, , , det A ¹ 0.
:
, | (1.10.1) |
X , B .
(1.10.2) |
:
(1.10.3) |
A1: .
, , : (A -1∙ A)∙ X = A -1∙ B, E ∙ X = A -1∙ B,
(1.10.4) |
, , A1, A, .
:
:
, , .
: det A =23≠0. det A ¹0, . A 1, A: Aik (i, k = 1, 2, 3) aik A: A 11 = 23, A 12 = 14, A 13 = 11, A 21 = 23, A 22 = 1, A 23 = 9, A 31 = 0, A 32 = 5, A 33 = 1.
, A 1:
A 1 . : , x 1 = 4, x 2 = 2, x 3 = 2. : 2×(4) (2) + 5×2 = 8 + 2 + 10 = 4, 3×(4) (2) + 5×2 = 12 + 2 + 10 = 0, 5×(4) + 2×(2) + 13×2 = 20 4 + 26 = 2.
( ) ,
(1.11.1) |
( , )
(1.11.2) |
, . (1.11.1) (1.11.2) , (1.11.2) .
.
:
x1 2 3 . 1 (2) 2, 1 (3) 3 :
. 3 x2 2 3:
x3 = 1, x 2 = (56 + x 3)/11 = (56 1)/11 = 55/11 = 5, x 1 = 22 +4 x 2 3 x 3 = 22 + 4×5 3×(1) = 1.
:
2 (2), 1 3 , 2 (5), 3 3 . x2 3 , 2 (2) 3 :
. , , .. .
:
|
|
x1 2 3 . 1 (1) 2, 1 (4) 3 :
2 3 , :
. . x 3 = 13 k, k . x 2 = (16/13) x 3 = 16 k, x 1 = 3 x 2 5 x 3 = 17 k.
, .
(1.11.3) |
(1.8.1) m ´(n +1), .
.
, .
1. AB 2 . AB 3 .
.
, .
2. 7/5 .
.
, . x 4 ,
,
, x 4 .
:
- ;
- , ,
, ;
- , .. .
1.12. m n .
(1.8.1), m £ n, .. . (1.8.1) .
( ).
, .
, . (1.8.1) (1.11.2), . , . (. 1.11) , (1.11.2) , , , . , , (1.11.2), (1.8.1) ( r).
.
, .. . , , . , .
|
|
:
, .. r = n, .
, .. r<n, (1.8.1) .
(1.8.1) . 1.1.
. 1.1.
1.13. n
, , . n .
n n ; , , .
n (), () .
n , , .. ,
n , , ..,
n l , l , .., ,.
n :
;
;
;
;
;
, .
, ;
.
, , , .
() , , , , .. . , , , .
x = y =
(1.16.1) |
:
1. .
2. .
3. α.
4. , x , , x .
() , , ( ), .
() x :
(1.16.2) |
:
1. , .
2. , λ .
3. .
4. .
φ x
(1.16.3) |
, .
, . , ,
e1, e2, , en n , , .. (ei,ej) = 0 .
|
|
n .
1.
2. :
3. (1, 2), (2, 3), (4, 5).
4. (1, 1), (3, 3), (0, 0) ?
5. , (3, 2) (1, 3).
6. (x 1, y 1), (x 2, y 2 ), (x 3, y 3 ).
7. , (x1, y1), (x2, y2 ), ( x3, y3 )
8. , ABCD (x1, y1), (x2, y2 ), ( x3, y3 ), ( x4, y4 )
9.
10. :
11. +?
12. :
13.