,U1 - V .2 . W (U1 W ) (U 2 W ) 10 14 (") 1 20109 : 8, 7 : 4 : 5 : 2007 : 1.6.07 : 3.6.07 : : . " . ( 15) 1 . V v1 , v2 ,..., vn , v . v S p{v1 , v2 ,..., vn -1} - v Sp{v1 , v2 ,..., vn } . vn Sp{v1 , v2 ,..., vn -1 , v} ( 20) 2 1 5 0 1 1 1 1 2 1 -1 1 11 W = Sp , , - U = Sp , , 5 0 1 1 1 -4 4 1 3 2 7 -2 . M 2R 2 .U W - U + W ,W ,U - . R U- , M 22 - . . M 2R 2 B 1 0 . 0 0 B . 11 ( 20) 3 . V U 2 - U1 , W , V . W U 2 {0} ( ) U 1 U 2 , U 1 + W = U 2 + W . U 1 = W , W U 2 = {0} - U 1 W , V = U 1 U 2 . . ( 25) 4 . V = {(z1 , z 2 , z 3 ) C 3 | z1 + (1 + i )z 2 - z 3 = 0} . [(-i,1,1)] B . B C V - . R V - , B = {v1 , v 2 ,... ,v k } ^ . R V B = {v1 , v 2 ,... ,v k , iv1 , iv 2 ,... ,iv k } . . . [(-i,1,1)] B ^ ( 20) 5 . m n B , A . - - A + B WA + B . B A . ( A + B ) ( A) + ( B ) : - : 2 2 B , A . ( A + B) > ( B) ( A + B ) > ( A) . . 12 02 (") 1 -20109 : 8 -6 : 2 : 19 : 2007 : 8.6.07 : " www.openu.ac.il/sheilta : . 1 . . 2 . 1 R n .1 . K = {a + b 2 | a , b Q} .2 .. 2 . Re 3 + i = 3 5 - 12i 5 - 12i 13 .1 . Im 3 + i = 41 i .2 169 3 . 1+ i 2 ( ) 3 = 3 3 .1 ( 2 - 8i ) . (1 + i 3 ) 3 2 = 10 5 .2 13 4 . z C .z = z + z + i z - z 2 2 .1 . Im(iz) = Re(- z) .2 5 . - 2 cos 5 + i sin 5 - 1 + i 3 .1 3 3 . 2(cos 3 3 + i sin ) -1 + i .2 4 4 6 z3 = 2 + 2i .1 . z3 = cos 17 + i sin 17 , z2 = cos 12 12 3 3 , z1 = cos + i sin + i sin 4 4 12 12 z 2 = -i .2 . z2 = cos 7 7 3 3 , z1 = cos + i sin + i sin 4 4 4 4 7 . b - a , - z2 + az + b = 0 - .1 1 . - - .2 1 . 9 - 8 8 . M Rn U = { A M R n | det A = 0} .1 n n . C n [x ] W = {P C n [ x] P(1) = P(2)} .2 14 . R Cn U = ( z1 ,..., z n ) C n z n = 0 { 9 } .1 . C W = , R , C .2 .V T - S 15 -10 .V U2- U1 ,U 10 . S T = SpS SpT = {0} .1 . Sp ( S T ) = Sp ( S ) Sp (T ) S T = .2 11 . T {v} , T- , v Sp T , v V .1 . u Sp({v}) , u SpT - u Sp(T {v}) , u , v V .2 12 . U1 = U 2 U1 U = U 2 U .1 . (U1 + U 2 ) U = {0} U 2 U = {0} - U1 U = {0} .2 13 . V - .U S- , S T ,S V T .1 . S T - V T S - U = SpS .2 14 . C C3 - {(1,0,0) , (0,1,0) , (0,0,1)} .1 . R C3 - {(1,0,0) , (0,1,0) , (0,0,1)} .2 15 .3 C4[ x] Sp({x 3 - 1 , x 2 + 1 , x 2 - x , x 3 + x}) - .1 U W - , dimU = dimW = 4 - R 7 - W - U .2 . 15 16 .A B .A B .1 .A B .2 17 . ( A) = n - 1 - B 0 , n B- A . ( B) = 1 AB = 0 .1 . ( B) = 1 BA = 0 .2 18 .V B = {v1, v2 , v3 , v4} ,V- B' = (v1 + v2 , v2 + v3 , v3 + v4 , v4 ) 1 1 . 0 0 0 0 0 1 0 0 1 1 0 0 1 1 B- B' - .1 . [v1 ] B ' = (1,-1,1,-1) t .2 19 . R 3 B2 = ((1,2,3), ( -1,0,1), (1,0,1)) - B1 = ((111), (0,11), (0,0,1)) ,, , . [(-1,0,1)] B1 = [(0,1,1)] B2 .1 . [(1,2,3)] B1 = (1,1,1) t .2 16 15 (") 1 20109 : 10,9 : 4 : 5 : 2007 : 15.6.07 : 17.6.07 : : . " . ( 25) 1 R R R R . X M 2 2 T ( X ) = AX :- T: M 2 2 M 2 2 A M 2 2 . T . A T . dim ker T 2 , T . s 0 0 s , ker T - t 0 0 t s 2 t R : . . . R . M 2 2 T -1 . A = 1 1 - 0 1 . ( 15) 2 . . , ker T = Sp{(0,1,1,0),(1,1,0,0)} - T : R 4 R 5 . . Im T = Sp{(2,3, 4,0,0),(1,1,0,0,0),(0,1,1,0,0)} - T ( - 1,1,0,1) = (1, - 1, 2,1) , T (1,1,1,0) = (2,3,1, - 1) T : R 4 R 4 . . T ( - 1,5, 2,3) = (3,7,0, - 3) - 17 ( 20) 3 . T 3 = 0 - T :V V V . T 2 (v ) 0 - v V . 1 - x3 = (1 - x )(1 + x + x 2 ) : . ( I - T )-1 I - T . {v, Tv, T 2 v} . [T ]B = 0 0 1 - V B ker T Im T dim V = 3 0 0 0 0 1 0 . . . ( 15) 4 . di mV = n . F V T1 0 , T2 0 - T1 , T2 :V F . N1 N 2 . N1 = KerT1 , N 2 = KerT2 . dim( N1 N 2 ) ( 25) 5 4 1 3 A = 2 3 3 T : R 3 [ x ] R 3 [ x ] 2 -3 1 . B = (1 + x,1 + x 2 , x + x 2 ) . kerT . . Im T . T ( a x 2 + bx + c ) . W = { p ( x ) R 3 [ x ] | T ( p ( x )) = 2 p ( x )} . R 3 [ x ] - W . . . 18 16 (") 1 20109 : 11 : 4 : 5 : 2007 : 22.6.07 : 24.6.07 : : . " . ( 20) 1 1 a 1 0 0 c . c , b , a , A = 0 1 b . ? A c , b , a D P , c, b, a . D = P - 1 A P - ( 20) 2 R R :- T: M 2 2 M 2 2 R . A M 2 2 T ( A) = A - At . . . Im T - ? Im T . ker T - . T . 1 - T . . . 19 ( 20) 3 . det ( A - I ) = 0 , ( A + 3I ) = 2 3 3 A . A . ? A - 3 I . A . . . ( 20) 4 . V , T : V V T v - T u - . . = , T u + v T [T ]B V B . . T (v ) = cv v V c ( 20) 5 . S , T Hom(V , V ) V . TS , ST = 0 , - ST v ST 0 . - TS Tv . TS - ST - . . . . . 20 17 (") 1 20109 : 12 : 3 : 5 : 2007 : 29.6.07 : 1.7.07 : : . " . ( 25) 1 . R 4 U = Sp{(1,1,1,1) , (1,2,-1,1)} .U- . . U - . .U v = (3,0,0,1) . ( 20) 2 . R 4 - W ,U . dim(U W ) 2 (1,1,0,0),(0,0,1,1) W - (1,0,0,0) ,(0,0,0,1) U - . U ,W ,W ( 15) 3 1 - 1 1 2 -2 2 . A = 0 1 - 3 T : R 3 R 5 2 1 -7 1 0 - 2 . ImT 21 ( 20) 4 . di m W1 = dim W2 - R n - W2 ,W1 . T (W1 ) = W2 - T : R n R n ( 20) 5 1 2 3 3 4 5 . A = 2 3 4 . Pt AP - P 22 03 (") 1 -20109 : 12-9 : 2 : 19 : 2007 : 29.6.07 : " www.openu.ac.il/sheilta : . 1 . . 2 . 1 . T(1, 2) = (21 + 2, |2|) - T: R 2 R 2 .1 , T ( z1, z2 ) = ( z1, z2 ) - T: C2 C2 .2 . C C2 2 . T(f(x)) = xf(x) - T : R 5 [ x] R 6 [ x] .1 C C . T ( X ) = 2 X + 3 X t - T: Mn n Mn n .2 3 T(1,2,3) = (0,1) , T(1,1,1) = (0,1) : T : C3 C2 .1 T(3,2,1) = (0,1) Im T = Sp{x 2 - 1, x 2 + 2 x + 2} - T : R 3 [ x ] R 3 [ x ] .2 . R 3 [ x ] = KerT Im T 23 4 : .S(x, y, z) = (z, x + y, x + y + z, 2x + 2y + z, x + y z) - S : R3 R5 . Im S = Sp{(1,0,0,0,0),(0,1,0,0,0)} .1 . KerT = Sp{(1,1,0),(0,0,1)} .2 T : V V - V {v1, v2, ..., vk} 6 - 5 . 5 . {v1, ..., vk} {Tv1, ..., Tvk} .1 .V {v1, ..., vk} V {Tv1, ..., Tvk} .2 6 . T: R n R m . Im T = R m n m .1 . KerT {0} n > m .2 . S:V V - T:V V 8 - 7 7 .S = T Im S = Im T - kerS = kerT .1 . Im T KerS Im ST = {0} .2 8 .kerTS = {0} kerS = {0} .1 .kerTS = {0} kerT = {0} .2 24 : T : R3 R3 10 -9 T(1,0,0) = (3,1,2) , T(0,1,0) = (1,0,0) , T(0,0,1) = (1,1,1) S : R3 R3 , 3 1 2 . 1 0 0 R3 1 1 1 .((1,1,1), (1,0,1), (0,1,1)) B - R3 E - 9 . [T ]E = ([ S ]E )t .1 . [T ]B = ([ S ]B )t .2 10 5 2 2 . [T ]B = 2 0 1 3 1 1 .1 9 3 7 . [ ST ]E = 3 1 1 .2 3 1 3 11 . .1 . .2 12 . B2 - A2 , B- A .1 . B t - At , B- A .2 13 . 2 2 B- A . B- A , | A| =| B| - trA = trB .1 . B- A , B- A .2 25 14 . A . A2 + I 10 A 3 .1 .A - 3 A 3 , A2 + I 10 .2 15 . A . A2 P(t 2 ) A P(t ) .1 .A + 2I P(t - 2) A P(t ) .2 16 2 0 0 . 0 2 0 - 0 0 1 2 0 0 0 2 1 .1 0 0 1 . 2 2 - 1 3 2 1 .2 0 2 17 . u, v R n . v- u u + v 2 = u 2 + v 2 .1 . u = v u + v - u - v .2 18 . R n W- U . U W U W .1 . (U W ) = U W .2 19 .1 - {( x, y, z) x - y = 0} - .1 . dim U = dim U - R 15 U .2 26 ...
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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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