n- . , j- Ej , , j =1, 2,..., n
(*)
n- .
(*) n- ; n- .
. , .
k1E1+k2E2+...+knEn =0.
E1, E2,...,En , kj, ,
(k1,k2,,kn) = (0; 0;...; 0),
k1=0, k2=0,,kn =0. , (*) .
, n- = (a1, a2,...,an) :
A= a1E1+a2E2+...+anEn = (a1, a2,...,an)
, = (3;-2; 4; -5) E1= (1; 0; 0; 0); E2 = (0; 1;0;0); = (0; 0; 1; 0); 4 =(0; 0; 0; 1) , :
= 1 - 22 + 4 - 54 = (3; - 2; 4; - 5).
, 1.3. :
- ;
- , ;
- () () ().
. , , . - () . .
2.5.
n :
= (aij) , X =(xj) - -, , = ( bi ) - -, ;
AX = B.
, ( ) . .
, - 1, .
, , , , , 1 11, ( ), , , , . , 1- 21 2- . 1- 31 3- .. 1 , , 1 , 1 .
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, , , , , . ., . . . , . .
1. i- , , , . . 0=bi¹0. , , .. .
2. i- . , i- , , .
3. , , , , .
, .
1.
A1, 2, 3, 4:
A1x1 +2x2 + 3x3 + 4x4 =B.
. , .
x1 : a11=5¹0; . 5. x1 , .
1 | 2 | 3 | 4 | 0 |
-2 | -2 | |||
2/5 -14/5 14/5 | 3/5 19/5 1/5 | 3/5 4/5 1/5 | 1/5 18/5 -13/5 | |
8/7 -19/14 | 5/7 -2/7 | 5/7 -9/7 | ||
3/7 3/56 1/4 | 3/7 -53/56 1/4 |
, .
: 1 = 3/7-3/7x4,
x2 =- 53/56-/5 4,
x3 = 1/4-1/4x4.
, .
A1, 2, 3 ; x1, 2, 3 , 4, A4 - ( ).
, , .
m n (m<n) , ( ) , n-m ().
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n . .
, .
, : 1 = 3/7, 2= - 53/56, x3 = 1/4, x4 = 0.
B A1, 2, 3
3/7A1 - 53/562 + 1/43 = B.
, , , .
, .
X=(x1,x2,,xn) , , .. xj³0 j=1,2,,n. .
, .
- , , , .