n- - . ., . - - .
1.
S - n- - S>n .
-:
11 __ as1
: = α1, ,: = αs
1n asn
-, k1, , ks 0.
k1α1++ ksαs=0
11 as1 0
: k1+ +: ks =:
1n asn 0
k1α11++ ksαs1=0
. . . S . k1, , ks n-
k1α1n++ ksαsn=0
.. S>n, . , .. k1, , ks .
2. __ __ __ __
n- (I) α1,..., αs (II) β1, , βs
I II, r≤S
-:
, r>S
α1=11β1++1sβs
αr=r1β1++rsβs
S- -:
11 __ ar1 __ .. r>S, 1 - , .. . ,
: = γ1,,: = γr k1γ1++ krγr=0
1s ars (k12++kr2≠0)
11 ar1 0
: k1++: kr =:
1s ars 0
1.
2- . .
-.
2.
. . - n- - . n--.
-, . . . - . . -.
. . .
. .
.
, .
|
-1 -2 1 -4 2
0 1 -1 3 1
4 -7 4 -4 5
:
1) . 2 , ≠0.
21= 2 -4 = 0 22= -4 3 = 2 ≠ 0 rang A≥2
1 -2 -2 1
2) 3 , 22 ≠0.
, rang =2.
2 -4 3 2 -4 -1
31= 1 -2 1 = 1 -2 -1 = - 2 -1 = 1 ≠ 0. rang A≥3
0 1 -1 0 1 0 1 -1
3) 4 ., . 31 ≠ 0.
2 -4 3 1 2 -4 -1 10 2 -1 10 2 1 12 _ 1 12
41 = 1 -2 1 -4 = 1 -2 -1 -1 = - 1 -1 -1 = - 1 0 0 = 1 12 = 0.
0 1 -1 3 0 1 0 0 4 -3 8 4 1 8
4 -7 4 -4 4 -7 -3 8 +I +I
+II +3III
2 -4 3 0 2 -4 -1 4 2 -1 4 -II 1 0 0 _ -1 4
42 = 1 -2 1 2 = 1 -2 -1 4 = - 1 -1 -1 = - 1 -1 4 = -3 12 = 0. => rang A=3.
0 1 -1 1 0 1 0 0 4 -3 12 4 -3 12
4 -7 4 5 4 -7 -3 12
+II -II
1.
.
. .
. . . ., .. . . .
-.
111+ 122+...+ 1nn=b1
211+ 222+...+ 2nn=b2
.
s11+ s22+...+ snn=bs
a11 ... a1n _ _ a11 ... a1n b1
A= , A=:::
as1 asn as1 ... asn bs
|
|
. . __
. <=> rang . (rang = rang )
-:
1) . =>
. k1, , kn ., ., S , ., . .
2) . <= __
ra ng A = rang A, . . . - . . . . __
.
11 a1n b1 , k1, , kn, . . ., .. .
: k1++: kn =: .
s1 asn bs