2005 II 3 I 18, 19 1. A, B AB BA 0 1 0 0 0 0 (1) A = 0 0 1,B= 1 0 0 0 0 0 0 1 0 (2) A = 3 1 1 2 6 4 1 2 5,B= 12 11 12 22 46 46 12 23 34 5 25 2. 3 A AB = BA 3 B 2 0 1 A = 0 3 0 1 0 2 3. 2 A (1) A 2 = O, (2) A 2 = E, (3) A 2 = A 4. F : R 3 R 3 ax 2 + bx + c x x +1 a u ux 2 + vx + w b v c w (1) F (2) T A = F A
2005 II 3 I 18, 19 5 25 1 (1) AB BA 1 0 0 AB = 0 1 0, BA = 0 0 0 0 0 0 0 1 0 0 0 1 (2) AB = 70 102 116 204 390 436 116 218 274, BA = 70 102 116 204 390 436 116 218 274 2 A A =2 1 0 0 0 1 0 0 0 1 + 0 0 1 0 1 0 1 0 0 A =2E + S AB =(2E + S)B =2B + SB, BA = B(2E + S) =2B + BS
3 2 3 B AB = BA SB = BS S B XY = YX X Y B (i, j)- b ij SB BS SB = BS = b 31 b 32 b 33 b 21 b 22 b 23 b 11 b 12 b 13 b 13 b 12 b 11 b 23 b 22 b 21 b 33 b 32 b 31 SB = BS b 11 = b 33, b 12 = b 32, b 13 = b 31, b 21 = b 23 A a b c d e d c b a a, b, c, d, e R 3 ( ) a b A = c d ( A 2 = a 2 + bc (a + d)b (a + d)c bc+ d 2 ) (1) A 2 = O a 2 + bc =(a + d)b =(a + d)c = bc + d 2 =0
3 3 b = 0 a 2 = d 2 = 0 a = d = 0 c c =0 a 2 = d 2 =0 a = d =0 b bc 0 a + d =0 d = a bc = a 2 A = ( ± bc b c bc ) bc 0, (2) A 2 = E a 2 + bc = bc + d 2 =1, (a + d)b =(a + d)c =0 b =0 a 2 = d 2 =1 a = ±1,d= ±1 a d c =0 c c =0 b c O.K. bc 0 d = a a 2 =1 bc A = ( ± 1 bc b c 1 bc ) A = ±E bc 1, (3) A 2 = A a 2 + bc = a, (a + d)b = b, (a + d)c = c, bc + d 2 = d b =0 a 2 = a, d 2 = d a =0, 1 d =0, 1 (a + d)c = c a + d =1 c =0
3 4 c =0 b c O.K. bc 0 a + d =1 a 2 a + bc =0 d 2 d + bc =0 a = 1 ± 1 4bc, d = 1 1 4bc 2 2 ( ) ( ) ( ) ( ) A = 1 0 1 a 0 0 0 a,,, a 0 0 0 a 1 0 1 A = 1 ± 1 4bc 2 c b 1 1 4bc 2 bc 1 4, A =O,E 4 (1) 1: F F F (a + b) =F (a)+f (b), F(ra) =rf (a) F 2 x +1 2 2 ((a 1 + a 2 )x 2 +(b 1 + b 2 )x +(c 1 + c 2 )) (x +1) = (2(a 1 + a 2 )x +(b 1 + b 2 ))(x +1) =2(a 1 + a 2 )x 2 + (2(a 1 + a 2 )+(b 1 + b 2 ))x +(b 1 + b 2 ) =(2a 1 x 2 +(2a 1 + b 1 )x + b 1 )+(2a 2 x 2 +(2a 2 + b 2 )x + b 2 ) =(2a 1 x + b 1 )(x +1)+(2a 2 x + b 2 )(x +1) =(a 1 x 2 + b 1 x + c 1 ) (x +1)+(a 2 x 2 + b 2 x + c 2 ) (x +1)
3 5 F a 1 + a 2 b 1 + b 2 c 1 + c 2 = F a 1 b 1 c 1 + F a 2 b 2 c 2 ((ra)x 2 +(rb)x +(rc)) (x + 1) = (2(ra)x +(rb))(x +1) =2(ra)x 2 + (2(ra)+(rb))x + rb = r(2ax 2 +(2a + b)x + b) = r((2ax + b)(x + 1)) = r((ax 2 + bx + c) (x + 1)) F ra 1 rb 1 rc 1 = rf a 1 b 1 c 1 F F 2: F ax 2 + bx + c x +1 (ax 2 + bx + c) (x +1)=(2ax + b)(x +1)=2ax 2 +(2a + b)x + b F a F b = c 2a 2a + b b = 2 0 0 2 1 0 0 1 0 F a b c 3: F R 3 2 R 2 1 F 1 : R 3 R 2
3 6 F 2 : R 2 R 3 x +1 P (x),q(x) r (P (x)+q(x)) = P (x)+q (x), (rp(x)) = rp (x) (x + 1)(P (x)+q(x)) = (x +1)P(x)+(x +1)Q(x), (x + 1)(rP(x)) = r(x +1)P(x) 2 F 1 (a + b) =F 1 (a)+f 1 (b), F 1 (ra) =rf 1 (a) 2 F 2 (a + b) =F 2 (a)+f 2 (b), F 2 (ra) =rf 2 (a) F F 1 F 2 F 2 F 1 F (2) 1: A (1) 1T A = F A 2 0 0 A = 2 1 0 0 1 0 A XA = E 3 X A A =(a b 0) 3 XA =(Xa Xb X0) =(Xa Xb 0) X XA 3 0 E 3 0 XA = E X A
3 7 2: F (1) 3 F = F 2 F 1 F 1 0 0 0 (( )) 0 0 F 0 = F 2 F 1 0 = F 2 = 0 0 c c 0 F 0 F T A = F A T A 1 F F A 1 8 1.1... 8 1.1.1... 8 1.1.2... 12 1.1.3... 13 1.2... 14 1.2.1... 14 1.2.2 1 :......... 14 1.2.3 2 :... 15 2 16 2.1... 16 2.2... 17 2.2.1... 17 2.2.2... 19 2.2.3... 20 2.3... 21 2.3.1... 21 2.3.2... 22 2.3.3... 23 2.4... 24 2.4.1... 24 2.4.2... 24 2.4.3... 26 2.4.4... 26
3 8 1 1.1 1.1.1 x y a y = ax a x y a x y a x y u, v x, y, z
3 9 y x x x 1 y y 1 x 2 y 2 x 1 + x 2 y 1 + y 2 x 1 r rx 1 ry 1 y = ax y x y 1 = ax 1,y 2 = ax 2 y 1 + y 2 = a(x 1 + x 2 ),ry 1 = a(rx 1 ) 1 2 3 2 1 2 x y y = f(x) f(x 1 + x 2 )=f(x 1 )+f(x 2 ), f(rx 1 )=rf(x 1 ) f(x) =ax a 1 y = ax x, y, z u, v u = f(x, y, z), v= g(x, y, z) 1 f(x 1 + x 2 )=f(x 1 )+f(x 2 ) f(rx) =rf(x) x 1 0 x 1 + x 2 = rx 1 r x 1 0 x 1 x 2 x 1 + x 2 = rx 1
3 10 f(x 1 + x 2,y 1 + y 2,z 1 + z 2 )=f(x 1,y 1,z 1 )+f(x 2,y 2,z 2 ) f(rx 1,ry 1,rz 1 )=rf(x 1,y 1,z 1 ) g(x 1 + x 2,y 1 + y 2,z 1 + z 2 )=g(x 1,y 1,z 1 )+f(x 2,y 2,z 2 ) g(rx 1,ry 1,rz 1 )=rg(x 1,y 1,z 1 ) (1) u, v x, y, z u, v x, y, z u = f(x, y, z) v = g(x, y, z) 1 g(x, y, z) f(x, y, z) (1) y = z =0 f(rx, 0, 0) = rf(x, 0, 0) x 1 f(x, 0, 0) a 1 f(x, 0, 0) = a 1 x f(0,y,0) = b 1 y, f(0, 0,z)=c 1 z b 1,c 1 (1) f(x, y, z) =f(x +0+0, 0+y +0, 0+0+z) = f(x +0, 0+y, 0 + 0) + f(0, 0,z) = { f(x, 0, 0) + f(0,y,0) } + f(0, 0,z) f(x, y, z) =a 1 x + b 1 y + c 1 z g(x, y, z) =a 2 x + b 2 y + c 2 z
3 11 (1) 1 linear (1) 1 linearity) u v x, y, z u, v x, y, z u, v u, v u R 2 x, y, z x R 3 u x 1 ( u v ) ( = a 1 x + b 1 y + c 1 z a 2 x + b 2 y + c 2 z x, y, z x, y, z ( ) ( ) x u a 1 b 1 c 1 = y v a 2 b 2 c 2 z u = Ax ) (2)
3 12 1.1.2 2 y = ax y = bx y =(a + b)x a b u = Ax u = Bx u = Ax + Bx u x Ax + Bx = = ( ( a 1 x + b 1 y + c 1 z a 2 x + b 2 y + c 2 z ) ( ) a 1 + 1 y + c 1 z a 2x + b 2y + c 2z ) (a 1 + a 1)x +(b 1 + b 1)y +(c 1 + c 1)z (a 2 + a 2)x +(b 2 + b 2)y +(c 2 + c 2)z ( ) a 1 b 1 c 1 a 2 b 2 c 2 ( + a 1 b 1 c 1 a 2 b 2 c 2 ) ( = a 1 + a 1 b 1 + b 1 c 1 + c 1 a 2 + a 2 b 2 + b 2 c 2 + c 2 ) a ra r ( u v ) ( = a 1 x + b 1 y + c 1 z a 2 x + b 2 y + c 2 z ) r ( u v ) ( = ra 1 x + rb 1 y + rc 1 z ra 2 x + rb 2 y + rc 2 z ) r r 2
3 13 r ( ) a 1 b 1 c 1 a 2 b 2 c 2 ( = ra 1 rb 1 rc 1 ra 2 rb 2 rc 2 ) 1.1.3 a b x z z = b(ax) u, v p, q, r, s 4 u, v B p q r s = k 1 u + l 1 v k 2 u + l 2 v k 3 u + l 3 v k 4 u + l 4 v = k 1 l 1 k 2 l 2 k 3 l 3 k 4 l 4 ( 4 2 B (2) 11 p q r s = = k 1 (a 1 x + b 1 y + c 1 z)+l 1 (a 2 x + b 2 y + c 2 z) k 2 (a 1 x + b 1 y + c 1 z)+l 2 (a 2 x + b 2 y + c 2 z) k 3 (a 1 x + b 1 y + c 1 z)+l 3 (a 2 x + b 2 y + c 2 z) k 4 (a 1 x + b 1 y + c 1 z)+l 4 (a 2 x + b 2 y + c 2 z) u v ) (k 1 a 1 + l 1 a 2 )x +(k 1 b 1 + l 1 b 2 )y +(k 1 c 1 + l 1 c 2 )z (k 2 a 1 + l 2 a 2 )x +(k 2 b 1 + l 2 b 2 )y +(k 2 c 1 + l 2 c 2 )z (k 3 a 1 + l 3 a 2 )x +(k 3 b 1 + l 3 b 2 )y +(k 3 c 1 + l 3 c 2 )z (k 4 a 1 + l 4 a 2 )x +(k 4 b 1 + l 4 b 2 )y +(k 4 c 1 + l 4 c 2 )z
3 14 k 1 l 1 k 2 l 2 k 3 l 3 k 4 l 4 ( a 1 b 1 c 1 a 2 b 2 c 2 ) = k 1 a 1 + l 1 a 2 k 1 b 1 + l 1 b 2 k 1 c 1 + l 1 c 2 k 2 a 1 + l 2 a 2 k 2 b 1 + l 2 b 2 k 2 c 1 + l 2 c 2 k 3 a 1 + l 3 a 2 k 3 b 1 + l 3 b 2 k 3 c 1 + l 3 c 2 k 4 a 1 + l 4 a 2 k 4 b 1 + l 4 b 2 k 4 c 1 + l 4 c 2 1.2 1.2.1 1.2.2 1 : AB BA z = By y = Ax y x, y, z x z 1
3 15 1 (2) B = A 2 AB = A(AA) =(AA)A = BA 1 t p(t) t A p(a) p(t) A Ap(A) =A ( a n A n + a n 1 A n 1 + + a 1 A + a 0 E ) = a n A n+1 + a n 1 A n + + a 1 A 2 + a 0 A = ( a n A n + a n 1 A n 1 + + a 1 A + a 0 E ) A = p(a)a p(t) E A E B A B 2 S 1 3 1 3 1.2.3 2 : y x 0 x y x y 0
3 16 y x x y A A 1 A A 1 A 1 A 1 A 1 AB = O A O B O A B 3 (1) a 2 =0 a =0 (2) a 2 =1 a 2 1=(a 1)(a +1)=0 a = ±1 (3) a 2 = a a 2 a = a(a 1) = 0 a =0, 1 A = O 3 2 2.1 1
3 17 1 0 0 1 0 0 1 1 1 1 2.2 2.2.1 X Y X X x X x X X x X Y X Y X Y
3 18 F : X Y X F Y F X Y X Y X x F Y x F F (x) F (x) Y F (x) Y y y = F (x) F : x y 1. 2. 3. X Y F : X X X X X X
3 19 2.2.2 Z Y Z G: Y Z F : X Y x X F (x) Y G G(F (x)) Z X x Z G(F (x)) X Z F G G F G(F (x)) X Y F (x) F (x)f G G F : X Z G G G(F (x)) G F (x) =G(F (x)) G F F G G F F G G F
3 20 G F O.K. 2.2.3 X Y F : X Y F X Y x 1 x 2 = F (x 1 ) F (x 2 ) F Y X y Y F (x) =y x X F : X Y Y X F Y Y Y X F F 1 : Y X F 1 F (x) =y F 1 (y) =x F 1 F X X F F 1 Y Y G: Y X G F X X F G Y Y G = F 1
3 21 2.3 2.3.1 n n C n n R n C n = R n = x 1 x 2. x n x 1 x 2. x n x 1,x 2,...,x n C x 1,x 2,...,x n R X = C n,y= C m X = R n,y= R m X = C n,y= R m X = R n,y= C m F : X Y F. x, y X r X = C n X = R n F (x + y) =F (x)+f (y), F(rx) =rf (x) C n C m X R n Y R m R n R m R n = R m n = m R n = R m
3 22 X X X = R n I I : R n R n, I(x) =x 2.3.2 F : R n R m G: R m R l F R m G R m G F G F G F G F (x + y) =G F (x)+g F (y), G F (rx) =r (G F (x)) x, y R n r R G F (x) =G(F (x)) F G G F G F (x + y) =G(F (x + y)) = G(F (x)+f(y)) = G(F (x)) + G(F (y)) = G F (x)+g F(y) G F (rx) =G(F (rx)) = G(rF(x)) = r (G(F (x)) = r (G F (x)) F R n R m G R m R l F G R n R l
3 23 2.3.3 F : R n R m F 1 F 1 F 1 (x + y) =F 1 (x)+f 1 (y), F 1 (rx) =rf 1 (x) x, y R n r R F F (F 1 (z)) = z F ( F 1 (x)+f 1 (y) ) = F ( F 1 (x) ) + F ( F 1 (y) ) = x + y F 1 F 1 (F (w)) = w F 1 (x)+f 1 (y) =F 1 ( F ( F 1 (x)+f 1 (y) )) = F 1 (x + y) F ( rf 1 (x) ) = r ( F ( F 1 (x) )) = rx rf 1 (x) =F 1 ( F ( rf 1 (x) )) = F 1 (rx) F F F n = m
3 24 2.4 2.4.1 (m, n) R n n m (m, n) R n R m A (m, n) R n R m T A 3 T A : R n R m, T A (x) =Ax T A T A (x + y) =A(x + y) =Ax + Ay = T A (x)+t A (y) T A (rx) =A(rx) =r(ax) =rt A (x) (m, n) A, B T A T B T A T B x R n Ax Bx x A B (i, j) a, b Ae j i a Be j j b Ae j Be j e j j 1 0 R n R m (m, n) 2.4.2 F : R n R m T A = F (m, n) A 3 A T A
3 25 F T A = F F F : R n R m F (m, n) O.K. A F = T A F (x) =Ax e 1,...,e n Ae j A j A F (e 1 ),...,F(e n ) A =(F (e 1 ) F (e 2 ) F (e n )) A F = T A x R n x i x i x = x 1 x 2. x n = x 1 F 1 0. 0 0 + + x. n 0 1 = x 1e 1 + x 2 e 2 + + x n e n F (x) =F (x 1 e 1 + x 2 e 2 + + x n e n ) = x 1 F (e 1 )+x 2 F (e 2 )+ + x n F (e n ) = x 1 (Ae 1 )+x 2 (Ae 2 )+ + x n (Ae n ) = A(x 1 e 1 + x 2 e 2 + + x n e n ) = Ax F = T A
3 26 (m, n) R n R m A T A 2.4.3 F : R n R m G: R m R l G F : R n R l (m, n) A (l, m) B (l, n) C F = T A, G = T B, G F = T C C = BA T B T A = T BA T B T A (x) =T B (T A (x)) = T B (Ax) =B(Ax) =(BA)x = T BA (x) 2.4.4 F : R n R n F 1 : R n R n n A, B F = T A, F 1 = T B
3 27 A B F F 1 = F 1 F = I E I T A T B = T B T A = T E T A T B = T AB, T B T A = T BA T AB = T BA = T E AB = BA = E B = A 1 A T A T A 1 = T AA 1 = T E = I, T A 1 T A = T A 1 A = T E = I T A T A 1 4 (2) 2