Документ взят из кэша поисковой машины. Адрес оригинального документа : http://matematika.phys.msu.ru/files/a_stud_gen/216/Lin_Alg_Zach.pdf Дата изменения: Fri Feb 19 17:53:32 2010 Дата индексирования: Tue Oct 2 00:38:17 2012 Кодировка:

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1. A. , : ; ; ; . ; 11323 1.1. A = 2 1 4 3 5 . 31547 1 1 3 2 -3 1.2. A = 2 1 4 3 -4 . 2 3 8 5 -8 1132 0 1.3. A = 2 3 8 5 -1 . 1 2 5 3 -1 2. R3 x1 , x2 , x3 , x4 . , : L(x1 ,x2 ,x3 ,x4 ); L(x1 ,x2 ,x3 ,x4 ); x1 , x2 , x3 , x4 . 2.1. x1 = (1, 0, 1)T , x2 = (5, 2, 3)T , x3 = (2, 1, 1)T , x4 = (4, 1, 3)T . 2.2. x1 = (1, 1, 1)T , x2 = (5, 3, 7)T , x3 = (5, 4, 6)T , x4 = (2, 1, 3)T . 2.3. x1 = (1, 0, 2)T , x2 = (3, 2, 8)T , x3 = (1, 1, 3)T , x4 = (2, -1, 3)T . 3. R3 x1 , x2 , x3 , x4 . , : L(x1 ,x2 ,x3 ,x4 ); L(x1 ,x2 ,x3 ,x4 ); L(x1 ,x2 ,x3 ,x4 ) R3 . 3.1. x1 = (1, 0, 1)T , x2 = (5, 2, 3)T , x3 = (2, 1, 1)T , x4 = (4, 1, 3)T . 3.2. x1 = (1, 1, 1)T , x2 = (5, 3, 7)T , x3 = (5, 4, 6)T , x4 = (2, 1, 3)T . 3.3. x1 = (1, 0, 2)T , x2 = (3, 2, 8)T , x3 = (1, 1, 3)T , x4 = (2, -1, 3)T . 4. P2 (R) ( 2) x1 , x2 , x3 , x4 . , : L(x1 ,x2 ,x3 ,x4 ); L(x1 ,x2 ,x3 ,x4 ); x1 , x2 , x3 , x4 . 4.1. x1 (t) = 1 + t2 , x2 (t) = 5 + 2t +3t2 , x3 (t) = 2 + t + t2 , x4 (t) = 4 + t +3t2 t R. 4.2. x1 (t) = 1 + t + t2 , x2 (t) = 5 + 3t +7t2 , x3 (t) = 5 + 4t +6t2 , x4 (t) = 2 + t +3t2 t R. 4.3. x1 (t) = 1 + 2t2 , x2 (t) = 3 + 2t +8t2 , x3 (t) = 1 + t +3t2 , x4 (t) = 2 - t +3t2 t R. 5. P2 (R) ( 2) x1 , x2 , x3 , x4 . , : L(x1 ,x2 ,x3 ,x4 ); L(x1 ,x2 ,x3 ,x4 ); L(x1 ,x2 ,x3 ,x4 ) P2 (R). 5.1. x1 (t) = 1 + t2 , x2 (t) = 5 + 2t +3t2 , x3 (t) = 2 + t + t2 , x4 (t) = 4 + t +3t2 t R. 5.2. x1 (t) = 1 + t + t2 , x2 (t) = 5 + 3t +7t2 , x3 (t) = 5 + 4t +6t2 , x4 (t) = 2 + t +3t2 t R. 5.3. x1 (t) = 1 + 2t2 , x2 (t) = 3 + 2t +8t2 , x3 (t) = 1 + t +3t2 , x4 (t) = 2 - t +3t2 t R. 6. L e1 , e2 , e3 . [x1 ], [x2 ], [x3 ], [x] x1 , x2 , x3 , x e. , x1 , x2 , x3 L. x x1 , x2 , x3 . 6.1. [x1 ] = (-1, 0, 1)T , [x2 ] = (-1, 1, 1)T , [x3 ] = (-1, 1, 0)T , [x] = (1, 1, 1)T . 6.2. [x1 ] = (-1, 1, -1)T , [x2 ] = (0, -1, 1)T , [x3 ] = (-1, 1, 0)T , [x] = (1, 0, 2)T . 6.3. [x1 ] = (-1, 0, -1)T , [x2 ] = (0, -1, 0)T , [x3 ] = (-1, 2, 0)T , [x] = (2, 0, -1)T . 7. L e1 , e2 , e3 .


2 [f1 ], [f2 ], [f3 ], [f1 ], [f2 ], [f3 ] f1 , f2 , f3 , f1 , f2 , f3 e. , : f1 , f2 , f3 L; f1 , f2 , f3 L. : f f ; f f . [x](f ) x f . [x](f ) x f . 7.1. [f1 ] = (1, 2, 1)T , [f2 ] = (1, 1, 0)T , [f3 ] = (1, 1, 2)T , [f1 ] = (0, -1, 1)T , [f2 ] = (1, 0, 1)T , [f3 ] = (0, -1, -1)T , [x](f ) = (1, 1, 1)T . 7.2. [f1 ] = (1, 1, 1)T , [f2 ] = (1, 2, 1)T , [f3 ] = (1, 1, 2)T , [f1 ] = (-1, 0, -2)T , [f2 ] = (0, -1, 1)T , [f3 ] = (0, 1, 0)T , [x](f ) = (1, 0, 2)T . 7.3. [f1 ] = (2, 1, 1)T , [f2 ] = (1, 2, 1)T , [f3 ] = (1, 1, 1)T , [f1 ] = (-3, -2, -2)T , [f2 ] = (-1, -2, -1)T , [f3 ] = (0, 3, 1)T , [x](f ) = (2, 0, -1)T . 8. L e1 , e2 , e3 . : [A](e) A e; (e, e ) e e . [A](e ) A e . 120 -1 -1 -1 1 1 . 8.1. [A](e) = 0 1 1 , (e, e ) = 0 201 1 1 0 110 -1 0 -1 8.2. [A](e) = 0 1 1 , (e, e ) = 1 -1 1 . 301 -1 1 0 103 -1 0 -1 8.3. [A](e) = 0 1 1 , (e, e ) = 0 -1 2 . 101 -1 0 0 9. L e1 , e2 , e3 . [A] A e. , : A; A. 112 9.1. [A] = 2 3 5 . 123 137 9.2. [A] = 1 1 3 . 125 102 9.3. [A] = 1 1 3 . 215 10. L e1 , e2 , e3 . [A] A e. , : A; A. 112 10.1. [A] = 2 3 5 . 123 137 10.2. [A] = 1 1 3 . 125 102 10.3. [A] = 1 1 3 . 215 11. L e1 , e2 , e3 . [A] A e. A. : ; -


3 ; ( . ); 201 11.1. [A] = -1 1 -1 . 001 2 -1 0 1 0 . 11.2. [A] = 0 -1 1 1 2 0 -1 11.3. [A] = -2 1 2 . 001 12. L e1 , e2 , e3 , e4 . Q e. Q e. , Q : Q e ; e e ; e e. 12.1. Q(x) = (x1 )2 +2x1 x2 +2(x2 )2 + x3 x4 . 12.2. Q(x) = x1 x2 +2(x3 )2 - 4x3 x4 +2(x4 )2 . 12.3. Q(x) = (x1 )2 +4x1 x2 - (x2 )2 - x3 x4 . 13. L e1 , e2 . F1 , F2 , F3 e. L? L, L? . 13.1. F1 (x, y ) = 3x1 y 1 +2x1 y 2 +2x2 y 1 +2x2 y 2 , F2 (x, y ) = 2x1 y 1 + x1 y 2 - x2 y 1 +3x2 y 2 , F3 (x, y ) = 1 y 1 +3x1 y 2 +3x2 y 1 +4x2 y 2 . x 13.2. F1 (x, y ) = x1 y 1 + x1 y 2 + x2 y 1 +2x2 y 2 , F2 (x, y ) = 2x1 y 1 +2x1 y 2 +3x2 y 1 +4x2 y 2 , F3 (x, y ) = x1 y 1 +2x1 y 2 +2x2 y 1 + x2 y 2 . 13.3. F1 (x, y ) = 2x1 y 1 + x1 y 2 + x2 y 1 + x2 y 2 , F2 (x, y ) = 8x1 y 1 +2x1 y 2 + x2 y 1 + x2 y 2 , F3 (x, y ) = 2x1 y 1 +2x1 y 2 +2x2 y 1 + x2 y 2 . 14. R2 ( (x, y ) = x1 y 1 + x2 y 2 ) f1 , f2 . , f1 , f2 R2 . f . 14.1. f1 = (1, 1)T , f2 = (0, 1)T . 14.2. f1 = (-1, 1)T , f2 = (1, 0)T . 14.3. f1 = (1, 2)T , f2 = (2, 1)T . 15. P1 [0, 1] ( [0, 1] 1; (x, y ) =
1

: e1 (t) = 1, e2 (t) = t t [0, 1]. , e1 , e2 P1 [0, 1] . e. 16. R3 ( (x, y ) = x1 y 1 + x2 y 2 + x3 y 3 ) x1 , x2 , x3 . , x1 , x2 , x3 . x1 , x2 , x3 ­ ( ). 16.1. x1 = (1, 1, 1)T , x2 = (1, 0, 0)T , x3 = (0, 1, 0)T . 16.2. x1 = (1, 0, 1)T , x2 = (0, 0, 1)T , x3 = (1, 1, 0)T . 16.3. x1 = (1, 1, -1)T , x2 = (0, 1, 0)T , x3 = (0, 0, 1)T . 17. P1 [0, 1] ( [0, 1] 1; (x, y ) =
1 0

0

x(t)y (t) dt)

x(t)y (t) dt)


4 : x1 (t) = 1, x2 (t) = t t [0, 1]. , x1 , x2 . x1 , x2 ­ ( ). 18. H e1 , e2 , e3 , e4 . [x1 ], [x2 ], x x1 , x2 , x e. : x L(x1 ,x2 ), x L(x1 ,x2 ). 18.1. [x1 ] = (1, 0, 0, 1)T , [x2 ] = (1, 1, 0, 0)T , [x] = (1, 0, 0, 0)T . 18.2. [x1 ] = (1, 0, 0, -1)T , [x2 ] = (1, 0, 2, 0)T , [x] = (1, 0, 0, 0)T . 18.3. [x1 ] = (1, 1, 0, 1)T , [x2 ] = (1, 0, 0, 0)T , [x] = (1, -1, 0, 0)T . 19. H e1 , e2 , e3 , e4 . [x1 ], [x2 ] x1 , x2 e. P L(x1 ,x2 ) e. 19.1. [x1 ] = (1, 0, 0, 1)T , [x2 ] = (1, 1, 0, 0)T . 19.2. [x1 ] = (1, 0, 0, -1)T , [x2 ] = (1, 0, 2, 0)T . 19.3. [x1 ] = (1, 1, 0, 1)T , [x2 ] = (1, 0, 0, 0)T . 20. H e1 , e2 , e3 . Q x1 - 2x2 +3x3 = 0. Q. . 21. H e1 , e2 , 310 e3 . A e: [A](e) = 1 3 0 . 004 : e A; e e ; e e; A e . 22. H e1 , e2 , 310 e3 . A e: [A](e) = 1 3 0 . : 002 e , A ; e e ; e e; A e . 23. E 2 O e1 , e2 . 2(x1 )2 - 4x1 x2 + 5(x2 )2 + 8x1 - 2x2 + 9 = 0. : ; «» «»; «» «»; . 24. E 2 O e1 , e2 . 2(x1 )2 - 4x1 x2 + 5(x2 )2 + 8x1 - 2x2 + 9 = 0. , .