. .
7
.
.
( )
.
.
n , (x1,x2, xn) 1. :
a1x1 + a2x2 + ajxj + anxn = b, (*)
= b.
aj, j = 1,,n, b (). aj ( ), b - .
(*) (, ,, ) , (.. xj j 1 n . , n n n . , 0, (*) . b = 0, n .
m (*), .. m n . i - , i = 1,2,,m, ai1, ai2, , ain bi, ..
ai1x1 + ai2 x2 + + aij xj + + ain xn = bi.
m n :
a11x1 + a12 x2 + + a1j xj + + a1n xn = b1
a21x1 + a22 x2 + + a2j xj + + a2n xn = b2
ai1x1 + ai2 x2 + + aij xj + + ain xn = bi (1)
am1x1 + am2 x2 + + amj xj + + amn xn = bm
, ,
= bi, i = 1,,m.
, (1) , ..
= 0, i = 1,,m, (10)
- . (10) (1).
(1) (, ,, ) , (1) (.. xj , j = 1,,n) , .. = bi i = 1,,m.
(1) , . .
(1) Xb (X0, ). , Xb = .
, , (1), , , . , . , - , .
|
|
, - , .
, . , , . .
(1) , .
1. (1) k, 1 k m, , akj = 0 j,
1) bk 0, (1) ;
2) bk = 0, k - (m 1) - n , .
, , k
0x1 + 0 x2 + + 0 xj + + 0 xn = bk.
2. (1) , , , .
. , , (1) . (1), , , ,
= b1 + b2.
, - (, ,, ) (1), (1). , (x1,x2,,xj, ,xn) (1) (1), (1) (1) , (1) (1), (- ).
.
2. (1) , , (1) .
2 2 , (1), :
) ( ) (1) , ;
) ( ) , .
) ) (1).
(1) , , , .
(1) :
x1 | x2 | xj | xn | b | ||
a11 | a12 | a1j | a1n | b1 | ||
a21 | a22 | a2j | a2n | b2 | ||
ai1 | ai2 | aij | ain | bi | ||
am1 | am2 | amj | amn | bm |
|
|
(2)
, aij (1), (1) A ( m n ), b. , aij b (1), (1) ( m (n+1) ), . = (A, b). i , i - (1), i = 1,, m. j A xj, (1).
aij . aij i - j - . , aij i - j - . () ( ) , , () ().
(1) (2):
) ( ) (2) , ,
) ( ) () , .
, , ( , ), , ( , ) . , .
(2) (1) , . .
( - )
, , ( , - ) . .
, , . , . , , . , .
- ( ). - , (1). , . .
(1). , n, m, aij, bi, i = 1,,m; j = 1,,n. .
I(0) (2) (2). , , . ( ai0j0. i0 , j0- .) . , .
k = 1, 2,. - .
|
|
I (2), () , , . , , . (, , , .. 1 I(0), () ai0j0 i0 , j0- .)
( k) .
1. (.. , ) 1 (.. , ), , . .
2. I(k), (k - ), , - , .
3. , I(k) () . , , .
, , () , . I(k), .. 1 3 I(k). I(k+1).
, .
. r , I(r), A(r) b(r), , .. . , , .. r min{m,n}.
? ? , r A(r) , (1),
) A(r) , .. , ,
) A(r), .
.
) r = m, m n. (.. ), I(r)
x1 | x2 | xr | xr+1 | xn | b | ||
a(r)1,r+1 | a(r)1n | b(r)1 | |||||
a(r)2,r+1 | a(r)2n | b(r)2 | |||||
a(r)i,r+1 | a(r)in | b(r)i | |||||
a(r)m,r+1 | a(r)mn | b(r)m |
(3)
, (3) i, r, . (3)
x1 + = b(r)1,
x2 + = b(r)2,
, (4)
xr + = b(r)r,
i, r, xr+1, ,xn, a(r)ij, j = r+1,,n, b(r)i, (3). xr+1, ,xn , .. . , (3), , , (4), , , (1)
xi = b(r)i - a(r)ijxj, i = 1,,r = m; xj j = (r+1),,n. (5)
|
|
(1)
Xb = {x=(x1, ,xn): xi = b(r)i - a(r)ijxj i = 1,, r = m; xj j =(r+1),,n.}.
) r < m, k, k > r, (, , )) , a(r)kj = 0 j. , b(r)k 0, k - , , , , .. (1) .
b(r)k 0, k - . , , (1) r n , r (3), . , ) (5).
. ( ) , ( ).
, () , , , - .
.
1) - , , . - r . . r , .. . r min{m,n},m - , n - . r < n, ( n r) .
2) , . .
3) (1) - , , - , , - , .
4) (1) , , , , .
5) . , , , . . , .
.
I.
2 x1 3 x2 + 3 x3 + 5 x4 = -1,
3 x1 + 4 x2 - 2 x3 + 6 x4 = 2, (6)
5 x1 4 x2 + 6 x3 + 10 x4 = 2
( - ).
. (6) - I(0).
x1 | x2 | x3 | x4 | b | |
I(0) | -3 | -1 | |||
-2 | |||||
-4 |
(0.1)
(0.2)
(0.3)
I(0) (6), ( , (6), (0.1)), - (6) ( (0.2)), - (6) ( (0.3)).
(6) x1, - (6) x2, I(0). (b) ( ) (6).
() . 2, 1- . , - . .
. I(0), 2, , , - .
|
|
1. I(0) I(1), 1 . I(0).
, , 3, . (0.2) () (0.1), 3/2. (0.2)
17/2 | - 13/2 | - 3/2 | 7/2 |
2 , ( 2) I(1)
-13 | -3 |
(1.2)
, I(0) (6), (6) , (0.1), (1.2) (0.3). , I(0) ( I(1)), ( ) 3 0.
I(0) , 5 0. (0.3) () (0.1), 5/2.
-3 | -5 |
(1.3)
I(0) () . (0.1) (.. - 2) (.. (1.1)) I(1).
-3/2 | 3/2 | 5/2 | -1/2 |
(1.1)
I(1) ( , )
x1 | x2 | x3 | x4 | b | |
I(0) | -3 | -1 | |||
-2 | |||||
-4 | |||||
I(1) | 1 | -3/2 | 3/2 | 5/2 | -1/2 |
0 | 17 | -13 | -3 | 7 | |
0 | 7 | -3 | -5 | 9 |
(0.1)
(0.2)
(0.3)
(1.1)
(1.2)
(1.3)
I(1), , . , , . , .
I(1) (-3), 3- 3 . , - .
() , , :
1. () , , () . I(0), - I(1).
2. ( ( ), - , . (), - .
3. I(1) , (6), I(0), .. , I(0), (6).
4. .
5. I(1) 2 , .. , , .
2. I(1) I(2), () . I(1).
, () , .
3/2 I(1). (1.3) 1/2 (1.1). (2.1) I(2).
(2.1)
I(1) , (-13) 0. (1.2) () (1.3), 13/3.
-40/3 | 56/3 | -32 |
3/8 I(2)
-5 | -12 |
(2.2)
I(1) () . (1.3) (.. - -3) (.. (2.3)) I(2).
-7/3 | 5/3 | -3 |
(2.3)
I(2) ( , )
x1 | x2 | x3 | x4 | b | |
I(0) | -3 | -1 | |||
-2 | |||||
-4 | |||||
I(1) | -3/2 | 3/2 | 5/2 | -1/2 | |
-13 | -3 | ||||
-3 | -5 | ||||
I(2) | 2 | 0 | 0 | 4 | |
0 | -5 | 0 | -12 | ||
0 | -7/3 | 5/3 | -3 |
(0.1)
(0.2)
(0.3)
(1.1)
(1.2)
(1.3)
(2.1)
(2.2)
(2.3)
I(2), , , , . , . I(2) (7), 4 - 2 . , - .
3. () I(3). I(2).
0 . (2.1), , I(3) (3.1). 5/3 I(2). (2.2) 5/21 (2.3). (3.3) I(3):
-8/7 | -1/7 |
(3.3)
(2.2) (.. - 7)
-5/7 | -12/7 |
(3.2)
(.. (3.2)) I(3).
I(3) ( , )
x1 | x2 | x3 | x4 | b | |
I(0) | -3 | -1 | |||
-2 | |||||
-4 | |||||
I(1) | -3/2 | 3/2 | 5/2 | -1/2 | |
-13 | -3 | ||||
-3 | -5 | ||||
I(2) | |||||
-5 | -12 | ||||
-7/3 | 5/3 | -3 | |||
I(3) | 2 | 0 | 0 | 4 | |
0 | -5/7 | 0 | -12/7 | ||
0 | -8/7 | 0 | -1/7 |
(0.1)
(0.2)
(0.3)
(1.1)
(1.2)
(3.3)
(2.1)
(2.2)
(2.3)
(3.1)
(3.2)
(3.3)
, I(3) , , . I(3) , .. , . , I(3) ( ) (). .
. I(3) , . ( x1, x3, x4) , ( x2) - .
, I(3).
x1 + 2 x2 = 4
(-5/7) x2 + x4 =-12/7 (7)
(-8/7) x2 + x3 = - 1/7.
(7) (6), .. . . , ,
x1 = 4 - 2 x2, x3 = -1/7 + (8/7) x2, x4 = -12/7 + (5/7) x2, x2 - . (8)
x = (x1, x2, x3, x4), (8), (7), , (6). Xb = { x = (x1, x2, x3, x4): x1 = 4 - 2x2, x3 = -1/7 + (8/7)x2, x4 = -12/7 + (5/7 x2, x2 }.
, (8) (6). , x2 = 0, (4, 0, - 1/7, - 12/7). (8) (6) ( ) .
, , . , . , , ( ).
2. ,
2 x1 3 x2 + 3 x3 + 5 x4 = -1,
3 x1 + 4 x2 - 2 x3 + 6 x4 = 2, (9)
4 x1 + 6 x2 - 6 x3 - 10 x4 = 3
2 x1 3 x2 + 3 x3 + 5 x4 = -1,
3 x1 + 4 x2 - 2 x3 + 6 x4 = 2, (10)
4 x1 + 6 x2 - 6 x3 - 10 x4 = 2.
, 1.
. I(0), b(9) b(10) ( , ).
x1 | x2 | x3 | x4 | b(9) | b(10) | |
I(0) | -3 | -1 | -1 | |||
-2 | ||||||
-4 | -6 | -10 |
(0.1)
(0.2)
(0.3)
, 1, 2, 1- 1 - . , - . .
. I(0), 2, , , - .
1. I(1), 1 . I(0). , I(1) 1. , 0 (-4) , (0.1) 2 () (0.3). I(1) ( , )
x1 | x2 | x3 | x4 | B(9) | b(10) | |
I(0) | -3 | -1 | -1 | |||
-2 | ||||||
-4 | -6 | -10 | ||||
I(1) | -3/2 | 3/2 | 5/2 | -1/2 | -1/2 | |
0 | 17 | -13 | -3 | 7 | 7 | |
0 | 0 | 0 | 0 | 1 | 0 |
(0.1)
(0.2)
(0.3)
(1.1)
(1.2)
(1.3)
I(1) , (9), - , (10). (9)
0 x1 + 0 x2 + 0 x3 + 0 x4 = 1,
(9).
(10)
0 x1 + 0 x2 + 0 x3 + 0 x4 = 0.
, , (10) 2 4 , (1.1), (1.2) I(1). (10), I(1)
I(1) | -3/2 | 3/2 | 5/2 | -1/2 | |
0 | 17 | -13 | -3 | 7 |
(1.1)
(1.2)
x1 | x2 | x3 | x4 | b | |
I(0) | -3 | -1 | |||
-2 | |||||
-4 | -6 | -10 | |||
I(1) | -3/2 | 3/2 | 5/2 | -1/2 | |
0 | 17 | -13 | -3 | 7 |
(0.1)
(0.2)
(0.3)
(1.1)
(1.2)
I(1) , , , . I(1) (-3), 4 - 2 . , - .
2. I(1) I(2), () . I(1) , 5/2 0. (1.1) () (1.2), (-5/6).
38/3 | -28/3 | 16/3 |
(2.1)
(1.2) (.. - (-3)) (.. (2.2)) I(2).
I(2) ( , )
x1 | x2 | x3 | x4 | b | |
I(0) | -3 | -1 | |||
-2 | |||||
-4 | -6 | -10 | |||
I(1) | -3/2 | 3/2 | 5/2 | -1/2 | |
-13 | -3 | ||||
I(2) | 1 | 38/3 | -28/3 | 0 | 16/3 |
0 | -17/3 | 13/3 | 1 | -7/3 |
(0.1)
(0.2)
(0.3)
(1.1)
(1.2)
(2.1)
(2.2)
, I(2) . .
. , , I(2), , , (10) . x1 x4 , x2 x3 . , I(2).
x1 + (38/3) x2 - (28/3) x3= 16/3
x4 - (17/3) x2 + (13/3) x3 = - 7/3 (11)
(11) . , .. . , (11) , , (10) Xb,
Xb = { x = (x1, x2, x3, x4): x1 = 16/3 - (38/3) x2 + (28/3) x3, x4 = - 7/3 + (17/3) x2
- (13/3) x3, x2 x3 .}
1 2 . , .
1) r. , .
2) , r, (n r). .
3) , . (n = r), .