.
10. .
(45) , , j .
20. a ¹ 0, b ¹ 0, .
48. , .
^ .
30. ^ , a ¹ 0, b ¹ 0, (a )^ (b ).
40. ^ ^ + ¹ 0, ^ ( + ).
50. n, , , .
50. (n 1)- n.
.
30 40 , L n. n , , , L . , Î L Û (, ) = 0 (*). n . = (1, 2, , n), = (1, 2, , n). Î L Û × × = 0 (**). (**) n . (n 1) . , (**) (n 1)-.
n. , , . Î Û (, ) = 0 Î . .
60. (n )- n.
50.
70. Ç = { 0 }.
80. .
. ^ . . , , .. a, b , a × + b × = 0. a ¹ 0, . a × 2 + b × (, ) = 0. (, ) = 0 2 ¹ 0, a = 0. . , .
90. 1, 2, , 1, 2, , s , ⠠ 1, 2, , , 1, 2, , s .
42. (1 £ £ n) n = Å .
. (1, 2,..., ) ( +1, + 2,..., n) . 90 , (1, 2,..., , +1, + 2,..., n) . n , n. , n = + . 70 n = Å .
|
|
n = Å . n, = 1 + 2, 1 Î , 2 Î . 1 . 2 .
7.4.
51. = (1, 2,..., n) n , .
. 1 7.2 . .
, , . , , .
43. .
. = (1, 2,..., n) n. . , .
11 = 1. 2 ^ 1, 21 = 2. 2 1. a , 21 = a 1 + 2 11. 21 ¹ 0, , (11, 21 ) = 0, .. (1, a 1 + 2) = 0. a 12 + (1, 2) = 0. 1 ¹ 0. 11 21 , . 31 堠 31 = a1 11 + a2 21 + 3. , 31 11 21, , (11, 31) = (21, 31) = 0.
( 43) . , |
31 . 11, 21, 31 , . 11, 21, , n11 , n1 n1 = b1 × 11 + b2 × 21 + + bn1 × n11 + n. n1 , b1, b2, , bn 1 (11, n1) = (21, n1) = = (n11, n1) = 0. , . , 1 = (11, 21,..., n1) . , .
44. .
, (, ) =1, (, s)= 0, ¹ s.
. (1, 2,, n), ½ ½= (47).
45. .
. ( )
= . | = (1, 2,..., n) . . (43) = × ×. 1 = | |×| |2. | |2 > 0, | | > 0. |
< 1, 2,..., > n , . , , .
|
|
. .
1. = | , . |
2. = | , . |
3. = | , | | = 81 < 0, . |
4. D = | D , , | D | = 5 > 0, = 7 > 0. , D . |
46. = (1, 2, , n) = (1, 2, , n), (, ) = 11 + 22 + + nn (48).
. ,
(, ) = × × = × = (1, 2, , n) × = 11 + 22 + + nn.
. 4 1 = (2, 1, 1, 2) 2 = (3, 2, 5, 1). L = < 1, 2 >.
. L ^, Î L ^ Û (1, ) = (2, ) = 0. = (1, 2, 3, 4). , (1, ) = 2 1 + 2 + 3 +2 4, (2, ) = 3 1 +2 2 5 3 + 4. , Î L ^ Û , ,
= ( 1 32, 1 8 2, 1, 7 2), 1, 2 .
, L ^ - ,
1 = (1, 1, 1, 0), 2 = (3, 8, 0, 7), .. L ^ = < 1, 2 >.
7.5.
52. 1 , , 1 , 1 (, ) = (1, 1).
47. 1 , dim E = dim E 1.
. Þ 1 . . , dim E = dim E 1.
Ü dim E = dim E 1 = n. 1 = (1, 2,..., n) 1 = (11, 21,..., n1) . 堠 j: 1 . Î = 1 1 + 2 2 + + n n, j () = 1 11 + 2 21 + + n n1. , , 1. , . Î = 1 1 + 2 2 + + n n . ࠠ j () = 1 11 + 2 21 + + n n1 . , (,)= 11 + 22 + + nn. 1 , (j (), j ()) = 11 + 22 + + nn. , (,) = (j (), j ()). , j - 1.
. , .
VIII.
, . : . : . .
|
|
8.1.
53. j ,
(, ) = ( j (), j ()) (49)
.
j .
10. | | = | j ()| .
| | = = = | j ()|.
20. = .
30. n , = (1, 2,..., n) , j . j () = ×, j () = ×, ; (, ) = × ×, (j (), j ()) = ( ×)× ×( ×) = ×( × × )× . (, ) = (j (), j ()), = × × . ,
= × × (50)
. n , , (50), .
= (1, 2,..., n) , = (50)
× = , = 1.
54. ,
= 1 (51).
48. , .
30 53.
49. , ( ) 1, ( ) .
51.
50. , .
. Þ j: n n = (1, 2,..., n) n. , . j () = × . ,
= . | j () = 11 + 22 + + nn. , (j ())2 = = ()2 = 1, .. j () . (j (), j ()) = 1 ×1 + 2 ×2 + + n × n = (, ) = 0, |
¹ , .. j () ^ j (). j () , , .. j () . , j () .
Ü j () . 1 = (j ())2 = 0 = (j (), j ()) = 1 ×1 + 2 ×2 + + n ×n ¹ . , 49, .
51. , | | = 1.
, , × = . | × | = | |, | |×| | = 1, | |×| | = 1, | |2 = 1, | | = 1.
52. 1 (1).
|
|
. l j.
(1, 2, , n), × = l ×, j. , . . . , . , . , .. | l |2 = 1. , 1 (1).
53. , .
. n (, ) = (j (), j ()) = (l1 , l2 ) = = l1 l2 (, ). l1 ¹ l2, , , l1 =1, l2 = 1. , (, ) = (, ). (, ) = 0. , .
8.2.
j - n.
55. j *: n n j, n
(, j ()) = (j * (), ) (52)
54. = 1×( *)× .
n = (1, 2,..., n), , j * j *. , , 1 * , , j () j * ()) , (, j ()) = × × 1, (j * (), ) = ( *) × × . (52), × × 1 = * × × . ( (36)), 1 = ×, * = * × . :
× ×( × ) = ( * ×)× ×, ×( × )× = ×(( *)× )× . × = ( *)× ,
= 1×( *)× (53)
1. * = 1× × à (54).
. (53) , ( *) = × × 1, * = ( 1)× × . , = . , * = 1× × .
2. .
53 54.
3. , * = .
, = .
. = (1, 2, 3, 4) 4 = . = j . .
. , . (54). * = 1× × . 1. , 1 = . ,
* = × × = .
55. L n j, L ^ j *.
. Î L, Î L ^. j () Î L , (, j ()) = 0. (, j ()) = (j * (), ). , (j * (), ) Î L. , j * () Î L ^ Î L ^. , L ^ j.
8.3. ()
56. , (j - Û j = j *).
(52) , j , (, j ()) = (j (), ) (55).
n = (1, 2,..., n), j. * = (53) : = 1× × . , = , .. . ,
56. , .
56 , .
.
|
|
10. .
, . .
20. .
, .
30. .
, , .
40. j j 1, j 1 .
.
50. j y , , , .. j × y = y × j.
, j y (j × y)* = y * × j * = y j.
j × y ( ), (j × y)* = j × y. , j × y , j × y = y j.
60. j . j, ^ j.
j *.
57. .
. n , j . . l0 | l |. ( l0)× = 0 , . . = (1, 2, , n) . = l0. , × × = l0 × × (*). ×, × = . , × , . , , , × × . × × . , .. × × = ( × ×) = × × = × × . . , , , .. × × = . , × × . (*) , l0 .
1. .
2. .
58. , , .
. j n, l1, l2, = l1 ¹ l2. , l1 l2 . j () = l1 , j () = l2 . (, j ()) = (j (), ). ࠠ (, l2 ) = (l1 , ), l2 (, ) = l1 (, ), (l1 l2)×(, ) = 0. l1 ¹ l2, (, ) = 0. , ^ .
59. j: n n , n , j.
. Ü = (1, 2,..., n) , j. j () = l ek = 1, 2, , n. , j , , .. j .
Þ j: n n . . n = 1 , . . .
, (n 1). n n. 2 54 j . 1 , l1. , , . 1 = < 1 >. , j (1) = 1. 1 j. dim 1 = n 1, 1 , j. (2,..., n). (1, 2,..., n) n.
.
60. .
. . Ÿ j = (1, 2,..., n) n. 56 n 1 = (11, 21,..., n1), j. 1 j 1, 1 . 1. 1 = 1× × .
.
= . | . = (l 1)3×(l + 3) |
l1 = l2 = l3 = 1, l4 = 3. . .
l = 1
1, 3- , , 1 = (1, 1, 0, 0), 2 = (1, 0, 1, 0), 3 = (1, 0, 0, 1). . . , = . |
:
11 = 21 = , 31 = .
l = 3
3, 1- , , 4 = (1, 1, 1, 1). , 41 = (1, 1, 1, 1). 11, 21, 31, 41 |
. 1
= . | 1 = . |
= 1 1 = × × .
.
62. j: n n j * j jj * , .. , .
. (j * j)* = j * (j *) * = j * j. , j * j . l j * j . ((j * j)(), ) = (j * (j (), ) = (j (), j ()) = =½ j ()½2. , ((j * j)(), ) = (l , ) = l ½ ½2. , ½ j ()½2 = l ½ ½2. ¹ 0, l ³ 0. jj * .
57. , (, j ()) > 0.
61. , .
. Þ j . (, j ()) > 0 , . , j () = l . , 0 < (, l ) = l (, ). (, ) > 0, l > 0.
Ü j . n = (1, 2,..., n), . = 1 1 + 2 2 + + n n, (, j ()) = ( = . ¹ 0, (, j ()) > 0, .. j .
. , (, j ()) ³ 0.
, (, j ()) < 0, (, j * j ()) = (j (), j ()) ³ 0. , j * j .
IX.
9.1.
Ln n- f Ln (f: Ln ).
58. f: Ln , Ln.
= (1, 2,..., n) Ln, Ln, = 1 1 + 2 2 + + n n, 1, 2, , n . f () = a, f () = a11 + a22 + + an n.
, a11 + a22 + + an n.
, f: Ln .
9.2.
Ln n- .
59. f: (Ln ´ Ln) ( ), Ln : 2018-10-15; !; : 232 | : , , . , .
<==
|
==>
| v- VI. . .
==> ...