高等代数习题线性变换.pdf

1、 1. R 3 s, t s (x 1 , x 2 , x 3 ) (x 1 , x 2 , x 1 x 2 ) t (x 1 , x 2 , x 3 ) (x 1 x 2 x 3 , 0, x 3 x 1 x 2 ) (1) st, ts, s 2 (2) s t, s t, 2s : (1) = - - - + = ) , 0 , ( ) , , ( 2 1 3 3 2 1 3 2 1 x x x x x x x x x s st , ( 3 2 1 x x x - + 0, ) , , ( ) 3 2 1 3 2 1 x x x x x x t = - + , t st = . ) 0

2、, 0 , 0 ( ) , , ( ) , , ( 2 1 2 1 3 2 1 = + = x x x x x x x t ts , 0 = ts ) , , ( ) , , ( 2 1 2 1 3 2 1 2 x x x x x x x + =s s = ) , , ( 2 1 2 1 x x x x + . s s = 2 (2) ) , , )( ( 3 2 1 x x x t s + = ) , , ( 3 2 1 x x x s + ) , , ( 3 2 1 x x x t ) , , ( 2 1 2 1 x x x x + = + ) , 0 , ( 2 1 3 3 2 1 x

3、x x x x x - - - + ) , , 2 ( 3 2 3 2 1 x x x x x - + = ) , , )( ( 3 2 1 x x x t s - = ) , , ( 3 2 1 x x x s ) , , ( 3 2 1 x x x t - ) , , ( 2 1 2 1 x x x x + = ) , 0 , ( 2 1 3 3 2 1 x x x x x x - - - + - = ) 2 2 , , ( 3 2 1 2 3 2 x x x x x x - + + - 2 ) , , ( 2 3 2 1 = x x x s ) , , ( 2 1 2 1 x x x x

4、 + = ) 2 2 , 2 , 2 ( 2 1 2 1 x x x x + 2. : s V , L , , 2 1 a a , m a V L + + ) ( ) ( 2 2 1 1 a s a s k k 0 ) ( = m m k a s . 1 - s : L + 1 1 a k 0 = m m k a . L , , 2 1 a a , m a , : L = = 2 1 k k = 0 = m k . L ), ( ), ( 2 1 a s a s , ) ( m a s 3. s V (1) s 0 (2) l s l 1 s 1 : (1) s 0, x x V, s (x

5、)=0 x = 0 . s , s -1 (s (x )=0, x = 0. (2) l s , (1) l s (x )=lx ,x s -1 (s (x )=ls -1(x ). s -1(x )= l 1 x , 4. 2 1 ,l l n A 2 1 , X X 2 1 ,l l 2 1 X X + A 2 2 2 1 1 1 , X AX X AX l l = = 2 1 X X + A ) ( ) ( 2 1 2 1 X X X X A + = + l l A 112212 XXXX llll +=+ 0 ) ( ) ( 2 2 1 1 = - + - X X l l l l 2

6、1 , X X 0 , 0 2 1 = - = - l l l l 2 1 l l l = = 2 1 l l 2 1 X X + A 5. B A, n AB BA A A AB A BA ) ( 1 - = 11 ()() EBAEAABAAEABAEAB llll - -=-=-=- AB BA A A n 0 t 0 t t 0 -tE A tE A- ) ( tE A B - B tE A ) ( - B tE A E tE A B E ) ( ) ( - - = - - l l tB AB E tB BA E + - = + - ) ( ) ( l l l t n 0 t t t

7、n t 0 = t AB E BA E - = - l l AB BA 6. . x x x x x x x x x x x f 11 6 4 9 2 3 2 2 ) ( 2 4 5 6 8 9 10 11 12 + - + - + - - - + = - = 0 1 0 1 1 0 2 0 1 A 1 A 2 1 - A 3 ) (A f 4 ) (A f 1 0 ) 1 )( 1 ( 1 0 1 1 0 2 0 1 2 = - + - = - - + - - = - l l l l l l l A E 2 5 1 , 1 3 2 1 - = = l l A 2 1 - A 2 5 1 ,

8、2 5 1 1 , 1 3 2 2 1 1 - = + = = = m l m l m 3 ) (A f - = + = 1 6 0 6 5 0 12 0 7 6 ) ( E A A f 4 ) (A f 5 3 2 , 5 3 2 , 7 3 2 1 - - = + - = = l l l 7. = 1 2 2 2 1 2 2 2 1 A 100 A 2 ) 1 )( 5 ( 1 2 2 2 1 2 2 2 1 + - = - - - - - - - - - = - l l l l l l A E A 1 , 5 3 2 1 - = = = l l l 5 1 = l ( ) 1 , 1 ,

9、 1 , 1 3 2 - = = l l ( ) ( ) 1 , 1 , 0 , 1 , 0 , 1 - - - - = 1 1 1 1 0 1 0 1 1 P - - - - = - 1 2 1 1 1 2 1 1 1 3 1 1 P - - = - 1 1 5 1 AP P = - - = - 1 1 5 1 1 5 100 100 100 1 P A P 1 100 100 1 1 5 - = P P A + - - - + - - - + = = - 2 5 1 5 1 5 1 5 2 5 1 5 1 5 1 5 2 5 3 1 1 1 5 100 100 100 100 100 10

10、0 100 100 100 1 100 100 P P A 8. s R n V R l V 0 ) ( = - = a l s a n V W s W V R k W “ , , , b a b l s a l s b a l s n n n ) ( ) ( ) ( ) ( - + - = + - W + b a 0 ) ) ( ) ( ) ( = - = - a l s a l s n n k k W W k , a V 0 ) 0 ( ) ( ) ( ) ( ) ( = = - = - s a s s sa l s a n n x W W ) (a s W V 9. t s , n V

11、) ( ), ( 2 1 x f x f t s , ) ( ), ( 2 1 x f x f 0 ) ( ker( ) ( ker( 2 1 = = s t f f 1 ) ( ), ( ( 2 1 = x f x f Q ) ( ), ( x v x u 1 ) ( ) ( ) ( ) ( 2 1 = + x f x v x f x u e t t t t = + ) ( ) ( ) ( ) ( 2 1 f v f u e t t = ) ( ) ( 1 f u 0 ) ( 2 = t f ) ( ker( 1 t a f “ a a t t = ) ( ) ( 1 f u 0 = a 0

12、 ) ( ker( 1 = t f 0 ) ( ker( 2 = s f 10. M 2 (F) s t s (X) X -1 1 1 1 , t (X) - 0 2 0 1 X, “XM 2 (F) s t, st E 11 , E 12 , E 21 , E 22 s t : - = - + - = + 0 2 1 2 0 2 0 1 1 1 1 1 ) ( 11 11 11 E E E t s = 12 11 2 E E + 22 21 0 2 E E + - - = - + - = + 2 0 0 1 0 2 0 1 1 1 1 1 ) ( 12 12 12 E E E t s = 1

13、2 11 0E E + 22 21 2 0 E E - + = - + - = + 1 1 0 0 0 2 0 1 1 1 1 1 ) ( 21 21 21 E E E t s = 12 11 0 0 E E + 22 21 E E + + - = - + - = + 1 0 0 0 2 0 1 1 1 1 1 ) ( 22 22 22 E E E t s = 12 11 0 0 E E + 22 21 E E - + t s + 22 21 12 11 , , , E E E E - - - = 1 1 2 0 1 1 0 2 0 0 0 1 0 0 1 2 A st 22 21 12 11

14、 , , , E E E E 12 11 11 ) ( E E E + = s , 12 11 12 ) ( E E E - = s , 22 21 12 11 21 0 0 ) ( E E E E E + + + = s , = ) ( 22 E s 22 21 12 11 0 0 E E E E - + + s - - = 1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 1 B B s s : - - = - 2 1 2 1 0 0 2 1 2 1 0 0 0 0 2 1 2 1 0 0 2 1 2 1 1 B t t 11. V s e 1 , e 2 , e 3 A - -

15、 1 2 1 1 0 1 3 6 5 s V s ? : l =2 ( ), l =2 : x 1 = 0 1 3 1, x 2 = - 1 0 3 1 , s 12. s t V s t i st ts q. i V q V : (1) V s(V)t (V) (2) s(V) Kert. : (1) “ x V, x =i (x )=(s t)(x )=s (x )+t (x ). V s (V)+t (V) x s (V) t (V). x =s (x 1 )+ t (x 2 ). x = i (s (x 1 ) = (s t)(s (x 1 ) = s (s (x 1 ) = s (t

16、 (x 2 )= st (x 2 ) = 0 (2 ) s (x )s (V), t(s (x )= 0, s (x )Kert x Kert t(x )= 0 x = (s t)(x )=s (x )+t (x )=s (x ) x s (V) 13. s F n(0) V a 1 , a 2 , , a r , a r 1 , , a n V a 1 , a 2 , , a r Kers s (a r 1 ), , s (a n ) Ims : a 1 , a 2 , , a r Kers , s (a i )=0, i=1,2, , r l r+1 s (a r 1 )+ l r+2 s

17、 (a r 2 )+ + l n s (a n )=0, s ( l r+1 a r 1 + + l n a n )=0, l r+1 a r 1 + + l n a n Kers l r+1 a r 1 + + l n a n =l 1 a1 + + l r a r a 1 , a 2 , , a r , a r 1 , , a n V , l r+1 = = l n =0 s (a r 1 ), , s (a n ) Ims = s (a 1 ), s (a 2 ) , s (a n ) = s (a r 1 ), , s (a n ) 14. R 3 s s (x 1 , x 2 , x

18、 3 ) (x 1 2x 2 x 3 , x 2 x 3 , x 1 x 2 2x 3 ) “(x 1 , x 2 , x 3 )R 3 s s (V) Kers : s ) 0 , 0 , 1 ( 1 = e , ) 0 , 1 , 0 ( 2 = e , ) 1 , 0 , 0 ( 3 = e : - - = 2 1 1 1 1 0 1 2 1 A ) 1 , 0 , 1 ( ) ( 1 = e s , ) 1 , 1 , 2 ( ) ( 2 = e s , ) (n s ) ( ), ( ( 2 1 e s e s L = . ), ( ker x s L = ) 1 , 1 , 3 (

19、 - = x , 1 ker dim = s 15. a 1 ( 1, 0, 2), a 2 (0, 1, 2), a 3 (1, 2, 5) b 1 ( 1, 1, 0), b 2 (1, 0, 1), b 3 (0, 1, 2) x (0, 3, 5) R 3 s R 3 s(a 1 ) (2, 0, 1), s(a 2 ) (0, 0, 1),s(a 3 ) (0, 1, 2) (1) s b 1 , b 2 , b 3 (2) s(x ) a 1 , a 2 , a 3 (3) s(x ) b 1 , b 2 , b 3 : - - = 5 2 2 2 1 0 1 0 1 1 T ,

20、- = 2 1 0 1 0 1 1 1 1 2 T . a 1 , a 2 , a 3 b 1 , b 2 , b 3 : - = - - - - = = - 0 1 0 1 2 1 0 0 1 1 2 2 2 3 4 1 2 1 2 2 1 1 T T T T 3 2 1 1 3 5 3 10 3 11 ) 1 , 0 , 2 ( ) ( a a a a s - + - = - = 3 2 1 2 0 3 2 3 1 ) 1 , 0 , 0 ( ) ( a a a a s + - = = 3 2 1 3 0 0 ) 2 , 1 , 0 ( ) ( a a a a s + + = = s )

21、, , ( 3 2 1 a a a : - - - 0 3 1 3 5 1 3 2 3 10 0 3 1 3 11 s b 1 , b 2 , b 3 : - = = - 2 1 1 1 0 0 0 0 1 1 AT T B - - - 0 3 1 3 5 1 3 2 3 10 0 3 1 3 11 - 0 1 0 1 2 1 0 0 1 = - - - - - 3 1 3 5 3 10 3 1 3 2 3 4 3 1 3 2 3 10 (2) = = ) 5 , 3 , 0 ( x 3 2 1 3 5 3 1 3 5 a a a + - ) (x s ) , , ( 3 2 1 a a a : - - = - 9 26 9 67 9 56 3 5 3 1 3 5 A (2) = ) (x s ) , , ( 3 2 1 a a a - - 9 26 9 67 9 56 =(b 1 , b 2 , b 3 ) -1 T - - 9 26 9 67 9 56 ) (x s b 1 , b 2 , b 3 : -1 T - - 9 26 9 67 9 56 = - 2 1 1 1 0 0 0 0 1 - - 9 26 9 67 9 56 = - - 9 71 9 26 9 56