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:::::::::::::::::::::::::::::::::::
x 1.
x 2.
I.
:::::::::::::::::::::::::::::::::::
1.
(9). 2.
(10). 4.
(13). 6.
x 4.
1.
1.
-
9
,
(27).
(24).
3.
5.
7.
(37). 10.
(38).
II.
.
,
(19).
(40).
(22). 3. (22).
2.
,
(29).
(33).
8.
2.
: : : : : : : : : : : : : : : : 21
(21). 2.
1.
1.
3.
(16).
(19). 5.
(20).
(35).
x 2.
(17).
(18). 3.
(30).
x 1.
(9). 3.O
(11). 5.
: : : : : : : : : : : : : : : : : : : : : : : : : : : 17
4.
x 3.
8
2.
(45).
(32).
4.
6.
(36).
9.
(37). 11.
: : : : : : : 24
,
-
: : : : : : : : : : : : : : : : : : : : 40
(44). 4.
(42).
: : : : : : : : : : : : : : : : : : 46
4
1.
(48).
x 3.
(47).
4.
x 3.
x 4.
x 1.
x 2.
x 3.
(46).
1.
(50).
6.
5.
-
: : : : : : : : : : : : 56
,
,
2.
(52).
(54).
(56). 3.
(57). 5.
(56). 4.
(58). 6.
(59). 8.
(60). 10.
9.
x 1.
x 2.
3.
(56).
1.
(58). 7.
(60).
(62). 11.
(63).
III.
1.
1.
,
(69).
(80).
3.
8.
1.
(90).
2.
,
(81).
(85).
(86).
4.
(91).
(73).
: : : : : : : : : : : : 65
: : : : : : : : : : : : : : : : : : : : 69
3.
(76).
: : : : 79
(79).
2.
(80). 4.
(84). 6.
5.
7.
-
(85).
: : : : : : : : : : : : : : : : : : : : 88
(88). 2.
(89). 3.
(90). 5.
(92). 7.
6.
(92).
-
-
IV.
1.
1.
4.
1.
4.
(96).
4.
(95).
(100).
: : : : : : : : : : : : : : : : : : : 95
2.
(95).
3.
-
(98).
: : : : : : : : : : : : : : : : : : : : : : 99
(99). 2.
3.
(102).
(103).
(106).
(107). 3.
: : : : : : : : : : : : : : : : : : : : : : 106
(111).
2.
(110).
5.
(109).
5
x 1.
x 2.
x 3.
x 4.
x 5.
x 6.
V.
: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 114
1.
(114). 2.
(116).
5.
P
1.
6.
1.
x 2.
x 3.
x 4.
(120). 2.
(123). 4.
(125). 5.
(129).
(139).
(140).
(134).
(136).
3.
4.
(143).
(148).
(146).
4.
(127).
.
3.
(135).
-
) : : : : : : : 146
(148).
(
3.
(149).
) : : : : : : : : : 149
(152). 3.
-
(161). 5.
-
: : : : : : : : : : : : : : : : : : : : : : 166
(168).
(183). 5.
-
5.
(158). 2.
(160). 4.
(165).
(166).
(177).
.
(142).
: : : : : : : : : : : : : : : : : : : : : : : : : : : : 158
(160). 3.
(164). 6.
(172).
2.
(
2.
(149). 2.
(152). 4.
(155).
5.
1.
3.
1.
3.
(121). 3.
: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 136
VI.
1.
(116).
: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 132
1.
1.
(117).
(132). 2.
(134). 4.
(155).
x 1.
3.
: : : : : : : : : : : : : : : : : : : : : : : : : : : 120
1.
1.
(115).
4.
2.
2.
-
: : : : : : : : : : : : : : : : : : : : : : : : 172
(176). 4.
(177).
(177). 6.
(174).
5.
: : : : : : : : : : : : : : : : : : : 179
(179). 2.
(181). 4.
(184). 6.
(180).
-
6
x 5.
x 6.
x 7.
x 1.
x 4.
2.
(192).
x 2.
-
(212).
(207). 2.
5.
VII.
(216).
(224).
(219).
(222).
10.
1.
6.
(217).
5.
(224).
(218).
(220).
(221). 7.
(223). 9. QR-
(226).
3.
(237).
3.
: : : : : 226
2.
(231).
-
(232).
5.
(236).
7.
(239).
2.
(240). 3.
1.
(208).
-
(209).
4.
(213).
2.
8.
,
(228).
1.
-
: : : : : : : : : : : : : : : : : : : : : : : 216
1.
(233).
(198). 3.
4.
: : : : : : : : : : : : : : : : : : : : : : : : : : : : 207
{
: : : : : : : : : : : : : 239
,
(241).
,
: : : : : : : : : : : : : : 243
(243). 2.
(247).
x 1.
3.
: : : : : : : : : : : : : : : : : : : : : : : : : 196
(196). 2.
(202).
(205).
4.
x 3.
(192).
(194).
1.
1.
3.
7.
: : : : : : : : : : : : : : : : : : : : : : : : : : : 192
1.
4.
6.
x 2.
(186).
(187). 8.
(188). 9.
(190).
(246). 4.
(245). 3.
-
VIII.
1.
: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 249
(251).
(249). 2.
-
: : : : 252
7
1.
x 1.
x 2.
x 3.
(252).
(255). 3.
(257).
IX.
1.
8.
10.
1.
3.
1. -
(263).
(265). 4.
(269). 6.
.
(277). 2.
(278).
(281). 2.
(284). 4.
2.
(256). 4.
-
: : : : : : : : : : : : : : : : : 263
2.
(270).
7.
(273).
(263).
3.
(268). 5.
9.
(275).
-
(272).
(274).
: : : : : : : : : : : : : : : 277
(277).
: : : : : : : : : : : : : : : : : : : 281
(285).
(283). 3.
: : : : : : : : : : : : : : : : : : : 289
: : : : : : : : : : : : : : : : : : : : : : : : : : : : 302
: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 306
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1.
:
1) a + b = b + a (
2) (a + b) + c = a + (b + c) (
3) a + 0 = a
4)
(;1)a
5) ( )a = ( a)
6) ( + )a = a + a
7) (a + b) = a + b
8) 1a = a.
(;1)a
;a.
a ;b.
= a, x = a ; b ( . 1).
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a ; b.
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1,
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a3 = a1 + a2 ,
b = ;a3
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1a1 + ::: + k ak = 0
,
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a1 ::: ak
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1 ::: k,
1a1 + ::: + k ak = 0,
: 21 + ::: + 2k 6= 0.
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a1 ::: ak
,
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0
1,
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b
= 1 a1 + ::: + k ak .
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14
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a1 ::: ak
b1 ::: bs,
.
b1 ::: bs
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a1 ::: ak
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x
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3.
-
0
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a1 ::: ak,
a
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x = 1a1 + :::
::: + k ak x = 1 a1 + ::: + k ak .
,
( 1 ; 1 )a1 + ::: + ( k ; k )ak = 0.
,
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1 ; 1 = 0, :::, k ; k = 0,
. .
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: 1a1 + :::
::: + k ak = 0.
x = 1 a1 + ::: + k ak
x
: x = ( 1 + 1 )a1 + ::: + ( k + k )ak .
.
4.
k>1
,
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a1 ::: ak
, . .
1 ::: k,
, 1
.
1 a1 + ::: + k ak = 0, ,
a1
:
a1 = ; a2 ; ::: ; ak :
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: a1 = 2 a2 + ::: + k ak .
,
a1 ::: ak
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2
k
1
1
1.
15
1.
2.
a.
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a 6= 0.
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a, b c
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a
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b,
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OC = OP + P C,
OP PC
a b.
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OP = a P C = b.
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= a + b.
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a, b c
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a, b c
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4.
a, b, c d.
a, b c
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a, b c
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, d
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O ( . 3)
D
d
,
c,
P
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OD = OP + PD,
OP
a b, PD
c.
OP
a
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c
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c
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a(1 0 1).
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5.
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a = 1 e1 + 2 e2 + 3 e3 ,
a = ( 1 e1 + 2e2 + 3e3) = ( 1)e1 + ( 2)e2 + ( 3)e3 :
a = 1e1 + 2e2 + 3e3 b = 1 e1 + 2 e2 + 3 e3 ,
a + b = ( 1 e1 + 2e2 + 3e3 ) + ( 1 e1 + 2 e2 + 3 e3) =
= ( 1 + 1 )e1 + ( 2 + 2 )e2 + ( 3 + 3 )e3 :
.
2.
C
,
1.
17
2 0 1]
OC = OA + (1 ; ) OB .
AB ?
2.
b(2
AC
3.
AB , AD.
3) c(;1 1).
a, b.
c
4.
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a
A, B
+ OB + OC = 0.
1.
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O, e1 e2 e3.
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2 1=2
2
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OB
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(x2 ; x1 y2 ; y1 z2 ; z1 ).
1.
2.
M
(x y z)
,
1.
(x ; x1) = (x2 ; x)
-
A B,
O, e1 e2 e3
x1 y1 z1 x2 y2 z2.
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AB = OB ; OA
OA
(x1 y1 z1) (x2 y2 z2)
5 x1
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AB,
(
= , . .
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jMB j =
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M,
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>0
AM = MB:
(1)
(x1 y1 z1) (x2 y2 z2) A B,
AM MB
(y ; y1 ) = (y2 ; y)
-
(z ; z1 ) = (z2 ; z):
2.
19
x = x ++ x
(2)
,
1
+ 6= 0:
z = z ++ z :
x, y z,
y = y ++ y
2
1
1
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2*
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j = j.
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20
,
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(r ') (r ' + 2 )
(r ' + 2k ),
06'<2
,
(r '),
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,
r = 0,
(r ')
r
(r ') (r1 '1)
r = r1, ' ; '1 = 2 k, k |
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, -
.
e1
y = r sin ':
l,
M.
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l.
n.
O,
l
,
l, h |
,
,
n,
O
MM
M|
M
r '|
(3)
-
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.
O,
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).
x = r cos '
1
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. 7,
5.
O
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O
1,
=2
,
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; <'6 .
r > 0,
,
k|
0
.
,
-
r, ', h.
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0
MM
( . 8).
0
n.
3.
(
2.
3.
|
: r = jOM j.
OM
l,
0
. 9).
1.
21
(r ' ).
|
,'|
OM
AOC
j OAj = 2, j OC j = 3,
B
O, OC , OA.
A(x1 y1 ), B (x2 y2 ), C (x3 y3 ).
ABCD.
,
: ) r = 2= cos ' ) r = 2 cos '.
OABC .
=3.
D
O, l, n |
4.
=2
e1
e2
.
O, e1 , e2 , n,
1.
.
l,
,
3.
x
e1
.
.
,
,
.
,
e1 , e2 e3
0
0
0
)
e1 e2 e3 .
e1 = a11e1 + a21e2 + a31 e3
e2 = a12e1 + a22e2 + a32 e3
e3 = a13e1 + a23e2 + a33 e3:
a
e1 e2 e3:
a = 1e1 + 2e2 + 3e3 :
.
-
-
0
(1)
0
0
0
0
0
0
0
0
0
0
0
1 2 3.
e1 e2 e3,
5 x1
1 0
1 0
1 0
1 = a1 1 + a2 2 + a3 3
2
0
2
0
2 0
2 = a1 1 + a2 2 + a3 3
3 0
3 0
3 0
3 = a1 1 + a2 2 + a3 3:
)
.
, a13
\a
-
".
(2)
.
-
. I.
22
(2)
,
(2)
1
2
,
.
0
1
(2),
0
0
0
0
2
3.
0
.
,
.
= a11 1 + a12 2
= a21 1 + a22 2 :
(2)
1
a1 a12 a13
a21 a22 a23 :
a31 a32 a33
0
-
:
(3)
:
(4)
e1 e2 e3
e1 e2 e3.
e1 e2 e3
.
2.
.
:
O, e1 e2 e3
O , e1 e2 e3.
M|
,
(x y z) (x y z ).
x, y
z
x ,y z ,
.
(a10 a20 a30)
O
O, e1 e2 e3
e1 e2 e3,
(4).
M
O O
OM = OO + O M,
OM = OO + x e1 + y e2 + z e3
(5)
x, y z |
OM
e1 e2 e3.
(5)
e1 e2 e3,
,
OM OO
M
O
,
(x y z) (a10 a20 a30).
x = a10 + a11 x + a12y + a13 z
y = a20 + a21 x + a22y + a23z
(6)
3
3
3
3
z = a0 + a1x + a2 y + a3 z :
(6)
.
3.
.
(6),
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3.
23
z:
0
x = a11x + a12 y + a10
y = a21x + a22 y + a20 :
,
.
'
e1
0
(
. 10)
0
0
0
0
(7)
e1
e1 e2 .
e1 = cos ' e1 + sin ' e2
e2 = cos ' 2 e1 + sin ' 2 e2 :
e2
,
,
0
0
0
e1 e2
. .
0
0
e1 e2 ,
'.
0
. 10.
-
e2
0
.
cos ' 2 = sin ', sin ' 2 = cos ',
x = x cos ' y sin ' + a10
y = x sin ' y cos ' + a20
1.
0
0
0
0
.
,
O |
2.
AB
0
O , O O, O B .
0
3.
0
0
0
OAB .
O, OB , OA
O, e1 , e2 , e3 .
O , e1 e2 e3 ,
x= 1;y ;z , y =1;x ;z , z = 1;x ;y .
0
0
0
0
0
0
0
0
0
0
,
(8)
.
?
-
. I.
24
x
1.
4.
,
,
.
.
.
,
-
.
,
(a b)
a b
-
,
(a a) = jaj2
.
ab.
(a b) = jajjbj cos'
a b.
'|
.
, -
.
,
:
-
,
.
,
,
.
.
.
(a b) = (b a).
a b
a.
:
0.
(e1 e1) = (e2 e2) = (e3 e3) = 1
(e1 e2) = (e2 e3) = (e3 e1) = 0:
1.
1
= (aje ej )
1
2=
1
2
,
1 = (a e1 )
2 = (a
,
a
(a e )
,
,
2
je2 j2
e2 )
3
= (aje ej ) :
3 = (a
3
3
2
-
e3 ) (1)
a = (a e1)e1 + (a e2 )e2 + (a e3)e3 :
.
a = a1 +
+ a2 + a3 ,
.
-
4.
2 x 1,
e1.
,
,
25
= ja1j=je1j,
+
;
,
. 11, ja1j = jaj cos '1 , '1 |
a e1 .
, 1 = jaj cos '1 =je1j = (a e1 )=je1j2.
.
.
a
1
|
2.
a0 , . . 1 ( .
.
,
a, b c
a1
-
a0 = a=jaj.
(3)).
( a + b c) = (a c) + (b c):
, ( a c) = (a c) (a + b c) = (a c) + (b c).
.
c = 0,
.
c 6= 0.
c
,
.
( a + b c2 )=jcj2 | 2
(a c)=jcj (b c)=jcj |
a + b.
a b.
5 x1
( a + b c)=jcj2 = (a c)=jcj2 + (b c)=jcj2:
.
,
.
,
(a b + c) = (a b) + (a c):
1.
,
a b
( 1 2 3)
( 1 2 3)
(a b) = 1 1 + 2 2 + 3 3 :
(2)
,
b
2:
(a b) = (a 1 e1 + 2 e2 + 3 e3) = 1 (a e1) + 2 (a e2 ) + 3 (a e3):
(1).
,
.
.
, ,
.
,
,
,
. I.
26
.
1
jaj =
q
cos ' = (jaajjbb)j = p
(3),
,
2+ 2+ 2
1
2
3
+
+ +
1
2
1
1
2
2
2
2
3
(3)
+
2
p
2
1
3
3
+ +
2
2
2
3
:
(4)
.
,
A B
(x y z) (x1 y1 z1).
p
jAB j = (x1 ; x)2 + (y1 ; y)2 + (z1 ; z)2 :
.
\
"
(5)
-
.
.
AB
l.
B
A
A B ( . 12).
(
)
AB
l AB.
,
0
AB
0
0
l
0
.
e|
l.
AB = e
AB
AB = A B = e + b
e.
(AB e) = (e e).
,
AB = ( AB e) e:
0
.
0
b=BB
e
0
00
= e=jej.
,
AB
0
,
00
e,
0
e,
(6)
jej2
l
e
0
e.
( AB e) e
(AB0 e)=jej |
je j = 1,
jej
jej
AB
0
0
e0 =
-
4.
,
27
AB.
0
,
0
e,
(AB e)=jej
e, > 0,
.
. 2.
AB
AB
AB
0
lAB
e.
e
(AB e)=jej
l,
eAB.
,
-
e(
l AB
e),
= jAB j cos ',
'|
.
,
(
,
(
)
.
.
,
.
(
),
2.
:
.
e
.
.
0
-
,
,
.
).
.
,
,
,
-
.
.
-
,
-
,
.
,
. 10,
-
-
. I.
28
,
,
. 10, |
,
.
, ,
:
.
,
.
,
.
.
(
e1
.
.
,
,
,
,
,
,
(
-
.
,
.
,
.
,
,
,
.
e2
,
,
,
-
.
. 14
|
e3
-
.
|
.
(
)
,
.
,
-
)
. 13).
,
,
.
, -
.
,
.
,
-
-
4.
,
.
29
a, b
,
|
.
,
.
,
3.
,
n.
,
.
.
,
,
,
(
,
,
,
,
,
,
-
:
),
a b,
,
S (a b).
,
.
.
,
.
|
,
-
-
-
:
|
,
,
.
,
-
,
.
.
.
:
.
,
.
,
-
.
,
-
.
,
,
n
a b n.
,
.
.
-
, -
. I.
30
,
4.
.
.
)
(
,
-
a, b
,
,
,
a, b c
,
.
c
-
(a b c).
, .
,
a, b c
,
( . . 14),
(a b c) = (c a b) = (b c a) = ;(b a c) = ;(c b a) = ;(a c b): (7)
.
(
b c
3.
.
d
a)
(a b c) = (a d):
,
.
(a b c) = 0.
a, b c
,
b c,
a
b c
.
a
(8)
d,
.
a, b c
d = 0.
,
,
n
-
b c ( . 15).
OH,
.
1,
b c
n b c. (
bcn
.)
(a n) |
a
n.
OH,
a b c.
, (a n) > 0
,
a n
, . .
abc ,
n b c.
, (a n)
abc
.
S|
.
(a n)S
,
(a n).
,
(a b c) = S(a n).
(8),
d = S n:
(9)
4.
,
31
b c
, a,
a
b
,
d,
(a b c) = 0 (a n) = 0,
(9)
(8)
.
,
,
(8)
a
b c.
,
b c
d1
d2
,
a
(a b c) = (a d1) (a b c) =
= (a d2).
,
(a d1) = (a d2)
(a d1 ; d2) = 0.
d1 ; d 2
,
,
.
,
d,
(8),
.
.
,
d
b c.
1.
b c
, d = 0.
2.
b c
, :
) jdj = S = jbjjcj sin', ' |
b c
)
d
b c
)
bcd
.
|
.
.
d,
, ,
,
(8),
b c.
,
,
,
.
, .
b c
b c]
b c.
,
(8)
(a b c) = (a b c]):
(10)
b c
c ,
(9).
,
.
.
1.
e1 e2 e3 |
e2 e3] = e1
c f1 f2 f3 |
f2 f3] = ;f1
4.
. .
e3 e1] = e2
f3 f1] = ;f2
.
e1 e2] = e3 :
,
f1 f2] = ;f3:
b c] = ; c b].
,
(a b c) = (a d),
(a c b) = ;(a d) = (a (;d)):
.
(11)
,
-
. I.
32
( a1 + a2 b c]),
( a1 + a2 b c) = (a1 b c) + (a2 b c):
(7)
.
,
(a b1 + b2 c) = (a b1 c) + (a b2 c):
,
,
,
.
5.
b1 , b 2 c
2
(12)
(13)
-
b1 + b2 c] = b1 c] + b2 c]:
(13)
(a b1 c]) + (a b2 c]):
2
(a b1 + b2 c]) = (a b1 c] + b2 c]):
a,
,
e1 e2 e3,
a
.
1
b1 + b2 c] b1 c] + b2
,
.
,
5.
.
a b
e1 e2 e3,
:
c],
.
-
a b] = ( 1e1 + 2e2 + 3e3) ( 1 e1 + 2 e2 + 3 e3)] =
= ( 1 2 ; 2 1 ) e1 e2] + ( 2 3 ; 3 2 ) e2 e3] +
+ ( 3 1 ; 1 3 ) e3 e1]: (14)
,
.
(14)
a b] = (
|
1
.
2.
-
2 3 ; 3 2 )e1 + ( 3 1 ; 1 3 )e2 + ( 1 2 ; 2 1 )e3:
.
,
(15)
4.
,
33
.
,
,
,
+
,
3.
3 1 2; 3 2
.
,
,
.
-
,
.
(15),
,
-
.
.
a, b c
( 1 2 3) ( 1 2 3 )
e1 e2 e3
(a b c) = ( 1 2 3 + 2 3 1 +
1 ; 2 1 3 ; 1 3 2 )(e1 e2 e3).
,
(a b c) = (c a b])
(14)
c = 1 e1 + 2 e2 + 3 e3 .
(
1
3),
2
(a b c) = 1 ( 2 3 ; 3 2 )(e1 e2 e3]) +
+ 2 ( 3 1 ; 1 3 )(e2 e3 e1]) + 3 (
(
,
,
,
(7)
.
6.
.
(
.
1 2 1 2.
,
1 2; 2 1
1
1
2
2
:
1
1
2
2
:
1 2 ; 2 1 )(e3
e1 e2]):
.)
,
-
,
.
)
:
-
-
:
a b] =
3
. .
2
2
3
3
e1 +
3
3
1
1
e2 +
1
1
2
2
e3 :
|
. I.
34
1
1
1
1
,
1
2
2
2
2
,
2
2
2
3
3
3
:
3
3
+
2
3
3
1
1
+
3
1
1
2
2
3
3
;
2
1
1
3
3
+
3
1
1
2
2
1
1
1
2
2
2
3
3
3
:
1
1
1
2
2
2
3
3
3
(e1 e2 e3):
(a b c) =
1
1
1
2
2
2
3
(a b c) =
,
2
a b] =
3
3
3
e1 e2 e3
1
1
2
2
3
3
(16)
:
(17)
:
:
(18)
.
. V,
-
,
.
a1x + b1y + c1z = d1
a2x + b2y + c2z = d2
a3x + b3y + c3z = d3:
a(a1 a2 a3), b(b1 b2 b3), c(c1 c2 c3) d(d1 d2 d3).
xa + yb + z c = d:
(19)
4.
,
a, b c.
,
,
b c].
35
, . . (a b c) 6= 0:
,
.
(19)
x(a b c) = (d b c), ,
a, b c
d1 d2 d3
b1 b2 b3
c1 c2 c3
(7)
,
a1 + a1
b1
c1
0
00
a1 a2 a3
b1 b2 b3 :
c1 c2 c3
.
.
(12)
a3 + a3
b3
=
c3
a1 a2 a3
= b1 b2 b3 +
c1 c2 c3
a2 + a2
b2
c2
0
00
0
a1 a2 a3
b1 b2 b3 :
c1 c2 c3
.
0
00
e1 e2 e3,
8.
(
,
3
3
00
-
.
e1 ,
-
.
2
2
00
.
7.
a b
-
,
.
e2 e3] + e3 e1] + e1 e2] = 0
,
,
.
(e1 e2 e3) = 0.
.
,
3*
0
.
6.
,
.
00
0
7.
d
,x
xyz |
,
=
3
3
1
(16),
.
2
1
1
3)
=
(
1
1
1
2
a b
2
2
(e1 e2 e3) 6= 0.
3) |
= 0:
(20)
. I.
36
.
:
a b],
.
,
(20),
e1 e2]
,
(14)
e2 e3], e3 e1]
9.
.
.
1
1
8.
p
S = j a b] j = (
,
,
.
a( 1 2) b(
( 1 2 0)
8,
0)
2
2
2
= 0:
,
.
3
=
3
1
2)
= 0.
,
,
-
2 3 ; 3 2 )2 + ( 3 1 ; 1 3 )2 + ( 1 2 ; 2 1 )2:
,
j a b]j2 = jaj2jbj2 sin2 ' = jaj2jbj2(1 ; cos2 '):
2
S 2 = (ajajb) (ajbjb2) :
(21)
(22)
.
n,
jnj = 1.
a b.
n
(14),
1
,
,
(14)
a b]
6
,
(
.
(20)
.
,
,
.
,
n a b]
.
,
n a b].
j a b]j, . .
a b] n
a b]
a, b
a b.
S (a b) = (n a b)
a b]
,
n,
.
-
4.
(
,
37
jnj = 1).
,
e1, e2 .
)
:
S (a b) =
0
1
1
,
S (a b) =
1 2 ; 2 1,
,
(16).
,
,
S (a b) =
9.
S (e1 e2):
2
2
1 2
;
e1 , e2 |
.
2 1:
(25)
.
.
a b c]] = (a c)b ; (a b)c:
a b c]] -
,
(26)
e1 e2 e3
b c.
-
.
.
,
e1 = (e e e e e] ) e2 = (e e e e e] )
.
2
1
3
2
6
e3 = (e e e e e] )
e1 e2 e3.
e1 e2 e3
\
"
3
3
1
.
,
1
2
3
(24)
-
,
e1
b, e2
b = e1 , c = 1 e1 + 2 e2 a = 1e1 + 2e2 + 3e3.
b c] = 2 e3
a b c]] = ; 1 2 e2 + 2 2 e1 :
,
(a c)b = ( 1 1 + 2 2 ) e1 (a b)c = 1 ( 1 e1 + 2 e2 ):
10.
(23)
1 2 ; 2 1j
.
-
0 1
2 0 (e1 e2 n):
2 0
1
1
S=j
(21).
(
n
1
1
-
2
2
3
,
. I.
38
(ei ej ) = 0
i 6= j.
.
,
,
, (ei ei ) = 1
,
i.
:
e1 e2 e3 |
,
e1 e2 e3,
a
:
a = (a e1)e1 + (a e2 )e2 + (a e3 )e3
(27)
a = (a e1)e1 + (a e2)e2 + (a e3)e3 :
(28)
(27),
a = 1e1 + 2 e2 + 3e3
e1 ,
e2
e3 .
1 = (a e1 ),
(28).
2 = (a e2 ), 3 = (a e3 ).
11.
e1 e2 e3 |
,
e1 e2 e3,
e1 e2 e3,
e1 e2 e3.
e1 e2 e3 ,
,
(28),
e1 e2 e3,
a = (a e1 ) e1 + (a e2) e2 + (a e3 ) e3 :
a
e1 , e2 e3
,
(ei ej ) = 0
i 6= j, (ei ei ) = 1,
e1 = e1 , e2 = e2
e3 = e3 .
(a e1), (a e2) (a e3 )
a
e1 e2 e3.
a
e1 e2 e3.
e1 e2 e3 |
.
,
,
.
11.
.
,
:
, ,
. .
,
,
.
,
.
,
|
,
.
,
.
.
,
.
,
.
,
.
.
10.
4.
,
39
,
,
,
(10).
,
:
, )
, , / , ).
(
,
,
(
,
),
1.
.
5.
,
6.
,
7.
8.
,
.
,
,
.
,
a
.
,
-
,
.
.
,
-
-
.
je1 j = 2, je2 j = 3 (e1 e2 ) = 2.
,
?
.
e1 e2 e3
, -
.
,
0.
.
|
2,5 ?
,
3.
4.
,
,
,
(2).
2.
.
.
,
( -
.
,
,
-
.
a(1
2) b(2 1).
,
?
e1 e2 e3 ,
-
II
x
1.
1.
.
.
r,
.
-
P
,
r.
(x y z)
M
jPM j = r:
p
(x ; a)2 + (y ; b)2 + (z ; c)2 = r:
(1)
,
(x ; a)2 + (y ; b)2 + (z ; c)2 = r2 :
(2)
,
, ,
,
.
(2)
.
.
L,
,
y = f(x).
,
,
,
y = f(x)
L.
,
S
S
, . .
,
,
(a b c).
f
.
|
,
\
,
F(x y) = 0,
,
,
|
.
|
:
,
"
-
. .
-
.
F (x y z) = 0,
.
F|
(2)
1.
,
r2
41
,
,
(2),
.
,
(x ; a)2 + (y ; b)2 + (z ; c)2 6 r2 :
,
.
-
,
-
.
,
-
,
,
(x y z) = F(x y z) ; jF(x y z)j = 0
,
F (x y z) > 0.
,
.
,
,
,
,
,
.
|
,
.
.
PS PT |
S \T
,
.
,
PS PT |
, FS (x y z) = 0 FT (x y z) = 0,
.
,
S T,
.
,
,
,
PS PT
PS ^ PT .
FS (x y z) = 0 FT (x y z) = 0:
PS PT |
S T,
S T |
,
,
PS
PT
.
PS _ PT :
,
PS PT |
,
, FS (x y z) = 0 FT (x y z) = 0,
T,
PS PT
PS
S
.
FS (x y z)FT (x y z) = 0:
|
S T
PT .
T
, . . PS
PT ,
.
(2) (1),
S
PT
,
PS .
(1) (2)
. II.
42
r > 0.
,
z ; c = r2 ; (x ; a)2 ; (y ; b)2
(2).
,
(2) (3),
p
z ; c = r2 ; (x ; a)2 ; (y ; b)2 :
,
(2)
(3),
,
p
z ; c = ; r2 ; (x ; a)2 ; (y ; b)2 :
(2)
(4).
,
|\
" \
"
,
(4)
2.
\
p
" \
.
,
"
|
,
,
)
,
.
,
-
.
,
.
(5)
(2),
-
(6)
.
,
-
,
,
.
.
.
, . .
-
.
,
(4)
(3)
.
-
.
,
A1 xk1 yl1 + ::: + As xks yls = 0
|
k1 + l1 ::: ks + ls
,
,
-
,
.
(2)
.
.
A1 xk1 yl1 z m1 + ::: + As xks yls z ms = 0
|
) k + l + m ::: k + l + m
1
1
1
s s
s
,
(3)
,
,
-
1.
43
.
-
x2 + y2 + z 2 + 1 = 0
,
,
.
p.
-
(5)
p.
-
.
,
,
O e1 e 2 ,
,
O e1 e2.
xy
x y
(7) x 3 . I:
x = a11 x + a12y + a10 y = a21 x + a22 y + a20:
-
0
0
0
0
0
x y
+ a12 y0 + a10 )k |
(a21 x0 + a22 y0 + a20 )l
k
|
Axk yl
.(
,
(6),
F(x y), . .
,
.)
(7)
, x y.
(a11 x +
x
y,
,
x
-
0
F (x y) = 0
0
.
p
2.
y.
.
p
2.
(6)
0
-
(5),
.
1.
0
(
.
?
0
-
,
2
2
2
2
(x + y + z ) (x ; 1) + (y ; 1)2 + (z ; 1)2 ] = 0
,
y2 + z 2 = 0
).
.
,
l.
0
.
0
0
k+l
,
G(x y ) = 0
0
G(x y )
0
.
.
0
0
0
0
0
0
0
,
.
,
G(x y ) = F(a11 x + a12y + a10 a21 x + a22 y + a20):
,
G(x y )
x y
0
0
0
,
0
0
0
0
0
x y,
. II.
44
(7),
G
F,
O e1 e2 e3
,
,
|
0
.
,
0
0
(
,
.
,
.
0
,
,
.
,
F (x y).
,
O e1 e2 e3
.
,
. .),
-
.
,
.
,
,
.
p
x2 + y2 = 1
,
,
".
,
\
.
,
1
:
x2 + y2 ; 1 = 0 (x2 + y2 ; 1)2 = 0:
(6),
(
).
,
\
,
,
"
.
-
.
,
,
.
.
.
.
,
.
-
|
.
3.
,
F(x y) = 0.
.
,
F(x y z)= 0,
,
,
M0 (x0 y0 z0)
x0 y0 z
,
,
, z,
M0
z
-
1.
45
e3 .
e3.
.
,
,
(
.
,
,
,
|
. 16).
,
,
-
,
-
,
,
,
.
+ y2 = r2
.
.
,
-
|
F(x) = 0?
4.
,
.
, . .
:
, s|
,
.
( x y z)
, F ( x y z) = s F (x y z).
s.
M0
M0
.
F
,
,
x2 +
,
,
F(x y z) |
.
(x y z)
, ,
-
,
F (x y z) =
= 0,
F |
. M
(x y z)
,
P ( x y z)
.
M P
,
P
OM ( . 17).
.
,
,
,
-
. II.
46
.
|
,
,
,
(
. 17).
.
,
,
,
F (x y z) = 0,
F|
.
A(1 0)
,
B
,
A.
2.
(x ; 2)2 + y2 = r2 , (x + 2)2 + y2 =
= r2
.
,
x = 0.
?
r = 3 r = 1.
3.
,
x2 + y2 + z 2 = 1, x + y + z = 1,
,
e3 .
4.
,
x2 + y2 = 4, z = 1,
.
1.
B (4 0).
2.
x
1.
,
,
Ax + By + Cz + D = 0
,
, . . A2 + B 2 + C 2 6= 0.
,
A2 + B 2 6= 0.
,
2.
,
(2)
(1)
(2)
1 2 x1
.
.
1.
,
,
,|
Ax + By + C = 0
(1)
,
.
,
.
-
(1).
.
-
-
(2).
2.
47
,
.
2.
(
,
-
.
.
)
,
,
,
-
a,
,
.
,
,|
.
,
(
,
,
,
.
-
r (
M0 M = r ; r0 ,
,
,
-
.
M0
. 18).
,
,
M
M
r ; r0 = ta:
t,
(3)
t,
,
M
(a1 a2 a3).
-
.
,
.
M0 ,
,
,
.
.
(3)
(3)
-
r
,
,
).
,
-
.
r0 a
t,
.
-
.
M0
.
,
M
-
.
,
,
(x y z) (x0 y0 z0 ) |
a
-
. II.
48
(3),
x ; x0 = a1 t y ; y0 = a2 t z ; z0 = a3 t:
,
,
x ; x0 = a1 t y ; y0 = a2t:
(4)
(5)
(4)
(5)
.
.
q
M
,
M
t1 t2 ,
r ; r0 = t1 p + t2 q:
.
t1 t2 .
,
t1 t2 ,
(6)
(x y z) (x0 y0 z0 ) |
,
p q
x ; x0 = t1 p1 + t2 q1
,
y ; y0 = t1p2 + t2q2
z ; z0 = t1p3 + t2 q3: (7)
-
,
M0
,
,
,|
,
(6)
.
M M0
(p1 p2 p3) (q1 q2 q3).
(6),
t,
3.
-
r0 |
M0 .
M
r |
( . 19). M0 M = r ; r 0 ,
,
,
.
p q
,
r ; r0
.
(
),
,
,
p
M
-
.
-
.
.
.
a.
M(x y), M0 (x0 y0), a(a1 a2).
,
-
2.
49
x ; x0 y ; y0 = 0:
a1
a2
(8)
1.
M0 (x0 y0)
a(a1 a2)
(8).
(8)
.
,
a2 x ; a1 y + (a1 y0 ; a2x0 ) = 0, . . Ax + By + C = 0,
A = a2, B = ;a1 C = a1y0 ; a2 x0.
,
Ax + By + C, A2 + B 2 6= 0,
M0 (x0 y0)
a(a1 a2),
Ax + By + C = x ;a1x0 y ;a2y0 :
(9)
,
x0 y0 ,
Ax0 + By0 + C = 0.
,
,
x0 = A;+ACB y0 = A;+BCB :
(10)
C = ;Ax0 ; By0 , Ax + By + C = A(x ; x0) + B(y ; y0 ), . .
(9)
a2 = A, a1 = ;B.
,
2.
(;B A)
(2)
,
(10)
.
.
,
n(A B)
(1).
,
(a n) = ;BA + AB = 0.
,
1 2
2.
Ax + By + C = 0
B
.
,
,
.
2
2
2
y = kx + b
k = ;A=B, b = ;C=B.
: k = a2 =a1 ( . 20).
.
a2=a1
.
(11)
,
k
,
4
. .
2
-
. II.
50
.
e1 e2 (
x=0
,
.
y = b.
b
A 6= 0.
x = x0 ,
4.
-
p q.
n = p q] |
,
(r ; r0 n) = 0:
(12) (13)
.
a,
M
.
-
,
(13) D = ;(r0 n),
(r n) + D = 0:
(13) (14),
(r ; r0 n) = 0
n,
,
(13)
-
a.
xyz|
.
(r ; r0 n)
n 26= 0 2 2
+ By + Cz + D (A + B + C 6= 0).
,
r0 n 6= 0,
Ax + By + Cz + D = (r ; r0 n).
:
r
(xe1 + ye2 + z e3 ; r0 n)
-
(14)
-
(r n) + C = 0:
r ; r0
-
r
3.
.
-
(12)
.
(12)
,
.
,
B=0
(11),
M0
(r ; r0 p q) = 0:
,
-
x0 = ;C=A |
,
,
-
. 21).
(11),
-
Ax +
-
:
-
2.
51
Ax + By + Cz + D,
D = ;(r0 n)
A = (e1 n) B = (e2 n) C = (e3 n):
A, B C
,
.
n
10 x 4 . I
(15),
A, B C
.
,
n = (eA ee ee ]) + (eB ee ee ]) + (Ce ee ee ]) :
r0
D = ;(r0 n).
r20 = n.
,
; (n n) = D,
r0 = ;Dn=jnj .
,
n r0
,
2
1
,
,
2
3
3
3
1
2
1
3
1
1
2
2
3
x(e1 n) + y(e2 n) + z(e3 n) ; (r0 n)
(r ; r0 n).
4.
(15)
n
(16)
,
1.
-
A, B, C
Ax + By + Cz + D = 0.
(15)
1 x 4 . I.
a = 1e1 + 1e2 + 3 e3
O e1 e2 e3.
,
(a n) = 1(e1 n) +
+ 2 (e2 n) + 3 (e3 n).
(15)
,
(a n) = A 1 + B 2 + C 3:
(
,
A, B,
C,
,
n,
, (a n)
,
.)
5.
a
1 2 3
Ax + By + Cz + D = 0
,
A 1 + B 2 + C 3 = 0:
(17)
.
,
(17),
.
5
,
,
,
,
. .
4*
. II.
52
,
,
a
6.
+ By + C = 0
,
, 1,
a|
,
2
Ax +
.
,
(
)
(
7.
,
, r0 |
,
(19)
-
-
r0 a.
,
)
.
,
A1 x + B1 y + C1 = 0
,
, . .
A1 = A
B1 = B:
, . .
(20)
,
(c
(20) (21) 6= 0,
(Ax + By) + C1 = 0
M0(x0 y0)
= 0 (Ax0 + By0 ) + C1 = 0.
C1 = C,
,
.
.
.
.
)
(21)
.
,
-
(20)
(;B A) (;B1 A1) |
,
.
-
-
C1 = C:
.
.
(18)
-
.
Ax + By + C = 0
,
-
2
A 1 + B 2 = 0:
r ; r0 a] = 0:
, . .
1
,
| B, A
5.
,
,
.
.
Ax + By + C = 0
, , Ax0 + By0 + C =
,
,
2.
53
8.
,
Ax + By + Cz + D = 0
,
A1 = A
A1 x + B1 y + C1z + D1 = 0
,
, . .
B1 = B
C1 = C:
, . .
)
(22)
-
(22)
,
-
(c
D1 = D:
(23)
.
,
n n1
,
,
n1 = n.
(15) A1 = (e1 n1) = (e1 n) = A. (22).
,
(22)
,
(16)
,
n1 = n.
.
,
7.
(20)
,
(A B) (A1 B1).
(22)
(A B C) (A1 B1 C1).
9 10 x 3 . I
A B =0
A1 B1
|
A B
= CC1 A
A1 = A1 B1 = 0:
,
B C
B1 C1
7
,
,
9.
Ax + By + C = 0
C C1).
C
,
.
(24)
A1x + B1 y + C1 = 0
(24)
(25)
|
(
-
.
C1
A B 6= 0
A1 B1
(x y).
.
. V.
. II.
54
6.
.
,
,
Ax + By + Cz + D = 0
|
,
2
B C
B1 C1
,
.
A1 x + B1 y + C1z + D1 = 0:
(25)
:
+ CC1 A
A1
(26)
t = y;y
0
2
,
(28)
,
,
|
(27)
,
1
(28)
3
(29)
3
(
y ; y0 = z ; z0 :
2
3
x = x0 ,
y = y0 :
.
(30)
,
.
x = x0
.
3
2
1,
.
,
-
(4).
,
x ; x0 = y ; y0 = z ; z0 :
x = x0
2,
1
,
x),
, 1,
x = 0.
(27)
x ; x0 = z ; z0
3
1
6= 0:
0
2
y ; y0 = z ; z0
,
t = z;z
0
1
2
B
+ A
A1 B1
.
t = x;x
,
2
,
(26)
,
, (31)
-
,
.
(26).
-
2.
55
AB1 ; A1 B 6= 0.
z
(x y),
x y,
,
z: x = 1z + 1 , y = 2z + 2 .
,
,
z t,
x = 1 t + 1 y = 2t + 2 z = t:
,
,
(26)
(
.
(26),
(A B C)
,
,
A 1 + B 2 + C 3 = 0,
+ Cz + D = 0.
,
+ B1 2 + C1 3 = 0,
(32)
1.
,
-
z.
|
.
-
M0 (
z = 0.
,
1
A B :
A1 B1
(32)
(26),
(
1
2
.
(32)
-
5
3)
Ax + By +
A1 1 +
, . .
-
,
(27).
0)
,
2
C A
C1 A1
.
2
1).
(A1 B1 C1)
1
,
B C
B1 C1
10.
.
9
.
.
.
x + y + z = 4 x ; y + 3z = 0:
x ; 2y + 3z = 1.
x=
= 1 ; t, y = 1 + t, z = 1 ; t x = 3t ; 1, y = 2t ; 2, z = 1 + t.
2.
3.
,
4.
5.
:
a) r a] = b, (a b) = 0
,
?
,
,
?
. 3.
-
. II.
56
) (r n1 ) + D1 = 0, (r n2 ) + D2 = 0, (n1 n2 ) = 0.
)
= 0.
.
x
3.
1.
M1 M2
.
,
.
(x1 y1 z1) (x2 y2 z2 ).
M1
,
M1 M2 ,
.
(29) x 2
x;x
y;y
z;z
x ;x = y ;y = z ;z :
1
2
(8) x 2
(n1 n2 ) =
1
1
2
1
, M1 M2
(1)
1
2
1
,
.
.
(x1 y1) (x2 y2 ),
x ; x1 y ; y1
x2 ; x1 y2 ; y1 = 0:
,
M1 , M2 M3 |
(x1 y1 z1), (x2 y2 z2)
c
.
M1
M1 M3
(12) x 2 (16) x 4
.I
x ; x1 y ; y 1
x2 ; x1 y2 ; y1
x3 ; x1 y3 ; y1
-
,
2.
3.
.
-
(x3 y3 z3)
.
A1 x + B1 y + C1z + D1 = 0
1
, M1 M2
-
z ; z1
z2 ; z1 = 0:
z3 ; z1
.
(r ; r0 n) = 0 (r ; r0 p q) = 0.
(
,
)
(a n) = 0
(a p q) = 0.
Ax + By + Cz + D = 0,
|
A 1 + B 2 + C 3 = 0:
a(
2
-
(2)
-
3),
,
A2 x + B2 y + C2z + D2 = 0:
5 x2
(3)
3.
57
10 x 2
(3)
A1 B1
1
+ B CC12 A
A2 + C A2 B2 = 0
1 C1
A B
B2 C2
,
(4)
A B C
A1 B1 C1 = 0:
A2 B2 C2
,
4.
n.
-
,
A B C
A1 B1 C1 6= 0:
A2 B2 C2
,
,
.
(5)
.
P
,
-
P n,
M0
M0 M
n
,
=2.
M, r0 |
M0 ,
,
(r ; r0 n) > 0.
.
,
M0 .
,
M1 (r1 ) |
a = r1 ; r0
,
-
r|
,
n,
.
,
,
(4)
M
,
,
(r ; r1 n) = (r ; r0 ; a n) = (r ; r0 n):
3 x2
,
+ D.
,
,
,
(r ; r0 n)
Ax + By + Cz +
-
Ax + By + Cz + D > 0:
,
P
n1 = ;n
(r ; r0 n1) > 0
(r ; r0 n) 6 0:
",
\
"
; r0 n) > 0.
|
.
(r ; r0 n) > 0,
\
(r ;
. II.
58
n.
"
,
:
+ Cz2 + D
,
(;1).
\
,
M1 (x1 y1 z1) M2 (x2 y2 z2)
,
Ax1 + By1 + Cz1 + D
,
.
M0 (x0 y0 z0)
+ A y0 + B z0 + C
,
"
\
.
,
ABC
"
.
,
Ax2 + By2 +
-
\
",
:
\
.
.
x0+
-
,
,
,
Ax + By + C > 0,
,
.
,
Ax + By + C = 0,
Ax + By +
+ C 6 0.
M1 (x1 y1 ) M2 (x2 y2)
,
(Ax1 + By2 + C)(Ax2 + By2 + C) > 0.
5.
.
(r ; r0 n) = 0
M
R. M0 M = R ; r0,
M ( . 22).
n, . .
h = j(R ;jnrj n)j :
(6)
M
(X Y Z),
(6)
3 4 x2 :
0
+ Dj
h = jAXp+ABY+ B+ CZ
+C :
2
6.
,
r ; r0 a] = 0,
R
.
,
R ; r0 a,
2
2
h
(7)
M
-
3.
(
. 23).
59
-
h = j R ;jarj a]j
(8)
-
0
.
Ax + By + C = 0
.
,
-
M0 (x0 y0 ) |
, M(X Y ) |
.
a(;B A).
(25) x 4 . I
,
S = j(X ; x0 )A ; (Y ; y0 )(;B)j.
(9) x 2 S = jAX + BY + C j
+ Cj :
(9)
h = jAXp+A BY
+B
,
(6),
,
n|
.
7.
.
p q
.
,
P Q,
p
P,
q
Q. (
r = r1 + a1 t r = r2 +
+ a2 t,
P
r1
a1 a2.
Q.)
h
P Q
p q.
p q
, P Q
h = 0.
h,
,
r2 ; r1 a1 a2 ,
( . 24).
h = j(r ;j ar aa ]j a )j :
2
2
1
1
.
r = r2 + a2 t
1
2
2
2
,
1.
(r2 ; r1 a1 a2) = 0
,
a1 a2] 6= 0:
-
r = r1 + a1 t
h = 0, . .
. II.
60
8.
,
.
,
,
,
.
,
.
. 0 6 6 =2,
cos > 0,
=2.
,
-
.
,
,
y = k1 x + b1 y = k2x + b2
.
'
,
.
,
tg '
tg ' = 1k+;k kk :
2
,
,
(1 k2) |
.
.
1 x2
,
.
2.
k1 k2
9.
-
1 + k1k2 = 0.
.
.
t
,
(1 k1)
-
,
1 + k1k2.
(10)
1
1 2
-
.
M
r = R + tn
M
. a)
(r ; r0 n) = 0 |
R,
M
,
.
(R ; r0 + tn n) = 0
.
-
r1 = R ; (R ;jnrj n) n:
0
2
:
-
-
3.
61
R
R ; r0
.
.
)
.
r ; r0 a] = 0
M
R. p = R ; r0 a]
,
M.
, p 6= 0,
a p] = a R ; r0 a]]
a
p.
,
.
,
r = R + t a R ; r0 a]]
,
M
.
,
,
a p]
R ; r0
a.
,
.
)
.
,
.
(r n) +
+ D = 0,
|
r ; r0 a] = 0,
a n] 6= 0.
(r ; r0 a n) = 0
.
,
:
(r ; r0 a n) = 0 (r n) + D = 0:
b|
a
.
a
:
(
a
n
)
b = a ; jnj n:
,
,
.
)
.
r = r1 +
+ ta1
r = r2 + ta2
,
. . a1 a2 ] 6= 0.
p = a1 a2]
.
,
(r ; r1 a1 a1 a2]) = 0
(11)
2
(
(r ; r2 a2 a1 a2]) = 0
. 26),
(12)
. II.
62
|
,
(12).
.
10.
.
| a1 a2 ].
-
,
|
A1 x + B1 y + C1 = 0 A2x + B2 y + C2 = 0 |
,
.
(A1x + B1 y + C1) + (A2 x + B2 y + C2) = 0
(13)
2 + 2 6= 0
.
3.
(13)
.
(11), (12).
,
,
( 2+
2
.
(13).
6= 0)
,
(13)
.
( A1 + A2 )x + ( B1 + B2 )y + ( C1 + C2 ) = 0:
,
A1 + A2 = 0 B1 + B2 = 0.
, A1 B2 ; A2 B1 6= 0
9 x2
,
= 0, = 0
,
.
.
,
(13)
.
x0, y0
.
A1 x0 + B1 y0 + C1 = 0 A2 x0 + B2 y0 + C2 = 0
x0, y0
(13),
.
,
,
,
M0 ,
(13).
,
.
M1 (x1 y1 ),
M0 ,
u = A1 x1 + B1 y1 + C1 v = A2 x1 + B2 y1 + C2:
,
u v
,
= ;v, = u.
M1
(13).
,
M1 ,
.
,
( 2 + 2 6= 0)
,
.
,
(x ; x0 ) + (y ; y0 ) = 0
3.
,
63
,
,
,
M0 .
|
.
x ; x0 = 0 y ; y0 = 0.
,
,
-
.
.
4.
,
.
-
,
.
.
.V
|
.
,
-
+ C1z + D1 ) + (A2 x + B2 y + C2z + D2 ) = 0
.
|
.
-
-
.
.
(A1 x + B1 y
2 + 2 6= 0,
,
-
-
,
-
(A1 x + B1 y + C1 z + D1 ) + (A2 x + B2 y + C2z + D2 ) +
+ (A3 x + B3 y + C3z + D3 ) = 0
2 + 2 + 2 6= 0,
,
.
,
.
11.
.
L,
-
,
,
A1 xk1 yl1 + ::: + As xks yls = 0:
(14)
-
x = x0 + a1 t y = y0 + a2t:
L
.
(15)
-
x
y,
t.
-
. II.
64
(15),
(14).
(15) (14):
k
l
k
1
1
s
A1 (x0 + a1 t) (y0 + a2 t) + ::: + As (x0 + a1 t) (y0 + a2 t)ls = 0: (16)
,
t
k1 + l1 ::: ks + ls .
,
,
.
k1 + l1 ::: ks + ls |
L.
(16)
.
,
,
,
,
.
,
,
,
.
5.
,
,
.
,
(x2 + y2 )2 ; 1 =
.
0.
1.
,
A1 x + B1 y + C1 = 0
?
6.
+ a0 t
.
,
A2 x + B2 y + C2 = 0.
,
y = 5 ; 5t, z = 3 + 2t.
5.
.
.
.
r= '
,
,
A(20 ;15), B (;16 0) C (;8 6).
2.
4.
x2 + y 2 = 0
|
|
,
3.
,
-
x = 1 + 2t, y = 2 + 3t, z = ;t x = 4t,
A(0 1 0)
x + y = 0, x ; y = 0 x + y + 4z = 0.
A(1 2 3), B (1 5 ;1) C (5 3 ;5).
,
,
,
r = r1 + a1 t
.
r = r2 + a2 t.
r = r0 +
III
1.
x
)
2
2
Ax + 2Bxy + Cy + 2Dx + 2Ey + F
A, B
,
,
=0
C
.
,
.
,
(1)
(1)
.
-
,
.
xy
x y
(8) x 3 . I
x = x cos ' ; y sin ' y = x sin' + y cos ':
(1)
A(x cos ' ; y sin ')2 + 2B(x cos ' ; y sin ')
(x sin ' + y cos ') + C(x sin ' + y cos ')2 + ::: = 0:
x y
,
.
xy
.
,
,
xy
B = ;A sin ' cos ' + B(cos2 ' ; sin2 ') + C sin ' cos ':
B=0,
.
B=
6
6= 0,
' ,
B
.
'
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2B cos 2' = (A ; C) sin 2':
A = C, cos 2' = 0,h
' = =4.
i
' = 12 arctg A2;BC .
)
2B , 2D
.
2E ,
,
(2)
A 6= C,
-
. III.
66
.
A x 2 + C y 2 + 2D x + 2E y + F = 0:
(3)
,
.
,
.
0
(1)
-
0
0
0
0
0
0
1.
0
0
(3)
-
0
(3)
,
,
,
2
D
2
A x + A x +D
A
0
0
0
.
, A 6= 0.
(3)
D
2
+ C y + 2E y + F ; A = 0:
,
0
02
0
02
0
-
0
0
0
0
02
0
-
x = x + D =A , y = y ,
A x 2 + C y 2 + 2E y + F = 0
.
.
, AC =
6 0, . .
.
1
00
0
0
0
00
0
0
00
0
0
00
0
00
00
0
-
A x 2 + C y 2 + F = 0:
0
00
0
A1. A C > 0 |
0
F
0
.
.
C
.
0
A C:
.
x +y =1
a
b
a2 = ;F =A , b2 = ;F =C .
,
a > 0, b > 0 a > b.
,
,
x =y y =x :
.
,
00
002
0
00
a=b
.
,
00
00
,
|
(5)
-
2
,
(5)
F
0
0
00
a > b,
0
00
002
2
00
(4)
F
00
A
0
A1a.
00
(5)
|
.
(6)
-
a.
1.
A1 .
F
67
A
00
C .
00
00
x 2 + y 2 = ;1:
a2 b2
00
00
,
(7),
.
A1 . F = 0.
00
a2 x 2 + c2y = 0:
x = 0, y = 0.
(8),
.
(10).
A C
00
0
0
00
F
,
00
0
00
0
,
00
.
.
0
.
0
00
-
00
a2x 2 ; c2 y 2 = 0:
00
00
(6),
A
0
1,
.
.
,
.
C
,
A C = 0, ,
.
,
A = 0.
,
0
0
C y 2 + 2D x + F = 0:
00
0
00
.
0
00
0
00
00
C 6= 0,
0
00
.
,
00
0
-
00
D 6= 0.
C y 2 + 2D x + 2FD = 0:
0
00
,
0
0
0
5*
(10)
ax ; cy ax + cy ,
,
(10)
00
,
1.
(9), -
|
A2 . F = 0.
,
(9)
0
,
,
-
00
00
-
(6),
A.
x 2 ;y 2 =1
a2 b2
2
2
a = ;F =A , b = F =C .
.
,
00
(8)
-
00
00
A2. A C < 0 |
F
A2a. F 6= 0.
,
(7)
.
-
:
. III.
68
-
x = x + F =2D , y = y .
00
00
0
00
C y 2 + 2D x = 0
00
0
y 2 = 2px
,
p = ;D =C .
0
0
.
2.
,
F
2 .C F <0|
00
0
.
p > 0,
,
: xe = ;x , ye = y .
,
p > 0,
(11)
(11)
,
,
|
D = 0.
0
C
00
F
0
.
.
00
y 2 ; a2 = 0:
0
0
00
00
F
0
y +a
.
00
0
00
2 . F = 0.
C
2
y = 0:
00
.
0
(13),
1.
y = 0,
00
,
.
.
(1).
,
:
x
y
1) a + b = 1 2) xa + yb = ;1 3) a2 x2 + c2y2 = 0
2
2
2
2
2
2
2
2
(13)
.
(14)
00
.
(12)
y ; a.
.
00
C,
y 2 + a2 = 0:
,
00
0
,
C
-
C,
00
2 .CF >0|
-
C y 2 + F = 0.
00
, ,
.
-
(14),
-
2.
,
69
4) xa ; yb = 1 5) a2 x2 ; c2y2 = 0 6) y2 = 2px
7) y2 ; a2 = 0 8) y2 + a2 = 0 9) y2 = 0.
7)
2
2
2
2
: 1)
) 4)
3)
5)
(
9)
2)
1.
(
-
6)
8)
).
-
.
3x2 + 10xy + 3y2 ; 2x + 2y ; 9 = 0:
2.
9x2 ; 24xy + 16y2 ; 34x ; 38y ; 9 = 0:
3.
,
?
-
4.
?
5.
je1 j = je2 j = 5, (e1 e2 ) =
x2 +
= 7.2
+ y = 1?
A+C
,
6.
(1)
-
.
x
2.
,
.
1.
.
jyj 6 b.
,
.
,
x2 + y2 = 1
a2 b2
a > b > 0.
(1)
,
,
(1)
jxj 6 a
,
.
.
(a 0), (;a 0), (0 b)
a b
.
2a 2b.
(0 ;b),
-
. III.
70
,
(x y)
,
-
. M1 ,
M
(;x y), (x ;y) (;x ;y)
M2 M3 ( . 27).
1.
|
.
a
: x2 + y2 = a2.
jxj < a,
x
,
p
2 =a2
b
1
;
x
p
a 1 ; x2=a2.
b=a.
c > 0.
(;c 0)
,
c = 0,
,
-
2.
,
c2 = a2 ; b2
F1 F2
( . 29).
" = ac
-
.
,
,
.
-
,
b=a (
. 28).
(2)
(c 0)
.
.
(3)
.
,
( .
" < 1.
M(x y),
. 29)
2.
,
71
x:
r1 = jF1M j = a ; "x r2 = jF2M j = a + "x:
,
r12 = (x ; c)2 + y2 .
2
y,
.
b
x
r12 = x2 ; 2cx + c2 + b2 ; a :
(2),
r12 = a2 ; 2cx + cax = (a ; "x)2 :
x 6 a " < 1,
,
(4): r1 = a ; "x.
3.
,
2a.
:
,
,
r1 + r2 = 2a:
.
M(x y)
(5), . . p
p
(x ; c)2 + y2 = 2a ; (x + c)2 + y2 :
2
(4)
-
2
2
2 2
2
:
= a2 b2,
p
xc + a2 = a (x + c)2 + y2 :
(2).
(1).
.
(
. 30)
x = a"
-
.
, -
(4)
(5)
-
(6)
,
b2x2 + a2 y2 =
-
,
-
x = ; a" :
,
4.
(7)
.
,
,
, -
".
.
F2(;c 0).
M
M(x y) |
-
. III.
72
x = ;a="
,
,
(4)
" = c=a,
.
M0
y0 < 0 |
(9) x 3 . II
d2 = x + a" = 1" ("x + a):
,
r2=d2 = ".
r2 =d2 = ", . .
p
(x + c)2 + y2 = " x + a" :
(6),
,
.
,
M0 (x0 y0 ) |
y0 6= 0. y = f(x), p
. ( y0 p
>0
f1 (x) = b 1 ; x2 =a2,
f2 (x) = ;b 1 ; x2=a2 .
y0 f(x).)
x + (f (x)) = 1:
a
b
x:
2x + 2ff = 0:
a
b
f(x0 ) = y0 ,
f
x0,
:
f (x0) = ; ab xy :
:
y ; y0 = ; ab xy (x ; x0 ):
,
,
b2x20 + a2 y02 = a2b2 ,
M0
xx + yy = 1:
(8)
a
b
(8)
(a 0)
y0 6= 0.
,
x = a x = ;a.
.
,
,
y
.
,
(8)
M0(x0 y0)
.
5.
M0(x0 y0)
,
,
.
2
2
2
2
0
x = x0
2
2
2
0
2
2
0
2
.
0
2
(;a 0),
x
,
0
0
2
0
0
2.
,
73
'1 '2 ,
F1M0 F2M0
n,
( . 31). 2
(8)
,
n(x0=a y0=b2 ),
(F1 M0 n) = xa (x0 ; c) + yb y0 =
= 1 ; xa c = a ;a"x :
(4),
,
cos '2 = 1=(ajnj).
2.
.
0
2
-
0
2
0
0
2
cos '1 = 1=(ajnj).
.
,
-
x2 ; y2 = 1:
a2 b2
,
. .
(9)
jxj > a,
. 32).
(;a 0),
.
.
2a (
(a 0)
,
.
.
.
b
a
,
6.
,
,
x=0
.
y = kx,
.
x ; k x = 1:
a
b
b2 ; a2 k2 > 0,
x = p ab :
b ;a k
2
,
-
,
.
|
-
2
2
,
2
2
2
2 2
(ab=v abk=v)
-
. III.
74
v = (b2 ; a2k2 )1=2.
(;ab=v ;abk=v),
ab=v
k(
,
,
b=a.
= bx=a
.
, k2
,
;b=a.
y = ;bx=a
.
,
.
,
,
y = bx=a y = ;bx=a
bx ; ay = 0
M(x y)
h1 = jpbx ; ayj
a +b
y = bx=a:
,
,
-
,
. 33.
.
M
2
2
2
2
2 2
2
2
,
c,
2
2 2
2
2
, ,
c2 = a2 + b2
-
bx + ay = 0.
h2 = jpbx + ayj :
a +b
, b2x2 ; a2 y2 = a2b2 ,
h1 h2 = jb xa ;+ ab y j = a a+b b :
7.
a2 b2=(a2 + b2 ).
.
8.
,
2
,
,
k
k
-
(a 0).
, x ,
k
y=
b=a
.
-
k = 0.
,
. 33).
,
h1
.
h2
,
.
(10)
2.
,
75
c > 0.
F1
(c 0) (;c 0)
" = c=a,
.
" > 1.
9.
x:
.
= 2a.
-
.
,
-
M(x y)
r1 = jF1M j = ja ; "xj r2 = jF2M j = ja + "xj:
2,
(11)
(x > a)
r1 = "x ; a r2 = "x + a
(x 6 ;a)
r1 = a ; "x r2 = ;"x ; a:
r2 ; r1 = 2a,
,
,
10.
,
jr2 ; r1j = 2a:
,
2a.
(11)
:
-
r1 ; r2 =
(12)
M
-
-
.
p
-
p
(x ; c)2 + y2 = 2a + (x + c)2 + y2 :
(10),
x = a"
,
F2
,
.
x = ; a" :
,
(2).
,
,
3
,
(13)
-
, ,
.
. III.
76
11.
"(
,
(9) x 3 . II
M
.
. 36).
-
4.
,
F2(;c 0).
M (x y) |
.
x = ;a="
d = x + a" = 1" j"x + aj:
,
r =d = ".
M0 (x0 y0),
,
(8)
xx ; yy = 1:
(14)
a
b
12.
M0(x0 y0)
,
0
0
0
.
5.
,
.
.
0
0
2
0
2
3.
-
0
(11)
,
,
,
,
,
(15)
y2 = 2px
p > 0.
(15)
-
,
.
,
y = ax2 .
,
,
2p = a 1.
;
F
.
(P Q
x > 0.
.
(p=2 0)
. 37).
x = ;p=2
2.
,
77
13.
,
M(x y),
r = x + p2 :
; p=2)2 + y2
: r2 = (x ;
-
y2
.
2
2
p
r2 = x ; 2 + 2px = x + p2 :
x>0
,
M
9 x 3 . II
d = x + p2 :
.
14.
M
r
,
(16).
-
,
.
:
,
.
M(x y)
x ; 2p + y2 = x + p2 :
2
-
(15).
.
,
r ="
d
.
y0 6= 0.
yp= f(x),
= ; 2px, 2
y0 .)
(f(x)) = 2px,
x = x0 f(x0 ) = y0,
.
(16)
M(x y)
" = 1.
,
-
.
M0 (x0 y0 ), M0
p
y=
. ( y = 2px
f(x)
2f(x)f (x) = 2p.
f (x0 ) = p=y0 .
0
0
y ; y0 = yp (x ; x0 ):
0
yy0 = p(x + x0):
,
y02 = 2px0.
(17)
. III.
78
,
y0 6= 0,
,
(17)
,
x = 0, . .
(17)
.
.
15.
M0
,
M0
( . 38).
,
,
,
-
-
M0 (x0 y0 ).
(17)
v(y0 p).
, (v e1) = y0
cos '1 = y0 =jvj.
FM0
x0 ; p=2 y0 ,
(F M0 v) = x0 y0 ; p2 y0 + py0 = y0 x0 + 2p :
, cos '2 = y0 =jvj.
jFM0j = x0 + p=2.
.
,
jFN j = jF M0j ( . . 38).
,
1.
(
2.
.
3.
4.
5.
6.
u v.
,
,
,
,
.
?
7.
8.
x 3.)
1=u2 + 1=v2 .
u
|
-
,
v|
,
,
,
,
,
1=u + 1=v
,
.(
.
)
,
.
; y + 7 = 0.
9.
.
y2 = 12x
,
(8)
,
?
.
,
x;
-
.
x0 y0
.
,
.
-
3.
x
3.
1.
,
79
,
,
2
Ax + 2Bxy + Cy2 + 2Dx + 2Ey + F = 0
,
x = x0 + t y = y0 + t:
t,
,
.
(1)
(2)
,
(2) (1):
A(x0 + t)2 + 2B(x0 + t)(y0 + t) + C(y0 + t)2 +
+ 2D(x0 + t) + 2E(y0 + t) + F = 0: (3)
,
2
(4)
P t + 2Qt + R = 0
P = A 2 + 2B
+C
2
Q = (Ax0 + By0 + D) + (Bx0 + Cy0 + E)
,
,
Q = (A + B )x0 + (B + C )y0 + D + E :
|
t = 0, . .
R = Ax20 + 2Bx0 y0 + Cy02 + 2Dx0 + 2Ey0 + F = 0:
,
(4)
,
,
,
(
),
(
\
"
,
P = 0,
2
2
A + 2B + C = 0
,
,
(4)
.
Q 6= 0,
Q=0
(
R = 0),
.
\
"
,
,
(9)
,
,
.
.
,
,
(9),
.
(5)
(6)
(7)
(8)
).
. .
(9)
,
.
.
-
-
. III.
80
2.
.
,
.
= BA BC
,
1.
.
A = C = 0.
2B = 0,
2)
C 6= 0.
,
k= = ,
< 0,
,
(0 1)
> 0.
Ck2 + 2Bk + A = 0. B 2 ; AC = ; .
,
< 0,
=0
> 0.
2,
,
= .
-
A 6= 0
,
-
,
.
= ;B 2 < 0.
(9)
(1 0) (0 1).
B 6= 0
1)
3)
= 0,
.
,
-
.
-
(9).
,
,
,
-
.
,
.
,
|
. 39).
(
,
= 0,
3.
,
,
,
> 0.
,
,
.
,
< 0,
,
,
-
-
.
3.
,
.
.
81
,
,
.
,
M0 (x0 y0)
.
,
4.
,
O
)
.
. .
(2)
,
(4)
-
( . 40).
Q = 0.
(7),
)(
)
(A + B )x + (B + C )y + D
.
(10)
,
(
,
,
,
,
,
.
,
,
:
?
,
A +B = 0 B +C
,
(9),
,
(10)
.
(1)
(x y)
.
O(x0 y0 )
(x y) = 0,
(x0 + y0 + ) = (x0 ;
,
,
.
,
,
6
,
.
.
,
-
, ,
.
-
,
+ E = 0:
(10)
-
):
.
(10) -
, . .
= 0:
|
.
.
a(
y0 ; ):
.
-
)
,
(x0 y0)
,
(11)
(11).
-
. III.
82
(xe0 ye0)
?
a ;a .
,
.
,
, -
O
,
:
,
,
,
,
,
e (xe ye) = (x y).
,
e
-
e (x
e ye),
.
y=0
y2 + 1 = 0.
,
-
.
(11).
A(x0 + )2 + 2B(x0 + )(y0 + ) +
+ C(y0 + )2 + 2D(x0 + ) + 2E(y0 + ) + F:
.
(x0 ; y0 ; )
(x0 + y0 + )
,
,
,
.
(Ax0 + By0 + D) + (Bx0 + Cy0 + E) = 0:
(12)
(11),
(12)
,
,
= 1, = 0
= 0, = 1.
,
(x0 y0)
Ax0 + By0 + D = 0
Bx0 + Cy0 + E = 0:
,
,
.
,
,
(11).
(13)
,
= BA B
6 0:
C =
6= 0
,
.
,
,
(13)
(13), , (12),
,
9 x 2 . II
.
(14)
, -
,
-
3.
,
83
,
=0
,
(
(
.
9 x 2 . II).
(
),
).
.
.
=0
,
|
2.
O(x0 y0 ),
,
-
|
M(x y)
M1 (x1 y1) OM1 = ;OM.
x = x0 + , y = y0 + ,
(11)
(x y) = 0
.
O
.
0
2)
6= 0):
M,
;t1 .
Q = 0.
6*
-
t = 0,
-
(4)
-
0
0
;
,
.
,
Q = 0.
,
O
(12)
,
.
( ) (
)
(Ax0 + By0 + D) + (Bx0 + Cy0 + E) = 0
(Ax0 + By0 + D) + (Bx0 + Cy0 + E) = 0
0
-
,
.
M1
OM ,
( )|
x1 = x0 ; , y1 = y0 ; .
,
(x1 y1) = 0.
3.
O(x0 y0 ), O
.
O,
:
1)
O
.
.
O|
.
.
,
,
,
(13)
O
0
.
(13),
0
0
.
t1 6= 0,
Pt21 + 2Qt1 + R = 0 Pt21 ; 2Qt1 + R = 0,
,
,
(13)
.
(
O, ,
O.
,
,
,
0
;
-
. III.
84
.
5.
.
,
(10)
(
1 x1
:
.
.
,
).
0
A
-
0
-
.
+ B ).
)(
0
0
+
) + C = 0:
( ) (
)
0
,
(
0
0
.
0
),
,
. 41).
(16)
-
0
4.
0
+ B(
(15)
0
0
.
(
0
(
(
(
)
-
0
(A + B ) + (B + C ) = 0
0
-
.
(
),
( ),
(
),
6 x 2 . II
0
-
,
0
0
),
)
,
(15)
,
;(B + C ) (A
+
-
(9)
:
A(B + C )2 ; 2B(B + C )(A + B ) +
+ C(A + B )2 = 0:
(AC ; B 2 )
(A 2 + 2B + C 2 ) = 0.
,
.
5.
( = 0),
( )
(
. 42).
|
( 6= 0),
3.
6.
,
,
85
,
.
,
.
,
.
-
.
(
)
(
.
,
0
0
)
-
,
,
( )
(10), . .
A +B =
B +C =
,
-
-
:
(17)
:
(A ; C) + B( 2 ; 2) = 0:
(18)
= cos ', = sin',
(18)
(2) x 1,
,
,
'.
6.
.
(18)
,
,
.
, A = C, B = 0.
: x2 + y2 = a2 , x2 + y2 = ;a2
x2 + y2 = 0.
,
.
7.
.
, ,
.
,
(1).
.
,
:
,
,
,
-
.
.
.
,
M0 ,
,
M0 (x0 y0 ),
,
.
(2).
.
L,
,
-
. III.
86
(4),
.
L
.
,
P = 0,
,
t=0
.
L,
,
,
,
Q = 0.
,
(4) R = 0,
-
,
Q = 0.
Q=0
,
:
(Ax0 + By0 + D) + (Bx0 + Cy0 + E) = 0:
(19)
M0
,
,
(19)
.
M(x y)
,
M0 M
a( ), . .
x ; x0 y ; y0
,
( ):
(Ax0 + By0 + D)(x ; x0) + (Bx0 + Cy0 + E)(y ; y0 ) = 0: (20)
L
M0 ,
.
(20)
,
,
M0
(1) ,
,
(Ax0 + By0 + D)x0 + (Bx0 + Cy0 + E)y0 + Dx0 + Ey0 + F = 0:
,
(20)
Axx0 + B(xy0 + x0 y) + Cyy0 + D(x + x0) + E(y + y0 ) + F = 0: (21)
8.
.
,
|
,
.
,
.
(x0 y0 )
Ax0 + By0 + D = 0 Bx0 + Cy0 + E = 0
Ax20 + 2Bx0 y0 + Cy02 + 2Dx0 + 2Ey0 + F = 0:
x0,
y0
.
-
Ax0 + By0 + D = 0
Bx0 + Cy0 + E = 0
Dx0 + Ey0 + F = 0:
p(A B D), q(B C E) r(D E F ).
x0p + y0 q = ;r:
(22)
(22)
(23)
3.
,
,
,
= 0.
A B D
= B C E = 0:
D E F
,
p q
(24)
7.
,
= 0.
< 0, = 0
> 0, = 0 |
(
,
(13)
= 0,
:
,
,
,
q
,
(23)
,
p, q
,
,
(23), . .
= =0
,
7 8
:
.
1.
3.
,
.
4.
?
)
,
=0
r
.
.
.
, ,
.
-
r.
.
,
=0
.
=0
-
-
.
,
p
.
-
,
,
|
).
-
,
,
.
,
3x2 + 10xy + 3y2 ; 2x + 2y ; 9 = 0.
,
(22).
,p q
,
,
,
|
= 0.
|
r
(24)
p q
9 x 2 . II,
,
,
,
2.
.
.
8.
(
87
,
,
.
-
. III.
88
,
3x + 10xy + 3y ; 2x + 2y ; 1 = 0:
5.
2
6.
7.
2
. 5?
x2 ; 2xy + 3y2 = 3
M0 (0 1).
x
4.
,
x2
,
. VIII.
-
,
S
d
. .
|
O e1 e2 e3
d,
P.
.
x2 + y2 .
,
,
.
,
.
L,
d(
.
e3
, O e1 e3 |
L
,
M
L.
. 43),
.
P.
f(x z) = 0.
M(x y z).
d
M1 (x1 y1 z1p)
, z1 = z jx1j = x2 + y2 ,
,
M.
L: f(x1 z1 ) = 0.
-
d,
.
,
p
-
.
1.
P,
-
M1
M,
e1
-
d,
M
M1 ,
P,
M1
-
y1 = 0.
x1 z1 ,
, ,
-
4.
,
89
M
.
f
2.
,
x2 + y 2 z = 0
f
p
.
,
x2 + z 2 = 1
a2 c2
c
x2 + y2 + z 2 = 1
a2
c2
0
0
0
,
.)
-
z 2 + x2 + y2 = 1 (a > c):
a2
c2
(
0
0
x2 + y2 + z2 =1
a2 b2 c2
b = a.
,
(4),
0
-
(3)
. 44).
<
x = x, y = y, z = z.
M (x y z ),
,
0
L
:
z 2 + x2 = 1:
a2 c2
M(x y z)
y=0 ,
< 1.
(2)
d.
.
,
(1)
(1)
-
p
x2 + y2 z f ; x2 + y2 z = 0
e3
(
S:
p
0
0
0
(4)
(
. 45).
-
. III.
90
,
.
,
(4)
b = c,
,
,
,
,
.
,
.
|
x2 + y2 + z 2 = a2
= b=a = c=a.
y=0 z=0
3.
.
.
,
,
O e1 e3
,
y=0
a2x2 + b2 y2 ; c2 z 2 = 0
,
(6) |
|
.
-
x 1 . II.
.
-
x ;z =1
a2 c2
2
2
(1)
(
(5)
. 46).
(6)
,
,
4.
-
a2 x2 ; c2 z 2 = 0.
(
,
.
. 47)
x + y ; z = 1:
a
c
2
2
2
2
2
y=0
x2 + y2 ; z 2 = 1:
a2 b2 c2
.
|
-
,
P
a2(x2 + y2 ) ; c2z 2 = 0
-
(7)
(8)
|
,
-
4.
91
.
,
(8)
x+z
a c
x ; z = 1+ y 1; y :
a c
b
b
x + z = 1+ y
a c
b
x ; z = 1; y
a c
b
( 2 + 2 6= 0).
,
|
(8),
,
.
0
0
,
=1
(9),
.
0
0
1 ; yb
1 + yb :
.
.
: x = z, y = 1.
,
.
.
|
,
(10)
,
x ; z = 1 ; y:
M0
M0 (10),
5.
(9)
(9)
,
.
x2 + y 2 ; z 2 =
M0
:2 =2 0 =0 .
,
.
-
x+z = 1+y
M0 ,
0
(9)
,
.
x+z =
a c
x;z =
a c
M0 (1 1 1)
(9),
-
.
0
0
,
,
,
: 2 = 0 2 = 0.
0
0
,
-
.
z 2 ; x2 = 1
c2 a2
-
. III.
92
,
.
y=0
,
(1)
z 2 ; x2 + y2 = 1:
c2
a2
(11)
-
z 2 ; x2 ; y2 = 1:
c2 a2 b2
(12)
(12),
. 48).
(
(\
,
.
,
6.
.
,
-
.
x2 = 2pz -
x2 + y2 = 2pz:
,
x2 + y2 = 2z:
a2 b2
,
7.
(14)
x=
O ( 0 0).
0
,
.
-
. 49).
-
x2 ; y2 = 2z:
a2 b2
.
(13)
(14)
(
.
,
.
y=0
,
")
-
.
(15)
(15)
-
O e2 e3
0
-
4.
|
O
; yb = 2 z ; 2a :
,
00
00
2
2
2
2
(0
( 0
,
e3,
O,
,
O
,
93
(
O e1 e2 e3
2=(2a2 )).)
e3.
00
,
2=(2a2 )).
(16)
,
,
,
.
O
00
O
00
.
,
z = 2xa
O e1 e2 e3.
2
,
,
.
. 50).
(
|
.
|
x;y =
a b
,
-
. 52).
. 51.
,
-
-
:
,
|
,
e3 .
,
-
y=0
2
y = 0.
(16)
, ,
x=
,
z=
,
(
x+y =2 z
a b
. III.
94
|
1.
,
0
x+y =
a b
,
.
0
0
x ; y = 2 z:
a b
0
,
.
-
,
?
,
:
) x = 1 + t, y = 3 + t, z = 3 + t ) x = 1 + t, y = 1 + t, z = 3 + t.
3.
,
,
,
.
4.
(15)
y = 0, x2 = 2a2 z x = 0, y2 = ;2b2 z .
A1 B1
A2 B2
z = 0.
,
A 1 B 2 , A 1 A2 , B 1 A 2 B 1 B 2
.
5. 2
2
2
2
2
2
;x + y ; z = 1
5x ; 3y + 4z = 0
z = 0.
6.
,
.
2.
IV
1.
x
1.
A
A,
.
,
2.
,
B = f (A).
A|
,
R
.
P
,
f:
P
,
.
,
P R
,
,
.
-
,
P
,
R
.
,
,
3.
B
.
,
2.
R.
,
,
1.
B
B.
,
.
R.
P ! R.
-
R
P
,
R,
R,
,
M
P
R.
.
.
,
p
.
,
R
,
:
,
.
> 0.
R
-
.
p
. IV.
96
N.
M
=M (
f
f (M)
N f (M) =
p,
. 53).
M
NM.
.(
-
f (M) =
p
> 1,
,
-
.)
x2
x 4 . III
.
4.
x y
x y,
,
x = x2 ; y2 y = 2xy
,
R
,
,
.
O
P
O,
f (M),
Of (M) = arctg j OM j OM:
j OM j
R.
5.
7.
.
g(f (A))
.
.
P
.
,
.
h,
P
-
f:
S,
-
p,
,
.
O.
,
.
R.
8.
! S.
p,
=2
,
,
6.
,
p|
,
|
P
,
,
|
P R
f (O) = O.
3.
.
-
. III,
A
P !R
g
: R!
P f
1.
97
g
,
,
,
gf .
.
,
,
,
,
,
,
,
, . . gf
,
f : P ! R,
P, fg |
.
,
,
fg.
g : R ! P.
g
a, f |
O.
. 54
g(f (A)).
f, g
.
h
(fg)h = f (gh):
,
A
f (g(h(A)))
f (g(h(A))).
P
-
.
7
. .
,
.
,
R
.
,
f: P
4{7 |
.
-
1
f:
.
R
.
-
,
.
.
fg
f
f
fe = ef = f :
,
,
.
,
e
,
7
-
|
,
g(h(A))
|
R.
,
gf
-
f (g(A))
h(A)
.
.
-
P !R
4 6
,
, . .
!R
R
,
,
2 3,
5, 6
-
-
. IV.
98
P.
f
f (A).
f (A)
A.
.
f.
.
f ;1 ,
,
f
f ;1 (f (A))
=A
A
,
,
P
-
,
,
P
P.
A
f ;1 f
= e, e |
.
,
f (f 1(f (A))) = f (A)
f (f 1(B)) = B
B
.
ff 1 = e.
,
,
f 1
(
),
f.
1.
f g
P
.
fg
(fg) 1 = g 1 f 1 .
,
f 1 g 1.
1
1
(fg)(g f ).
f (gg 1)f 1 .
fef 1 = ff 1 = e.
,
fg
.
;
,
;
;
;
;
;
;
;
;
f:
A = f (A)
Q p1 p2,
A |
(x y)
,
P ! R.
O e1 e2,
A
(x y ).
P R
(x y ).
R.
p1, p2
:
Q
,
-
.
.
A
P
R
,
,
,
x = '(x y) y = (x y):
,
;
fg
.
4.
-
;
;
;
P
,
;
;
,
-
e1 e2.
,
4.
-
,
(x y),
-
:
(1)
P R
O,
-
2.
.
99
,
(1)
P R
,
.
P R.
.
1.
)
)
)
)
)
2.
3.
,
-
,
?
f, g
,
,
,
.
x
1.
,
,
?
?
?
.
-
,
.
h
fgh.
-
x+y = 5
2.
.
,
A
,
, . .
B
f
jAB j = jf (A)f (B)j.
,
,
.
O e1 e2.
e2 = OB (
. 55).
A B
: e1 = OA,
OAB
OAB.
M(x y).
M
(x y )
(x y).
OM = xOA + yOB.
O M = xO A + yO B .
,
1,
OM
7*
(x y ).
OA
OB
.
,
100
. IV.
OM e1 e2,
.
OM = OO + O M = OO + xO A + yO B :
(1)
O A e1.
jO A j = 1
'
e1, e2
(cos ' sin '). OB
( sin ' cos '),
,
OA OB
,
e1 e2.
O
(c1 c2).
(1)
:
x = x cos ' y sin ' + c1
(2)
y = x sin ' y cos ' + c2:
,
1.
(2), ' |
,
, c1 c2 |
.
,
,
,
.
1.
c
M
(x y)
M
x = x + c1 y = y + c2
c1 c2 |
c.
2.
'
,
.
O=O ,
, c1 = c2 = 0.
.
,
x = x cos ' ; y sin ' y = x sin ' + y cos ':
3.
.
.
M(x y)
M
x = x y = ;y:
c1 = c2 = 0 ' = 0
(2).
2.
.
,
.
.
f
P
,
P
,
2.
f
101
x = a1 x + b1y + c1
y = a2 x + b2y + c2 :
.
(3),
a1 b1
a2 b2
(4).
.
9 x 2 . II.
(x y )
.
x ; c1 y ; c2 (
,
1,
(4)
.
.
4.
.
(x y)
,
(4)
(3)
,
(4).
x
-
,
3 x 1)
(
-
-
,
|
5.
|
|
.
M
y)
.
,
,
6 0:
=
,
x y,
,
(3)
.
2.
,
(3)
-
.
,
,
(3)
,
6.
,
x = x y = y:
.
(
,
x = x y = 0:
.
,
,
OM = OM.
M
6 x 1)
O|
x = x
,
y = y:
OM
.
,
M
. IV.
102
|
7.
C,
c1 c2 |
.
,
C.
,
.
3.
.
(3)
O e1 e2.
0
O e1 e2.
M(x y)
(x y )
(7) x 3 . I:
x = 1 x + 1 y + 1 y = 2x + 2 y + 2 :
(5)
M
M
,
,
(x y )
(x y ).
,
,
:
x = 1 x + 1y + 1 y = 2 x + 2 y + 2:
(6)
(x y ) M
(x y )
M.
(3):
(6)
x y
x = 1(a1 x + b1y + c1) + 1 (a2x + b2y + c2 ) + 1
y = 2 (a1x + b1y + c1) + 2 (a2 x + b2y + c2 ) + 2 :
,
|
1
x y:
x = A1 x + B1 y + C1 y = A2 x + B2 y + C2:
(7)
x y
(5),
:
x = A1 ( 1x + 1 y + 1 ) + B1 ( 2x + 2 y + 2 ) + C1
y = A2 ( 1x + 1 y + 1 ) + B2 ( 2x + 2 y + 2 ) + C2:
,
|
1
x y.
.
,
,
.
,
,
,
(4).
3.
.
3
,
0
,
x = c1 , y = c2 ,
,
.
,
(3)
.
, (3).
-
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2.
4.
103
1
.
.
1
.
|
.
g
x = a1x + b1y + c1
g(f (M))
x = d1 x + e1 y + f1
(9) (8)
M.
1,
1 x1
4.
M1
.
(4)
y
M1 M2 .
O e1 e2
f
.
M
f
f (M)
y = a2 x + b2 y + c2
(8)
-
f (M)
y = d2x + e2 y + f2 :
g(f (M))
,
.
,
-
(9)
.
-
x y.
,
b2,
| b1
(a1b2 ; a2b1 )x = b2 (x ; c1) ; b1(y ; c2 ).
, x|
x y. .
.
M1 M2
x1 y1 x2 y2 ,
x2 ; x1 y2 ; y1 .
(3)
.
M2
f
(3)
-
.
5.
,
-
(3),
M 2 M1
x2 = a1x2 + b1 y2 + c1 x1 = a1 x1 + b1y1 + c1 :
,
M1 M 2
x2 ; x1 = a1(x2 ; x1) + b1(y2 ; y1 ):
y2 ; y1 = a2 (x2 ; x1) + b2(y2 ; y1 ):
,
M1 M2
M1 M2 ,
M1
. IV.
104
M2
,
.
.
,
-
,
.
6.
.
1
2
,
:
f
.
|
a
(10)
a b
,
1,
2
,
,
,
f (a + b).
(10)
-
,
f
ef
,
a
f
f (a).
f (a + b) = f (a) + f (b)
f ( a) = f (a):
,
-
2
= a1 1 + b1 2
= a2 1 + b2 2:
,
,
1
,
f
(11)
.
1
2
|
= a1( 1 + 1 ) + b1 ( 2 + 2 ) 2 = a2 ( 1 + 1 ) + b2( 2 + 2 )
a b.
1 2
1 2 |
=
(a
+
b
)
+
(a
+
b
1 1
1 2
1 1
1 2) = 1 + 1
1
2 = (a2 1 + b2 2 ) + (a2 1 + b2 2 ) = 2 + 2 :
|
.
(11)
.
(11)
,
f .
,
, f (0) = 0.
a+ b= 0
f (a) + f (b) = 0.
,
.
,
f (a) + f (b) = 0, 2 + 2 6= 0,
a + b = 0.
,
.
7.
f
O e1 e2
(3).
c1 c 2 |
f (O), a1 a2 b1 b2 |
f (e1 ) f (e2 )
O e1 e2.
1
2.
105
y=0
c2 .
O
=0
a1 a2.
1
= a1 ,
2
(10)
= a2.
8.
,
f
N
, L LM LN |
L , a1 a2 b1 b2 |
.
x = a1 x + b1 y + c1
.
.
f (O)
,
f (e1 )
.
9.
M
,
LM
L M
,
x=0
c1
= 1, 2 =
b1 b2.
L M N,
1
N
L = f (L), M =
.
.
.
c1 c2 |
LM LN
LN
y = a2 x + b2 y + c2
f,
,
,
7,
.
,
N
, . .L ,M
f (O)
e1
, f (e1)
f (e2 )
,
= f (M) N = f (N).
,
L ,M
(4),
(3)
,
LM
,
LN
,
-
.
,
f
f (e2 )
M
f
f (O) f (e1 ) f (e2)
O e1 e2.
OM = xe1 + ye2
,
xy|
O e1 e2.
f
,
f (O)f (M) = xf (e1) + yf (e2),
, x y|
M
f (O) f (e1 ) f (e2 ).
M
.
M
,
1.
) x = x + y ; 1, y = x ; y + 1
) x = x ; y ; 1, y = ;x + y + 1.
2.
x;y = 2
3.
,
.
4.
f (A) = A.
A
-
:
a)
(1),
a)
. 1.
f,
. 1.
. IV.
106
,
5.
,
,
,
,
6.
O
7.
|
e1 e2
) O e1 2e2 ?
,
8.
.
,
?
) O e2 e1
,
:
,
,
(3).
,
x = x cos ' + y sin ' y = x sin ' ; y cos '
.
9.
,
10.
12.
.
.
,
f
O e1 e2
.
A(1 0), B (;1=2 1) C (;1=2 ;1).
,
-
.8
?
1.
y = ax + c2 :
,
x
-
?
x = x + by + c1
,
11.
.
,
3.
.
g
f
.
,
g
'
f
x = a1 x + b1 y + c1 y = a2 x + b2 y + c2
a1 b1 6= 0:
a2 b2
+ at
.)
c|
(11) x 2
f.
(
M
:
OM = Of (O) +f (O)M = c + f (r):
Of (O), r |
OM = c + f (r0) + f (a)t:
(1)
(2)
r = r0 +
M
M.
(3)
3.
f
|
6= 0,
f (a)
,
107
(3)
,
.
f
(3)
a 6= 0,
r = r0 + at
,
t,
a
(3).
M
1.
M
,
-
,
t1 6 t 6 t2 .
,
.
2.
,
2.
.
.
,
.
AB CD
.
AB = CD.
AB = CD.
j AB j = j A B j = j j:
j CDj j C D j
C
C
AB
AB
.
f (q)
f
9 x2
.
.
-
(23) x 4 . I:
S = S (p q) = (p1 q2 ; p2 q1)S (e1 e2):
(1).
-
-
AB
O e1 e2
p q,
,
(p1 p2) (q1 q2)
f (p)
(q1 q2),
-
:
.
,
AB CD
,
.
,
(p1 p2)
f (e1 ) f (e2 )
p q
e1 e2.
f (p) f (q),
S = S (f (p) f (q)) = (p1 q2 ; p2 q1)S (f (e1 ) f (e2)):
7 x2
.
f (e1) f (e2 )
(a1 a2) (b1 b2).
. IV.
108
S (f (e1 ) f (e2 )) = (a1 b2 ; a2 b1)S (e1 e2)
S = (p1 q2 ; p2q1 )(a1b2 ; a2b1 )S (e1 e2):
,
S = a1 b1 :
a2 b2
S
,
a1 b2 ; a2 b1.
,
,
|
(4)
,
,
,
.
(4)
.
,
a1 b2 ; a2 b1 > 0,
S =S = ja1b2 ; a2 b1j:
(5)
a1 b2 ; a2 b1 < 0,
-
,
,
.
.
-
(5).
.
.
-
(5)
,
.
.
,
S,
,
,
.
. x 2 . II
.
,
,
Sn
Sn ,
-
.
,
-
,
a
b=a.
S.
(5)
,
: -
a b
4 x2
x = x, y = y. , . .
3.
b=a.
109
,
b=a,
S = ab:
3.
.
.
.
,
3.
O e1 e2
f
,
S = (b=a) a2.
,
p.
L
f (O) f (e1 ) f (e2 )
O e1 e2.
f (O) f (e1 ) f (e2 )
p.
,
9 x2
.
,
.
.
,
,
,
4.
,
2)
,
,
-
.
.
,
|
,
,
,
,
1
.
-
,
.
|
,
.
,
.
1)
,
-
,
.
x 1 . III
-
.
L
3,
-
,
.
.
.
.
.
.
.
-
.
, . .
.
,
,
,
-
. IV.
110
3)
|
.
.
4)
,
),
),
(
(
,
.
1 x 1 . III
.
0
0
0
.)
,
-
a b :::
,
0
-
,
.
,
x = x=a, y2 = y=b
x + y 2 = 1,
-
,
-
,
x2 =a2 + y2 =b2 = 1
a b. (
-
,
:
1) x2 + y2 = 1 2) x2 + y2 = 0 3) x2 ; y2 = 1 4) x2 ; y2 = 0
5) y2 = 2x 6) y2 ; 1 = 0 7) y2 = 0.
.
9 x2
,
,
.
4.
, .
.
1.
.
4ABC |
,
.
f
A.
4A B C
,
,
q (
rqp
-
|
f
A.
)
ABC A B C .
|
.
,
,
rqp
r
,
f,
f.
p,
,
-
,
-
3.
(
q
,
A=A ,
111
A A
A
p(B)
,
,
).
B (
).
C
q(p(C))
.
,
.
(
e1 e2)
,
-
,
-
-
.
,
-
,
.
,
.
,
,
,
.
,
,
-
,
5.
.
,
AB:
.
.
. .
,
AA
p
p|
,
,
.
,
|
.
6.
y
,
(4)
(1) x 2
(1) x 2.
y,
.
.
|
5.
.
.
,
+1,
,
,
.
,
;1.
-
, : | -
. IV.
112
7.
.
|
-
,
|
-
,
.
:
.
:
.
.
-
,
-
.
2.
.
-
f,
.
ABC
A ,B
f.
1,
C
g,
,
-
.
-
C
.
.
,
.)
-
,
|
:
.
.
,
,
.
-
,
f
1.
,
.
AB
g(A) = A
A C .(
,
AB AC
f.
,
g(B)
-
g(C)
,
-
= jA B j=jA g(B)j, = jA C j=jA g(C)j.
p1
AC
g(B) p1 (g(B)) = B
A g(C).
,
p2
AB
g(C) p2 (g(C)) = C
AB.
,
p2 p1 g
A, B
A,B C
,
8 x2
p 2 p1 g = f ,
.
3.
,
1.
2.
A ,B
3.
,
x ; y + 1 = 0, x + y ; 1 = 0 2x + y = 2
.
A, B C
C.
,
4ABC
4A B C .
,
ja j = jaj.
)
4.
.
,
5.
6.
)
)
'
7.
a(0
. .
a).
a
-
y = 8x + y
:
.
.
,
,
.
-
,
x = 4x + 7y
,
)
8
113
-
,
a
O.
:
-
V
x
1.
n
:
.
mn
.
1.
,
m
a11 a12 ::: a1n
a21 a22 ::: a2n
: : : : :: :: ::: : : : : :: :
am1 am2 ::: amn
,
,
,
(i j)
,
|
,
,
,
m 1
m n
.
|
.
,
-
.
I = f1 2 ::: mg J = f1 2 ::: ng:
I J
(i j)
i 2 I j 2 J:
I iJ . .
,
aj :
,
,
,
.
,
,
.
,
.
,
|
,
.
.
1 n
,
n
.
m
.
.
1.
a1 =
,
115
.
a11
a21
a12
2
a2 = a..2
.
am2
..
.
am1
a1n
2
::: an = a..n :
.
amn
k a1 a2 ::: an k :
a1 = a11 ::: a1n ::: am = k am1 ::: amn k
a.1
.. :
am
A
m n
- i1 ::: ir s
j1 ::: js
,
: i1 < i2 < ::: < ir j1 < j2 < ::: < js :
A
r s
A
,
A:
,
aij11 ::: aij1s
A = :: ::: :: : : :: :: : :
aijr1 ::: aijrs
,
aii
,
.
2.
.
a11 a12 ::: a1n
a2 n
A = : :a:21: : :::a::22:: : :::
: : :: :: :: :
am1 am2 ::: amn
m
n
.
B n
m
.
A
B
.
a11 a21 ::: am1
a22 ::: am2
B = ::a12
:: ::: :: :: :: ::: :: :: :
a1n a2n ::: amn
A
AT :
T
A A
.
r
0
0
8*
. V.
116
,
,
mn
ii,
.
A
AT = A:
,
.
2.
A
,
AT =;A:
,
,
3.
A
> j:
i < j:
4.
4.
(1),
aij = aji
-
i j|
.
,
,
: aij = 0
i>
: aij = 0
.
,
A
,
i 6= j:
: aij = 0
m n:
m n
A B
cij = aij + bij
.
.
,
,
i j|
aij = ;aji
.
A
(AT )T = A:
,
A B:
bij = aji (i = 1 ::: m j = 1 ::: n):
.
.
3.
1.
B
A:
aij
cij = aij
1.
.
A B|
cij
(i = 1 ::: m j = 1 ::: n):
A B
C
A + B:
C
cij
A
A:
(i = 1 ::: m j = 1 ::: n):
ABC
A + B = B + A (A + B) + C = A + (B + C)
(A + B) = A + B ( + )A = A + A
( )A = ( A):
,
,
.
O|
m n
C
aij bij
(1)
-
-
(2)
1.
A + O = A:
(;1)A
;A:
117
B ;A
B ; A:
,
,
A + (;A) = O:
A
-
B A
-
,
,
1 x 1 . I.
A1 ::: Ak
1 ::: k
A
+
:::
+
A
:
1 1
k k
,
-
q
,
,
1.
.
p1 ::: pk |
n:
,
::: k
q = 1 p1 + ::: + k pk
,
1
q
p1
p1
.. = 1 ..1 + ::: + k ..k :
.
.
.
qn
pn1
pnk
1
n
5.
.
1
. -
-
q1 = 1p11 + ::: + k p1k
:: :: :: ::: :: :: :: ::: :: :: :
qn = 1pn1 + ::: + k pnk :
.
m n
.
.
,
A1 ::: Ak
1 = ::: = k = 0:
::: k
(3),
1A1 + ::: +
, . .
k Ak = O
, . .
-
,
.
,
(3)
k
. V.
118
2.
(
1
0
e1 = ..
.
0
ei i-
0
1
e2 = ..
.
0
:::
1,
)
.
1e1 + ::: + nen = o
1
0
0
..
0
1
.
.
+
+
:::
+
1 .
2 .
n . =
.
.
0
0
0
1
,
1 = 2 = ::: = n = 0:
,
n
e1 :::
1 0 ::: 0
E = ::0:: ::1 ::::::: : :0: :
0 0 ::: 1
,
.
.
,
2.
(
.
0
0
.. = .. :
.
.
0
n
en:
,
)
.
.
.
.
,
1:
A1 = ; 2 A2 ; ::: ; k Ak :
1
-
-
,
k>1
(3),
-
n
x1 . I
2.
,
-
1
2
.
,
,
n
)
,
(4)
:
.
.
(4):
(
0
.
en = ..
0
1
1
-
1.
,
,
(3),
4.
A1 ::: Ak
119
A1 ::: Ak
,
,
-
,
.
,
1.
.
,
5.
,
6.
.
-
.
,
.
,
,
A1 ::: Ak
B
-
.
.
-
,
B = 1 A1 + ::: + k Ak
B = 1 A1 + ::: + k Ak :
,
O = ( 1 ; 1 )A1 + ::: + ( k ; k )Ak :
A1 ::: Ak
,
, i; i = 0
i = 1 ::: k:
,
1.
)
1 3.
)
?
)
2.
)A B
3.
1 2 3
4 5 6 :
7 8 9
,
A = 14 25 36
:
) AT B ) A B T
A = 11 12
1
?
3
.
-
2 3
B= 5 6 :
8 9
) AT B T ?
B = ;21 11
C = 41 35 :
. V.
120
2A + 3B ; C:
4.
D = 14 25
A B C
1 3
7 9
5.
)A B
)A B C
6.
3,
?
:
3?
a = k 1 2 3 4 k b = k 2 3 4 5 k c = k 3 4 5 6 k?
,
7.
,
x
P
1.
.
n
X
k=1
k
1
n
X
k=1
.
.
. 3,
-
2.
,
,
.
.
,
,
n
-
,
ak = a1 + a2 + ::: + an
k
,
n
X
k=1
k k
=
1 1 + ::: +
.
n
n
X
Pk =
,
1 ::: n 1 ::: m
,
.
n
X
:
Pk
k=1
n
n
n
X
X
X
(Pk + Qk ) = Pk + Qk :
k=1
k=1
k=1
k=1
n n:
.
(1)
(2)
,
-
2.
121
,
n X
m
X
i=1 j =1
(
-
:
Pij :
.)
,
Pij
,
.
a
b
n X
m
X
Pij =
.
ai (i = 1 ::: n)
,
a b
n p:
A
C
B:
C
A
A
B
m X
n
X
j =1 i=1
Pij :
(3)
a
bj (j = 1 ::: n):
.
-
b
,
, . .
ab = a1b1 + ::: + anbn :
A
m n
,
(
(
)
.
B:
mp
C
m p:
,
A
C
,
B:
i = 1 ::: m j = 1 ::: p
cij =
.
. ,
i = 1 ::: n j = 1 ::: m
i,
.
,
,
,
i=1 j =1
2.
n
X
k=1
B
)
,
aik bkj :
C
A B
AB:
,
,
|
(4)
(4),
,
-
.
-
.
,
B
AB
AB = Ak b1 ::: bp k = k Ab1 ::: Abp k:
:
(5)
. V.
122
,
,
n
j-
-
bj :
A
AB |
A
a1
a1.B
.. :
AB = ... B =
am
am B
1.
.
A
x
m n
x1
a11x1 + ::: + a1nxn
2
x = a21x1 + ::: + a2nxn :
..
: :: :: :: : : : : : : : :: : : : :
.
n
am1 x1 + ::: + amn xn
x
a11 ::: a1n
2
a2n
Ax = ::a:1:: : :::
: :: :: :: :
am1 ::: amn
m:
:
xA
.
m 6= 1
1 x 1).
A(
Ax
,
Ac
B:
,
x:
Ax = x1a1 + ::: + xnan :
2.
m
B
m n
n:
b11 ::: b1n
m
m
X
X
2 ::: b2
b
i
1
n
k x1 ::: xm k : :: :: ::: :: :: : =
xi b1 ::: xi bin :
i=1
i=1
bm1 ::: bmn
3.
m n:
m
-
n
x1
x1 a1 x1a2 ::: x1an
x2
x2 a x2a ::: x2a
.. k a1 ::: an k = : :: ::1::: :: : : 2:: : :: : : : : : : :n: : :
.
xm a1 xm a2 ::: xm an
xm
4.
A|
n:
m n ei | im ej | jeTi Aej |
1 1c
-
2.
123
aij :
0
..
.
1 = aij :
..
.
0
a11 ::: a1n
21 ::: a2n
T
ei Aej = k 0 ::: 1 ::: 0 k ::a:::
: : :: :: ::: ::
am1 ::: amn
1. j iB
j-
Ac
B:
AB
,
,
B(
i-
(5)).
3.
.
A
BA
. .
,
AT :
m BA |
n:
,
A B|
1 1
0 0
(
A:
-
A
E
0
,
1).
0 0
1 1
A B
0
0
,
-
AB
n m
.
1 1 = 0 0 :
0 0
1 1
.
,
E=E:
.
|
.
.
AO = O
i-
.
m n
,
B
AB |
,
AB = BA
(6)
0
i.
:
AE = EA = A:
BE EC
BE = B
EC = C:
,
=E
)
-
A
A
0 0 = 1 1
1 1
0 0
.
AB = BA
AB
,
n (6)
-
,
E E=E E E=
.
0
O(
OB = O :
00
0
:
-
. V.
124
(
O O
2.
O
0
,
00
.)
, . .
AB (AB)C
(AB)C = A(BC):
BC A(BC)
,
AB C
mA nA mB nB mC nC :
AB
nA = mB
AB
mA nB :
(AB)C nB = mC :
AB
nA
X
k=1
,
,
!
(AB)C
aik bkl cls (i = 1 :::
l=1 k=1
nB = mC
nB
X
bkl cls (k = 1 ::: mB
l=1
nA = mB
!
nA
nB
X
X
aik
bkl cls (i = 1 :::
k=1
l=1
(1)
.
3.
(3)
mA s = 1 ::: nC):
A(BC)
mA s = 1 ::: nC):
(7)
(8)
,
,
asi(bij + cij )
m
X
i=1
s
.
B C
A|
p m (p
A(B + C)
i=1
(1)
(8)
A(B + C)
(B + C)A
A(B + C) = AB + AC
(B + C)A = BA + CA:
m
X
(7)
BC:
s = 1 ::: nC):
:
m n
-
aik bkl (i = 1 ::: mA l = 1 ::: nB)
nB X
nA
X
.
,
,
A B C:
-
).
(s = 1 ::: p j = 1 ::: n):
asi bij +
j:
m
X
i=1
:
asi cij :
AB AC
.
-
:
2.
4.
125
AB
(AB) = ( A)B = A( B):
5.
B T AT
-
AB
-
(AB)T = B T AT :
.
A B
m n n p:
AB
,
j-
n
X
k=1
j-
AT
n
X
k=1
BT
(9),
1)
2)
)
ii
.
j
.
,
= ATk ::: AT2 AT1 :
.
,
)
B TAT
-
.
.
,
(9)
bkj aik (j = 1 ::: p i = 1 ::: m):
.
.
-
i-
b1j ::: bnj i-
ai1 ::: ain:
j-
(A1 A2 ::: Ak )T
4.
.
,
aik bkj (i = 1 ::: m j = 1 ::: p):
|
,
,
,
:
.
-
,
.
,
,
:
-
.
,
,
a b.
,
:
,
-
126
. V.
) ba ! ba ! b +a a ! b +a a
) ba ! a +b b ! b ;a +a ;b b =
= a;+ab ! ;ba ! ba :
.
,
.
:
.
,
A
B
,
B A:
A
m n
A
S
m:
S
A
,
.
,
S1 |
,
E
m
i6= 0:
S1 A
A ,
i:
S2 |
,
E
ij.
A
S2
ji- .
.
.
S2 A
k 6= i:
(1),
|
A
kE:
,
(
1)
kA
kS2 A
kA:
i:
i- j1.
, iS2 A
i- jA:
5. 1 0
a b = a b
0
c d
c d
a b = a
b
1 0
1 1
c d
c+a d+b :
,
,
.
-
2.
S
127
,
,
,
,
5.
,
,
,
,
.
,
.
(
-
2 x 1).
,
-
-
.
.
-
.
.
,
-
,
.
6.
|
-
SE = S:
.
,
.
:
.
,
,
,
.
,
,
a1 a2 ::: an
-
,
:
1a1 + 2 (a1 + a2 ) + ::: + nan = ( 1 + 2)a1 + 2a2 + ::: + nan = o:
, 1+ 2=0 2=
= 0 ::: n = 0:
,
,
,
1
,
.
i6= 0
.
.
,
.
,
,
.
.
.
.
,
7.
|
-
. V.
128
.
c
.
.
SA
-
.
S a1 ::: S an:
: 1S a1 + ::: + nS an = o:
,
.
-
6,
.
.
-
8.
.
A
-
.
n:
,
i = 2 ::: n
i-
(1 s1)
,
.
(k + 1),
,
n-
,
:
.
1.
,
s1 ::: sk |(k)
A
sk+1 ,
ais1 :
|
,
s1 -
1.
A ,
A(k+1):
-
A(k)
k
.
.
.
.
-
s1 -
A(1) :
.
,
-
a1 ::: an:
,
s1 . .
A(k)
,
A = ka1 ::: ank
nan = o:
1a1 + ::: +
A
S|
SA
S
-
,
-
s1 ::: sk |
sk+1 :
k + 1-
A(n)
(1- , 2- ,..., ns1 ::: sn).
-
2.
A(n)
i i-
129
si -
).
,
.
A
,
,
(
,
,
".
\
.
8
.
A
E
,
10.
,
, . .A
.
6.
.
XA = AX = E,
,
,
.
.
: X1 X2 :
9
. .
.
,
,
10
,
A
-
,
A
A T
A
A|
|
.
.
AT :
,
,
-
A
.
A
.
,
X1 = X1 (AX2 ) = (X1 A)X2 = X2 :
-
-
.
7
.
X
E|
11.
.
,
,
.
.
,
6
,
(10)
-
S1 ::: SN
S1 S2 ::: SN E = A:
,
A.
.
.
.
T1 ::: TM
TM ::: T1 A = E:
,
.
-
{
9.
-
,
.
. V.
130
12.
,
.
.
,
Y X = E:
,
X
,
,
X
,
YX =E
XA = E:
,
A .
Y
-
X
Y (XA) = Y:
(Y X)A = A:
Y =A
AX = E:
, . .
2 + :::
1 a1 + ::: + n an = o
1
1
::: + 2n 6= 0:
v = k 1 ::: nk
A|
: vA = o: A
X
X
: vAX = oX:
,v=o
v:
.
A
A 1:
;1
.
A
,
(10)
X:
;
.
A
A
(A 1 )k
A:
.
A
;
k
;
k
;
.
Ak
-
Ak Al = Ak+l
k l:
.
(A 1 ) 1 = A:
,
;
;
(AB) = B
(AB)(B 1 A 1) = A(BB 1 )A 1 = AA 1 = E:
A 1A = E
AT (A 1 )T = E:
(AT ) 1 = (A 1 )T:
.
,
A
, 1
A
(10)
TM ::: T1E = TM ::: T1 = A 1 :
:
D
n 2n
A
.
D ,
.
A 1:
;
1
;
1 A;1
A
1
;
;
;
;
;
;
;
;
;
;
;
;
2.
1.
,
131
A|
n
.
,
b = AA 1 b:
A
;
,
1.
,
1
)
x = c1 x + d1 y
y = c2 x + d2 y:
)
1
2 k2k =
3
)
AB =
9*
-
,
:
,
n
X
i=1
A
2
4
6
2.
?
b1
::: bn |
-
ai bi:
) (A + B )2 + (A ; B )2 = 2(A2 + B 2 )?
X 2 + E = O:
I = 01 ;10
.
.
n:
c1 d1 :
c2 d2
1 1c
,
,
A
-
f g
) (A + B )2 = A2 + 2AB + B 2
5.
,
.
2.
:
-
,
A|
n
a1 b1
a2 b2
4.
A
6 x 1:
f g
k2k |
1
2
) k2k 2 = 4
3
6
3.
a1 ::: an |
B:
,
-
,
,
x = a 1 x + b1 y
y = a 2 x + b2 y
,
b
,
A 1b ,
A:
;
.
.
,
n:
A
1?
. V.
132
z = a + bi
6.
A(z ) = ab ;ab :
,
A(z1 ) + A(z2 ) = A(z1 + z2 ) A(z) =
= AT (z ) A(z1 )A(z2 ) = A(z1 z2 ) A(z 1 ) = A 1 (z ):
;
7.
;
1 0 0
1 1 2 :
1 1 3
.7
8.
x
1.
.
r
A
.
3.
A
,
.
r:
.
,
.
r1
,
r:
. 1:
,
r
,
)
,
1.
.
r
,
2:
|
1.
.
(
-
,
.
.
.
j1 ::: jr 1:
;
, . .
-
,
.
r;1
j1 ::: jr 1
r-
.
,
r1
r
.
r
.
-
.
,
,
r ; 1:
;
1 x2
1
::: r 1:
;
.
-
3.
133
.
1
,
::: r 1:
;
j1 ::: jr 1;
,
r-
j
.
j1 ::: jr 1:
;
,
-
rarj
,
, .
+ ::: + r 1ajr;1 + aj = o
.
r
arj 6= 0
10 + ::: + r 1 0 + aj = 0:
=0
a
+
:::
+
a
1 j1
r 1 jr;1 = o:
(1)
j1 ::: jr
j: (
.)
,
j
1
;
,
1aj1
;
-
;
;
j1 ::: jr
,
.
,
j1 ::: jr 1 j
(1)
;
,
r
,
.
r
,
.
,
.
A
,
A
0
A:
.
m n
,
A
,
.
,
.
A:
-
1
,
,
.
.
|
A |
,
.
n;1 ,
.
.
-
1
;
-
.
.
A
,
A
0
,
,
,
A
0
. V.
134
2.
.
:
1.
1
.
,
r:
.
AT
,
,
|
,
\
"
A
,
r
A:
,
,
.
,
-
.
,
-
.
.
-
,
:
,
.
.
,
1ai1 + ::: + r air + aj = o:
,
6= 0
.
.
A
.
".
.
1,
aj |
ai1 ::: air :
,
p
,
.
3.
A
Rg A:
2.
x2
r
,
AT
\
|
.
p>r
-
.
,
.
r+1
-
6 7
. -
2.
9 x2
3.
AB CA
4.
A
Rg AB = Rg B
.
Rg CA = Rg C:
-
3.
.
D
AB |
AB:
A AB:
, Rg AB 6 Rg D:
AB |
1 x2
,
A:
,
135
.
-
,
,
.
, Rg AB 6 Rg A:
,
Rg AB 6 Rg B:
D
B AB:
.
.
m n
),
r
m ,
m;r
|
.
5.
Rg D = Rg A:
,
-
0
4.
(
r
>r
x2
.
(r = 0).
,
,
,
.
.
.
.
m>
,
|
.
,
,
,
r- ,
.
A
k = 1 ::: r:
0
,
.
,
8 x 2.
,
.
.
r
, . .
.
r
,
.
.
-
,
,
8
-
.
.
A
,
aj1 ::: ajr (j1 < ::: < jr )
e1 ::: er:
,
ajk ! ek
.
A:
,
r
0
. V.
136
,
Rg A = r
.
,
,
.
,
.
,
,
,
,
A
,
-
A
,
8 x 2: (
,
,
-
.)
1 2 3
7 8 9
-
.
.
.
.
A|
,
,
aij =
r
n
.
,
r:
,
.
x
4.
,
-
?
aij i = 1 ::: m j = 1 ::: n Rg A =
1 :::
i j:
r
i j
.
1.
,
A= 4 5 6 :
1.
5.
-
.
.
,
4.
.
6.
,
)
)
)
)
2.
n ; 1:
3.
= 1:
,
A
j1 ::: jr
1 ::: r
,
,
.
n
m
1
:::
n
r
,
,
A B
,
,
:
.
4.
|
137
,
|
a11 + ::: + ann
,
.
)
n,
.
,
.
n
.
,
.
,
A
1.
,
f
A|
,
-
(
-
-
.
.
A
,
f(A) = h1 ai1 + h2ai2 + ::: + hnain
h1 ::: hn |
,
iai1 ::: ain
.
2.
3.
1.
A
det A ,
,
,
.
1,
.
,
,
,
, . .
,
.
,
,
,
.
.
,
.
,
.
1
.
1.
f
n
,
:
i-
-
(1)
.
.
,
-
A
-
p+
. V.
138
+ q
p q
p+ q
pk qk |
f(A) = f(Ap ) + f(Aq )
Ap Aq
A
.1:
f
(1).
iA
k
aik
(2)
p q:
-
pk + qk
p q:
,
f(A) = h1 ( p1 + q1 ) + ::: + hn ( pn + qn):
,
f(A) = (h1 p1 + ::: + hn pn) + (h1 q1 + ::: + hnqn):
h1 ::: hn
i,
h1p1 + ::: + hnpn = f(Ap )
h1 q1 + ::: + hn qn = f(Aq ):
,
(2).
2:
.
iA
i-
ai1 e1 + ::: + ain en:
(2),
f(A) = ai1f(A1 ) + ::: + ainf(An )
A1 ::: An
A
i.
A
f
.
1
.
,
,
1p1 + ::: + sps
det A =
A1 ::: As |
,
.
,
2.
.
,
.
-
det A
-
A
.
,
s
,
1 det A1 + ::: + s det As
A
p1 ::: ps:
,
A
,
(3)
4.
139
.
A
iai + aj , i 6= j:
A
det A = det A + det Aj
Aj
A
ij- .
aj
: ij.
, det Aj = 0:
, det A = det A :
3.
(;1):
.
A
A
j.
A
ai
-
0
0
0
,
-
0
i2:
-
..
..
..
..
..
.
.
.
.
.
a.i
ai +. aj
ai +. aj
ai +. aj
aj
..
..
..
!
=
! ... :
A= .. !
aj
aj
aj ; ai ; aj
;ai
;ai
..
..
..
..
..
.
.
.
.
.
A
A:
,
3,
.
4.
f
0
,
,
.
.
2
A|
,
a1
,
::: n:
.
,
.
.
a2 ::: an
det S1 = det E =
.
-
,
-
(2),
f(A) = 2f(A2 ) + ::: + nf(An )
A
A2 ::: An
2- , ...,n.
f(Ai ) = 0 i = 2 ::: n:
f(A) = 0
2.
.
.
S1
6 0
=
-
,
,
S2
.
-
. V.
140
2
,
det S2 = det E = 1:
5.
,
-
d1 d2
-
,
,
d1 (S) = d2(S):
,
,
A
S
det(SA) = det S det A:
(4)
,
,
SA
A
,
S E:
det(S1 A) = det A:
det S1 =
(4)
.
,
det(S2 A) = det A
det S2 = 1:
1.
.
n
,
.
d1(A) = d2(A)
d2 :
,
A|
= d2 (A) = 0:
-
d1
A:
d1(A) =
,
9 x2
.
A:
(4),
d1(A) = d1(S1 ::: SN ) = d1(S1 ) d1(S2 ::: SN ) = ::: = d1(S1 )::: d1(SN ):
, d2(A) = d2 (S1 )::: d2(SN ):
5
d1(A) = d2(A)
.
,
:
A
,
det A = det S1 ::: det SN :
(5)
,
6= 0
, . .
.
(5)
6.
,
.
.
,
3.
.
i-
.
.
,
aij |
j-
Dij
.
,
.
A
n;1
n
-
.
-
-
4.
A
141
i-
aij
,
,
2.
j.
dij = det Dij :
-
n;1
.
,
.
,
.
A
fj (A) =
dkj |
,
(6)
n
X
k=1
,
A:
i-
.
aij
i)
(6)
i-
,
,
.
(
, dkj |
,
k=i
-
i 6= k)
-
| fj
,
fj (A) = 0
.
A
(l > i).
(6)
,
k 6= i k 6= l
Dkj
,
dkj
.
,
fj (A) = (;1)i+j aij dij + (;1)l+j alj dlj :
,
aij = alj
.
j
i
fj (A) = (;1) aij ((;1) dij + (;1)l dlj ):
Dij Dlj
,
:
i l
. -
,
.
4
akj
i-
.
.
(6)
,
aij (;1)i+j dij
Dij
akj
ifj
akj (;1)k+j dkj
(
-
n;1 n
-
n
.
1.
.
,
,
j
2.
-
.
.
,
-
(7)
-
. V.
142
,
Dlj
iDij | (l ; 1)- .
Dij
l ; 1 i,
.
(l ; 2)- , (l ;
; 3)- , ..., i.
(l ; 2) ; (i ; 1) = l ; i ; 1 .
,
dij = (;1)l i 1 dlj :
(7),
,
fj (A) = 0:
3.
fj (E)
E |
n:
(6)
fj (E) = (;1)j +j djj :
Djj |
n;1
1.
fj (E) = 1
.
.
1
fj
j
,
:
n
X
det A = akj (;1)k+j dkj :
(8)
,
; ;
k=1
j-
,
.
4.
A
aij
7.
|
-
.
,
(1).
dij |
,
(8)
8.
n
n
X
det A = aij (;1)i+j dij
j =1
.
(1),
,
det A = hj aij + q:
-
i
(9)
aij :
q:
hj
,
j:
det A = aij (;1)i+j dij + r:
hj
i, q
aij :
,
k
Dkj
j, ,
, dkj
aij :
, dij
aij :
,
r
.
,
A0
,
A
aij
0,
,
det A0 = q
4.
det A0 = r:
A1
A
aij 1:
det A1 = hj + r = (;1)i+j dij + r:
hj :
9.
det A = det AT :
f(A) = det AT :
AT . .
AT (
f(A) = det AT = 0:
= 1:
,
, ET = E
.
(;1):
,
4 x3
A
,
10 x 2),
, f(E) = det E T = det E =
.
,
-
A
,f
,
,
7
A:
9
,
5.
143
.
.
.
10.
,
,
,
,
.
,
-
11.
det AB = det A det B:
.
A
.
.
AB = S1 :::SN B: (4),
det AB = det S1 ::: detSN det B:
(5)
.
A
n
, RgA < n:
Rg AB < n:
,
AB
det AB
,
det A det B:
.
n
.
.
1 ::: n
,
-
. V.
144
.
: 1, 2 2, 1.
,
1 2
1 ::: n
ik
n=4
:k<s
2 3
.
2, 4, 3, 1
,
4|
i1 ::: in
,
.
ik > is :
.
i1 ::: in:
i1 ::: in ,
,
-
N(i1 ::: in):
N(i1 ::: in) |
,
-
:
a11 ::: a1n
X
det : : : :: ::: :: :: :: : =
(;1)N (i1 ::: in) a1i1 a2i2 ::: anin : (10)
an1 ::: ann
(i1 ::: in)
.
,
1 ::: n
.
i1 ::: in
:
1i1 , 2| i2 . .
.
.
,
.
(10)
.
n=2
a11 a12 :
a21 a22
1,
2
2, 1
,
, (;1)N (1 2)a11a22 (;1)N (2 1) a12a21:
a11a22 ;
; a12 a21 . .
.
,
n;1
A
n:
det A
: n
X
(11)
det A = (;1)k+1 a1k d1k :
k=1
k-
d1k
n;1
D1k :
d1k = det D1k =
k-
X
(i1
(;1)N (i1 ::: in;1 ) a2i1 a3i2 ::: anin;1 :
::: in;1 )
in;1
i1 :::
2 ::: n
A
.
-
,
k
,
D1 k
4.
k(;1)k+1 a1k
(;1)k+1 a1kd1k =
145
(11)
X
(i1
:
(;1)N (i1 ::: in;1 )+k+1 a1k a2i1 a3i2 :::anin;1 :
::: in;1 )
k i1 ::: in 1
1 ::: n
N(k i1 ::: in 1) = N(i1 ::: in 1) + k ; 1
k
k;1
,
k:
,
N(k i1 ::: in 1)
,
N(i1 ::: in 1) + k + 1
;
;
;
(;1)k+1a1k d1k =
(10),
.
(11)
;
X
(i1
(;1)N (k i1 ::: in;1 ) a1k a2i1 a3i2 :::anin;1 :
::: in;1 )
,
(11)
(10) ,
.
1.
det A:
2.
,
A
m + k > n:
6.
A C|
)
)
8.
. .
n
P
1
1
2
1
,
0
4
0
5
n
.
8
0
9
4
P = OA B
C
,
k b:
| A(t):
det A(t) = kt + b
A
1
2
1
1
.
det A
AT = ;A:
,
-
m k
4
:
:
k n;k O |
det P = det A det C:
k b
n:
::: 1
::: 1
::::::::::::::::::::::: :
;1 ;1 ::: 2 1
;1 ;1 ::: ;1 2
2
;1
k
1
3 :
3
1
det A = 0:
(n ; k) k:
7.
k
n:
2n + 1
4.
5.
,
A|
A|
det A = 0:
,
3.
10
;
t:
t:
. V.
146
9.
.
ax2 + bx + c
,
x2 + x +
a b c 0
0 a b c = 0:
0
0
(5, 4, 3, 2, 1)?
10.
x
1.
5.
(
.
a11x1 + a12 x2 + ::: + a1n xn = b1
a21x1 + a22 x2 + ::: + a2n xn = b2
: :: :: :: :: : : ::: :: :: :: : :: :: :: :: :: : :
am1 x1 + am2 x2 + ::: + amn xn = bm
m
x1 ::: xn:
.
,
-
)
(1)
n
-
a11 a12 ::: a1n
A = :: :: ::: :: :: :: ::: :: :
am1 am2 ::: amn
.
,
b
,
,
A :
-
a11 a12 ::: a1n b1
A = : :: : : : :: :: :: :: : : : :: :: ::: :
am1 am2 ::: amn bm
.
(1),
.
n
1
::: n
x1 ::: xn:
(1)
x1
,
a11
a1
b1
.. + ::: + xn ..n = ..
.
.
.
am1
amn
bm
1
::: n
,
-
5.
(
1 x 1)
(
,
,
x1a1 + ::: + xn an
a1 ::: an |
.
(
|
,
.
x1 + x2 = 1
x1 + x2 = 0
,
.
,
,
|
.
1
2.
,
(1)
,
.
(n = 2 m = 1) x1 + x2 = 0
,
,
.
,
6 x1
,
:
.
.
3.
,
6= 0
:
(1)
,
,
(1)
i-
i-
j- .
,
.
-
-
.
A
,
10*
.
-
(1),
.
,
.
,
Ax = b
1 x 2).
147
=b
, b|
.
1.
:
)
j- ,
.
.
-
. V.
148
2.
,
,
,
,
.
,
::: n
: m = n:
,
n .
.
,
,
(
,
m=n
,
1
.
,
|
1
,
.
,
).
,
,
;
1.
.
n
a11x1 + a12 x2 + ::: + a1n xn = b1
a21x1 + a22 x2 + ::: + a2n xn = b2
: : :: ::: : : : : :: :: ::: : : : : :: ::: :: : : :
a1nx1 + an2 x2 + ::: + ann xn = bn :
,
,
A|
b
3.
.
,
,
.
.
n
(2)
,
.
1,
n
,
1 x 2:
.
,
n
-
n det A 6= 0:
A
.
,
-
n
,
i
b:
i = det ka1 ::: ai 1 b ai+1 ::: ank :
, b = x1 a1 + ::: + xnan
x1 ::: xn |
i-
-
;
-
i = x1 det k a1 ::: ai;1 a1 ai+1 ::: an k + :::
::: + xi det k a1 ::: ai;1 ai ai+1 ::: an k + xn det k a1 ::: ai;1 an ai+1 ::: an k :
6.
(
,
i- ,
.
,
)
149
i = xi
det A:
(3)
xi = det A (i = 1 ::: n)
n=3
. 6 x 4 . I.
4.
.
A
,
. ,
A 1
A:
ej | j1
.
,
jA
j
A 1 ej :
xj
Axj = ej :
ii|
: xij = i= det A
,
A i
ij.
,
,
ej
j1,
.
, i = (;1)i+j dji
dji |
j
ai
A:
,
,
,
xij :
,
1)i+j dji :
(4)
xij = (;det
A
(4),
,
,
.
i
;
;
;
1.
y1 y2 y3
x1 x2 x3
(4),
2.
1.
.
x
6.
,
a b :
c d
.
m
,
, (x1 y1 ) (x2 y2 ) (x3 y3 ):
(
n
= b1
= b2
: : :: ::: : : : : :: :: ::: : : : : :: ::: :: : : :
am1 x1 + am2 x2 + ::: + amn xn = bm
a11x1 + a12 x2 + ::: + a1n xn
a21x1 + a22 x2 + ::: + a2n xn
)
,
x 5.
.
-
. V.
150
Ax = b:
,
.
1.
(1),
{
-
m n
b
,
b
|
-
A
A0
.
, Rg A0 = Rg A
.
.
Rg A > Rg A = r:
m;r
|
.
,
r + 1k 0 ::: 0 1 k:
,
A
r + 1-
.
(1) -
.
,
-
A
A
|
-
,
,
.
3 x 5).
,
-
A
,
k 0 ::: 0 1 k:
,
:
m
A
A
2 x 3 Rg A0 = Rg A :
.
Rg A = Rg A
1.
A
(
b:
-
,
.
,
A:
.
Rg A = Rg A
A:
,b
A:
A
A
A
,
A
.
(1)
.
.
.
0=1
-
.
-
6.
(
,
n
m
(1).
.
A
)
.
(1)
a11y1 + a21y2 + ::: + am1 ym = 0
a12y1 + a22y2 + ::: + am2 ym = 0
:: :: :: :: :: ::: :: :: :: ::: :: :: :: : ::
a1n y1 + a2ny2 + ::: + amn ym = 0
,
AT
y|
o|
2.
(3)
AT y = o
,
m
yT A = o
n:
151
(2)
,
,
(2)
(3)
(1)
,
-
yT b y1 b1 + ::: + ym bm = 0:
(4)
. 1:
(1)
, . . x
n
A
x
=
b
:
T Ax = yT b:
y
m
y
y
|
(3), yT b = (yT A)x = ox = 0:
2:
,
(1)
.
1
k 0 ::: 0 1 k
A = kA j bk ,
,
.
y1 ::: ym
y.
yT k A j b k = k 0 ::: 0 1 k
(
1 x 2).
:
yT A = o yT b = 1:
,
(3),
(4).
.
.
A1 x + B1 y + C1 = 0 A2 x + B2 y + C2 = 0:
,
y1 y2
y1 A1 + y2 A2 = 0 y1 B1 + y2 B2 = 0 y1 C1 + y2 C2 6= 0:
,
y1
y2
.
= ;y2 =y1
:
,
A1 =
= A2 B1 = B2 C1 6= C2 :
7 x 2 . II.
. V.
152
m
2.
r.
r,
.
n
(1).
-
.
6 x 3).
(
r
r
.
kEr jA jb k,
0
.
,
-
0
kEr jA kx = b :
0
,
|
0
:
x1 = 1 ; ( 1r+1 xr+1 + ::: + 1nxn )
: : : : :: ::: : : : : :: :: ::: : : : : :: ::: :: : : : : :
xr = r ; ( rr+1 xr+1 + ::: + rnxn ):
A, i|
b.
i|
j
(5)
0
0
,
.
(
|
)
.
(5)
x10 ::: xr0, xr0+1
xr0+1r ::: xn0
1
x0 ::: x0, . .
3.
-
,
-
,
(1).
.
::: xn0
,
,
(5)
(5)
,
2.
y
.
y = x ; x0 :
(1)
(5),
,
,
-
.
,
.
.
(1)
Ax = o:
.
(1).
,
(6),
x = x0 + y:
x|
(1).
Ay = Ax ; Ax0 = b ; b = o:
x0 |
-
-
:
(6)
x
-
-
6.
(
,
y|
= Ax0 + Ay = b + o = b:
.
(6)
,
.
,
.
, . . Rg A = n:
(
2 x 5)
.
A
3.
.
,
.
,
,
Ax = o
.
.
F
,
,
.
,
n,
4.
()
,
.
-
x
.
,
-
()
,
.
F.
( ){
,
( ).
.
-
m n,
.
.
p
,
.
F,
.
,
()
Ax =
-
x1 x2 |
|
.
Ax1 = o Ax2 = o
A( x1 + x2 ) = Ax1 + Ax2 = o.
A
) AF = O,
)
F
)
153
(6), x = x0 + y.
.
,
)
x = F c:
|
c
F
Ax = o
p,
(7)
()
A(F c) = (AF )c = 0.
. V.
154
.
b = 0:
,
.
0
Rg A = r < n
c1 ::: cn r |
(5),
,
-
;
x1 = ; 1r+1 c1 ; ::: ; 1ncn r
: : : : :: ::: : : : : :: :: ::: : : : : :: ::: :: : : : : :
xr = ; rr+1 c1 ; ::: ; rncn r
(8)
xr+1 =
c1
: : : : :: ::: : : : : :: :: ::: : : : : :: ::: :: : : : : :
xn =
cn r
(7),
c = kc1 :::
n
r
T
::: c k ,
F
; 1r+1 ; 1r+2 ::: ; 1n
::: :: :: :: :: ::: :: : : : : ::: :: :: :
; rr+1 ; rr+2 ::: ; rn
1
0
::: 0 = E;nA r :
(9)
F1 =
0
1
::: 0
::: :: :: :: :: ::: :: : : : : ::: :: :: :
0
0
::: 1
,
(9)
()
.
c,
F1 c |
.
c
jn ; r,
, jF1
.
,
F1
( ): AF1 = O.
n;r
|
.
(9)
,
.
,
(9)
.
,
5.
r
n,
n;r
.
(9) {
.
.
A
.
,
5.
n;r
;
;
;
;
0
;
,
,
F,
.
P,
(6),
-
()
n;r
( ).
V = kF jP jxk.
6.
(
n ; r,
P
x
P,
F.
V,
,
P.
-
(9)
A
c
x
(10).
ee
.
2 4.
(1), F |
,
(10)
|
x0 |
x = x0 + F c
(1).
c
,
-
.
4.
3.
155
|
,
,
.
)
,
(10),
f1 ::: fn r |
,
(10)
:
(11)
x = x0 + c1f1 + ::: + cn r fn r :
3
,
,
. x0 |
, (10)
(7).
1 x5
,
n
cn
,
,
. .
7.
A|
n
cn
.
det A = 0
.
, c1 ::: cn r |
;
;
;
,
,
.
,
.
5.
.
.
.
n=3 r=1
.
;
det A = 0
,
.
Ax + By + Cz + D = 0
A 6= 0
: y = z = 0:
,
,
.
3
1,
Rg A < n
,
,
x = ;D=A:
.
(12)
-
. V.
156
,
: y = 1 z = 0 y = 0 z = 1:
x
,
;B=A
;C=A:
,
(12)
x
;D=A
;B=A
;C=A
y =
0
1
0
+ c1
+ c2
:
(13)
z
0
0
1
.
,
,
k ; D=A 0 0 kT
(
)
, ,
,
.
(10)
x0
.
.
2 x 2 . II
A 1+B 2 +C 3=0
. .
.
(
)
.
,
(13) |
,
.
,
.
A
1.
A
.
)
)
2.
,
(a b) = 0:
a
{
b a 6= 0:
a x] = b
3.
n
.
5.
Ax = 0
6.
F|
A
F
7.
Q
p
,
.
: ) Fy = 0
.
) F T z = 0?
1 2 3
4 5 6
7 8 9
n p|
.
,
-
.
k 1 1 1 k:
k Er j B k |
4.
,
:
1
1 :
1
,
F
,
F = FQ:
0
0
-
-
6.
8.
9.
,
(
(
y
a1
)
a2
A
FT:
,
,
)
,
D
D
:
.
157
b
,F|
A
.
.
-
VI
x
1.
.
(
,
f
F(x)
-
,
1 x 1 . I.
:
. .
1.
,
.I
),
.
.
1 x 1 . V,
,
,
.
,
:
,
1.
,
C:
f|
C:
C
g
f(x)
F (x) = f(x):
0
.
,
,
,
-
,
, . .
0
-
0 1]:
,
f
,
,
.
F (x) = 0:
-
.
C|
.
:
,
,
F0
F(x) = F0(x) + C
|
,
:
:
-
C|
-
.
.
,
, -
1.
.
,
(
x y
L
x
159
|
L
),
x
L
;x
-
L
-
):
;x
2.
,
,
L|
L|
L:
L:
.
.L
-
x + (;x) = o
,
x . . 1x = x:
,
L
,
x
.
,
L
x
,
.
-
L
x
xy z
-
:
),
1) x + y = y + x
2) (x + y) + z = x + (y + z)
3)
o
,
x+o = x
4)
x
5) (x + y) = x + y
6) ( + )x = x + x
7) ( x) = ( )x
8)
x
1
,
,
x+y
(
(
,
.
L
.
,
-
L
o ,
.
n:
n ,
L
,
,
.
.
-
.
3.
-
,
-
. VI.
160
4.
5.
.
.
2.
,
,
-
o+o =o
,
.
fog:
,
,
;x1 ;x2
;x2:
;x1
,
.
o = o:
.
o1 o2 :
: o1 + o2 = o1 = o2 :
x
(;x1) + x + (;x2 )
o+o = o
-
,
x + (;x) = o
;x
x:
y ;x
y;x
y x:
,
0x = o
x:
,
0x = 0x + x ; x = (1 + 0)x ; x = o:
,
(;1)x = ;x
x:
,
(;1)x + x = (;1 + 1)x = 0x = o:
,
,
o = (x ; x) = x ; x = o:
x=o
=0
x = o:
,
6 0
=
1
x=o
1x = o:
:
.
.
x
+
:::
+
x
1 1
k k
x1 ::: xk
1 ::: k:
3.
.
;
,
.
.
,
,
,
.
.
,
,
,
,
,
,
-
L
,
1.
161
,
,
x1 . I x1 . V
.
.
)
,
-
.
1.
,
2.
,
k>1
.
a1 ::: ak
4.
5.
.
-
.
3.
.
,
,
,
,
,
:
,
L
-
.
e1 ::: en 1
|
, ::: n
1
.
,
e|
x
..
.n
.
. .
-
L
=
11
-
.
,
.
.
a1 ::: ak
,
.
)
)
= k e1 ::: en k
,
(
x 1 . V.
4.
,
,
.
.
.
: e=
:
. VI.
162
:
1
n
X
i ei = k e1 ::: en k
x=
..
.
=e :
n
i=1
5
.
6.
,
-
.
.
:
x + y = e + e = e( + ) x = e = e( )
|
x y:
|
3 4 x 2 . V.
6
,
.
7.
,
n
,
.
m > n:
m>n
.
e1 ::: en
n m
.
8.
,
,
,
f1 ::: fm
n:
1.
,
n
.
.
-
f1 ::: fm:
n
,
,
f1 ::: fm
n
.
-
.
.
,
n-
,
.
,
m
,
L
,
n|
dim L :
-
-
.
.
m
1.
163
:
n
6.
,
,
n-
7.
n:
t
.
2 x1
,
|
,
.
-
n
.
-
,
m
m:
t0 = 1 t t2 ::: tm 1 |
.
;
0 + 1t + 2t
.
.
2 + ::: +
m;1 tm;1
.
,
.
.
n-
n
.
x1 ::: xn |
y
y x1 ::: xn
,
y + 1x1 + ::: + n xn = o
x1 ::: xn
.
10. nk<n
,
k+1
.)
.
,
.
-
.
.
6= 0
-
.(
,
-
,
.
9.
8
,
.
,
,
n
0 1]
,
,
,
.V
n
.
m
-
,|
,
8.
n+1
8.
.
,
11*
.
,
k + 1<n
-
. VI.
164
.
,
n
,
5.
.
e1 ::: en e1 ::: en
0
0
j
i
:
n
X
je
=
i j
j =1
(i = 1 ::: n):
::: n1
2
S = : : : : : :::::::: ::n: :
n
n
1 :::
n
|
(1)
1
1
2
1
e:
.
e
.
-
n-
0
ei
.
(1)
e:
0
e1 ::: en
, det S 6= 0:
0
, je
ej
0
0
k e1 ::: en k = k e1 ::: en k S
0
0
e = eS:
0
e=eS
0
,
1
;
e e:
e = eS e = e T:
0
0
00
,
,
0
1
;
(2)
|
e e
0
0
e
(3)
-
0
e
n
0
,
=S :
0
e:
0
S
x=e
e
0
.
e:
x=e :
(2)
x
e
0
0
(2)
-
00
00
e:
e e:
,
S
e
e = eST:
11.
, det S 6= 0
det S 6= 0
x = eS :
,
0
,
.
S
,
x
(4)
1.
165
1
::: 1
.. = : :: ::: :: : : : n: :
.
n
n
n
1 :::
n
,
,
1
1
n
i = X i 0j
j
j =1
6.
e = e0T det T < 0:
12.
0
E
;
e
(e0 ):
1
..
.n
0
(i = 1 ::: n):
x4 . I
.
0
e0
(5)
x 3 . I.
.
,
.
E+ (e0)
e = e0S det S > 0:
,
e 2 E (e0)
0
;
-
E+ (e0) E (e0)
e0:
;
.
f0
f0 2 E+ (e0 )
e = f0S
. . f0 = e0P det P > 0:
e 2 E+ (f0 )
det S > 0 e = e0P S
det P S = det P det S > 0:
, e 2 E+ (e0 ):
E+ (f0) E+(e0):
e0 2 E+ (f0)
det P 1 > 0:
,
f0
e0
E+(e0) E+ (f0)
E+ (f0) = E+ (e0): E (f0) E (e0)
,
E+ (f0) E+ (e0)
.
,
E+(f0) = E+ (e0) E (f0) = E (e0):
,
f0 2 E (e0)
.
,
E+(f0) = E (e0) E (f0) = E+ (e0):
,
E1 E2:
.
,
E1 E2
.
.
,
(
)
.
;
;
;
;
;
;
;
;
. VI.
166
1.
i-
.
,
A
2.
Eij
j-
m n
1,
aij
,
3.
t
?
.)
n
63
1 t ; 1 (t ; a)2 (t ; a)3 :
: 1 t t2 t3
p(t)
4.
-
mn
m n: (
.
.
f1 ::: fn
e
,
.
5.
?
f
x
1.
L
.
,
,
,
,
L:
L
L:
0
0x
;x = (;1)x
L:
.
L
,
1.
L:
P:
L
,
-
.
L
0
L
0
0
: f1 =
,
.
.
L
)
)
2.
e1 ::: en
,
= e1 + e2 f2 = e2 + e3 f3 = e3 + e4 f4 = e4 ; e1 ?
0
,
0
,
L:
.
0
L
x
0
L:
0
0
L
L
0
.
-
0
P
,
0
L
L
0
:
-
2.
L:
167
x y
L x = 1p1 + :::
::: + k pk y = 1 q1 + ::: + m qm X pi qj X
2 P (i = 1 ::: k j =
= 1 ::: m):
,
x+y =
i pi +
j qj . . x + y
P
:
X
,
x = ( i )pi:
,
0
L
P:
p1 ::: pm |
,
P
,
P
P|
,
m
1.
2.
n
P
.(
,
,
p1 ::: pm
-
0
.)
P:
P
p1 ::: pm
p1 ::: pm
.
,
-
,
,
m:
3 x6
.
-
.V
n:
3.
L
,
.
2.
L:
L
L
0
|
dim L 6 n:
0
,
,
L
L:
L |
0
3.
,
,
.
4.
L:
-
.
L:
e1 ::: ek ek+1 ::: en
,
x = 1 e1 + ::: + k ek ,
0
L
n-
dim L = n
L
0
m>n
.
L
L
0
0
,L
e1 ::: ek
0
L
k+1 = 0
0
n-
L:
.
L:
-
0
n
L
0
::: n = 0:
k+1 = ::: = n = 0
x
, x2L:
,
0
. VI.
168
L
x = 1 e1 + ::: + k ek :
k+1 = ::: = n = 0:
0
x
,
x:
k+1 = 0
,
,
.
n
X
k+1 0 i = 0
.
.
2.
L
L L
00
:
0
L +L
0
,
00
00
x=
:
k-
L:
i pi +
Xx
00
-
L
L +L
0
j qj
00
(
0
L +L
x 2L x 2L :
L L
k l:
e1 ::: ek f1 ::: fl:
e1 ::: ek f1 ::: fl
00
0
0
0
00
00
00
e1 ::: ek f1 ::: fl:
,
,
00
.
-
,
,
L L
0
,
00
0
pi
x
,
.
|
00
0
,
X
L
L +L :
.
L \L
.
L \L
0
.
0
00
-
0
L
,
00
0
0
(5) x 1
i
n-
x =x +x
L +L
.
i=1
00
L
a qj |
,
-
0
n
X
n 0 i = 0:
(k < n)
n ; k:
0
L
x
0
L L
.
L +L
)
00
L
0
00
0
4.
n;k
,
L
,
:::
i
i=1
e1 ::: en
::: n = 0
.
,
x y
,
,
L \L
0
00
-
00
2.
L \L
0
00
x
:
169
L
,
,
Li
00
:
L
L
0
x+y
00
.
,
.
s>2
L
0
,
L 1 ::: L s
s
.
.
|
x1 + ::: + xs
xi
dim L 1 + ::: + dim L s:
L 1 ::: L s
,
xi 2 L i (i = 1 ::: s):
L 1 ::: L s -
,
dim(L 1 + ::: + L s ) 6 dim L 1 + ::: + dim L s :
,
s=2
,
.
.
L 1 ::: L s
,
, . .
,
,
:
,
L
1
:::
Ls
)
)
x1 + ::: + xs
)
.
,
.
5.
,
m6s
0
:
Li
:
.
-
,
L
L i (i = 1 ::: s)
x2L
xi 2 L i (i = 1 ::: s)
-
,
.
,
0
,
. VI.
170
L i (i = 1 ::: s)
)
L:
0
.
.
1.
,
xi1 ::: xim
,
),
,
.
,
,
,
L i:
)
), )
)
).
-
,
,
.
k = dim L 1 + ::: + dim L s
.
L
,
(
).
L
,
k
,
.
2.
,
)
).
,
)
x
x=
= x1 + ::: + xs
x = y1 + ::: + ys
xi yi 2 L i (i = 1 ::: s):
(x1 ; y1 ) + ::: + (xs ; ys ) = o:
,
).
3.
,
)
1
).
,
,
L
L 2 +1 ::: + L s:
z = x1 2 L
x2 + ::: + xs:
x1 = x2 + ::: + xs
,
z
,
L i:
4.
,
,
)
).
,
L i (i = 1 ::: s):
L
,
,
)
.
.
,
,
L i (i = 1 ::: s):
,
,
.
x1 + ::: + xs = o
.
,
,
x1 : x1 = ;x2 ; ::: ; xs:
,
x1 2 L 1
L 2 + ::: + L s:
|
0
0
0
2.
171
).
,
,
.
.
),
-
.
-
.
, (L 1
L 2) (L 3 L 4) = L 1 L 2 L 3 L 4:
L L L +L =L :
,
L +L =L :
6.
L
L
L =L L :
.
e1 ::: ek
L
L
ek+1 ::: en: ek+1 ::: en
L:
5
L =L L :
0
00
0
0
0
00
00
0
0
L
00
0
00
0
,
00
0
1.
00
,
,
:
.
-
.
L1 L2 |
.2
6
M
L = M (L 1 \ L1 2): 2 L1 1 + L 2 = L 1 + (L1 1 \ L2 2) + M1:
,
L +L = L +M
(L \ L ) L :
,
L 1 + M |1
.
z 2 L \ M: z 2 M L 2
z 2 L1 \
2
1
2
\L
, z 2 (L \ L ) \ M:
z=o
L1\M
.
dim(L 1 + L 2 ) = dim L 1 + dim M:
2
1
2
, dim L = dim(L \ L ) + dim M:
,
.
L
1.
2.
3.
4.
)
1
2
3
4
e1 e2 e3 e4 :
L
0
5
6
7
8
9
10
11
12
L:
-
0
L
M
L =L M :
. 1,
L
,
-
L
00
0
0
)
.1.
-
0
0
.1
b1 b2
L +L
0
a1 a2 a3
1 1 1 2 2 2 2 5:
L \L
0
00
:
0
00
:
-
. VI.
172
5.
)
)
:
)
?
x
1.
L
A:
3.
.
.
L L
L
L
L ! L:
.
x y
|
A:
L
(1)
,
L
.
F(t)
,
n-
A
-
|
,
.
.
.
L
-
).
,
;1 1] 0 2]:
,
:
L
C 0 ;1 1] C 0 0 2] |
,
5.
.
,
L
f(t)
(1)
-
x:
(
4.
.
Rn
|
,
L:
.
,
(11) x 2 . IV |
3.
n.
n-
A(x):
.
.
2.
L:
x
,
L L
|
L
1.
x
,
A( x) = A(x):
+
,
L
,
-
A
L !L
A(x + y) = A(x) + A(y)
,
.
,
'(s) = f(s ; 1)
.
C 0 ;1 1]
F (0) = 0:
m n:
3.
173
2 Rn
A:
6.
L
m:
R n R m:
L
.
.
,
.
n m
L L
,
.
= 0A(x) = o: (
L
6,
A(L )
0
1.
L
L
0
-
.
-
,
,
.)
,
,
-
L
A(o) = A(0x) =
L
.
A:
dim A(L ) 6 dim L :
0
.
-
L !L
-
0
.
k > 0:
e1 ::: ek |
L:
x2L
x = 1 e1 + ::: + k ek
A(x) = A( 1 e1 + ::: + k ek ) = 1 A(e1 ) + ::: + k A(ek ):
(2)
,
A(L )
L
A(e1 ) :::
::: A(ek ):
,
,
,
L: ,
A(L ) |
A(e1 ) ::: A(ek ) ,
,
.
k
1 x 2:
:
L
A(L ) L :
Im A:
.
.
L
A
m A(L )
L
L:
, ,
.
.
,
-
L
0
0
0
0
0
0
Ker A:
A
2.
0
A
L:
. VI.
174
,
.
,
= A(x) + A(y) = o:
:
A(x) = o
: dim Ker A > 1:
.
A(x + x0 ) = y:
y2L
A
A(L )
= A(x) o 6= x0 2 Ker A
,
A
A(x1 ) = A(x2 ) = y
:
.
,
,
,
|
x1 6= x2
L ! L: 1
A( x + y) =
,
y=
-
.
, -
A(x1 ; x2) = o z = x1 ; x2 |
.
3.
,
.
:
x1 ::: xk
A( 1x1 + ::: + k xk ) = o:
1x1 + ::: + k xk = o ,
.
2.
A:
A(y) = o
L L
,
-
.
,
1A(x1) + ::: +
kA(xk )
-
= o:
, x1 ::: xk
n m
L:
.
e1 ::: en |
x = e1 + ::: + n en
A(x) = 1 A(e1 ) + ::: + nA(en ):
, A(x)
x
A(e1 ) ::: A(en ):
L:
= k f1 ::: fm k:
f:
m
X
p f (i = 1 ::: n):
A(ei ) =
i p
(3)
f =
-
p=1
(3)
A(x)
:
1
::: m
m
X
p fp = X i p fp :
i
p=1
ip
n
p=X p i
i
i=1
(p = 1 ::: m):
(4)
3.
175
p
i
A
,
=A
,
1
:
m n
f)
x(
A:
,
L
4.
.
A
.
.
x=e
B
B
A(ej1 ) ::: A(ejr ):
.
.
-
L
-
A(ej1 ) ::: A(ejr )
A(ei ) (i = 1 ::: n)
A(x) -
,
A:
,
,
5.
A:
m n
! L:
j1 ::: jr |
A:
,
Im A
L !L
.
=B
.
ei . . iB
5
-
(
A(e1 ) ::: A(en )
,
i-
.
A
.. :
.n
,
f
.
1
1
1
e:
.
e f
)|
f:
(5)
L
(5)
1
n
.. = : :: :: : :::
::
::
::
::
.m
m ::: m
1
n
x
A(ei ):
(4)
.
(5)
.
-
A =o n r: d = n;r
. VI.
176
,
, -
.
,
r=n
, . .
,
,
,
L
. .
,
.
r = m:
3.
6.
: n = m = Rg A:
.
L !L
L L
nL n.
1.
,
,
-
L !L
.
6
.
.
,
.
:
.
.
L L|
n
,
A:
,
,
(
,
r=n
7.
.
, -
y2L
.
,
,
.
,
:
,
.
,
n,
,
L
-
.
(5)
,
.
,
.
,
6.
,
.
-
.
|
)|
,
-
3.
4.
.
e
S P
|
0
0
0
0
:
,
0
,
2.
,
L
e1 ::: er
12
. .
;
0
(5)
0
(6)
.
L L
.
A:
L !L
A = EOr O
O
r
Rg A = r
.
n;r
r
(7)
,
.
er+1 ::: en
n ; r),
r
fr+1 ::: fm
r
m:
L !L
A
A
,
L
A(x) B (x)
L
, C (x)
(7).
.
B: L ! L:
A+B
C (x) = A(x) + B (x)
,
C|
A:
B :
0
,
.
A
e e
f
0
0
L
.
A:
;
L
6.
L
x
0
0
A:
-
0
0
y
(4) x 1 = S
=P :
P = AS :
, = P 1AS :
.
Ker A (
A(e1 ) ::: A(er ):
e f
A
A A:
L L
).
.
-
L ! L:
A = P 1AS:
5.
(Er |
0
A:
A
0
(5),
0
r
0
0
=A :
0
f
e f
f
e
x
y = A(x):
f
177
C :L !L
x 2 L:
,
r
-
. VI.
178
A + B = (A + B) :
|
A + B:
A+B
A
B
,
A|
,
,
-
x
A(x):
A:
,
-
L L
,
m n:
(
B:
L !L
0
, BA
f
e f g:
,
B|
L L
LL L
A
0
0
00
).
A
L:
:
.
4 x3
(
8.
L L:
A:
E E|
,
B (A(x)) = x
9.
,
L L:
A|
,
=A
A(x)
=
. V)
-
.
,
B: L ! L
BA = E AB = E
.
e
f g:
BA
B
BA
= B = BA
BA
00
7.
.
L !L
L L L: A: L ! L
,
00
B (A(x))
B:
0
00
e g:
e f
1:
A
|
A|
.
L ! L:
A
A
A:
.
A
L !L
A|1
A :
A 1
;
;
1
;
x2L y2L
A(B (y)) = y:
A
f e:
.
-
(8)
.
-
4.
,
179
(8),
A:
2.
A
.
L
u = A(A 1 (u)) 2 A(L ):
z 2 Ker
A:
A
= A 1 (A(z)) = A 1(o) = o:
r<m
u
,
;
r < n:
A(L ):
-
2
. 1 x 1)
-
:
z 6= o
:z=
1
;
;
;
f e
A 1:
;
1.
C=
A
2 3:
)
)
2.
Ker A
,
2
1 2 3 :
2 4 6
:
(
Im A:
)
,
,
. 1,
?
Ck |
C
2 3
?
,
k
0 1]:
C k C k 1: ,
|
.
:
)
)
?
4.
A: L ! L M = A(L ):
A : L !M
A (x) = A(x):
, :
) Ker A = Ker A ) Rg A = Rg A ) A
.
5.
L = L 1 L 2 x = x1 + x2 x1 2 L 1 x2 2 L 2:
P1 P2
L
P 1 (x) = x1 P 2 (x) =
= x2 (
).
,
2
P 1 + P 2 = E P 1 P 2 = P 2 P 1 = O P i = P i (i = 1 2)
O|
, E|
.
6.
2,
(7).
7.
A|
.
, :
a) A(L \ L ) = A(L ) \ A(L ) ) A(L \ L ) A(L ) \ A(L )?
3.
;
0
0
0
0
0
x
1.
,
,
12*
00
0
0
00
4.
0
00
0
00
.
.
,
|
.
.
-
. VI.
180
,
L
L
0
,
0
,
.
.
e = k e1 ::: en k
,
(7) x 3
\
"
A
,
(6) x 3
,
,
|
2 x3
,
.
,
.
,
.
L
,
2.
,
,
AB = BA:
,
,
,
.
,
AA
.
L
,
, . .
2
,
,
.
,
,
,
,
.
,
(1)
A
S P:
.
,
-
.
,
x3
| -
e:
(6) x 3
;
0
L !L
A:
A = S 1 AS:
A
-
,
A(e1 ) ::: A(en )
0
(1),
L !L
A:
L L:
,
,
AB
A B
A2
L
.
,
.
.
.
-
,
-
.
A
BA:
B |
,
-
-
4.
181
Ak = AAk 1:
A
;
E:
0E
p(t) =
p(A):
A
B
,
,
3.
BA
p(A)
L
L
0
1.
A
.
,
A
,
-
, A(L )
0
,
,
,
2.
.
L
L:
3.
,
.
.
0
A:
L
L
x
,
,
,
,
A
A:
L
L:
0
L
p:
.
p
.
k-
A(x)
0
.
4.
n-
-
p(A):
p
5.
e1 ::: ek
.
BA
0
,
-
A
A
, ,
.
A(x) = x:
,
A
,
.
e
,
A
k tk
0 + 1t + ::: +
-
A
+ 1A + ::: + k Ak
-
L
A
.
e1 ::: en
L
0
,
A
-
. VI.
182
A1
(n ; k)
i
j
, ,
,
:
A
A
A = A1 A2 :
3
4
A2 A3 A4
k k k (n ; k) (n ; k)
(n ; k)
.
,
A3 = O . .
A
j = 1 ::: k i = k + 1 :::
,
k
A |
A(e1 ) ::: A(ek ):
L |
,
L
ek+1 ::: en
.
,
,
,
0
0
k
n:
-
A
1e
= j 1 + ::: +
e1 ::: ek
,
2
A = AO1 A
A4
e1 ::: ek
- p
i < p+1
p + 1 ::: p + k
k
,
.
,
i
j
A
i > p + k:
L
L = L1
.
:::
A=
,
,
2.
=0
-
k
p + 1 ::: p+ k
j = p + 1 ::: p+ k
, . .
s
d1 ::: ds
Ls
d1 ::: ds
A1
A2
O
-
.
.
-
e1 ::: ek |
.
j = 1 ::: k
k ek
j
(2)
1.
(2)
A:
A(ej ) =
-
-
.
:
O
...
-
(3)
As
-
,
-
4.
L
183
A
0
A:
0
.A
0
L L:
,
.
0
L
,
0
0
,
A
,
4.
A ; E.
,
(
.
0
,
|
|
x
(A ; E )(x) = o
A
.
,
B
-
Ker (A ; E )
,
,
:
.
A
= 0:
A(x) ; E (x) = A(x) ; x = o ,
A(x) = x:
4.
0
x 2 Ker A
A(x) = o
, B (x) 2 Ker A:
z
,
x = A(z):
,
B (x) 2 Im A:
.
,
A
A
|
.
,
)
,
:
L
.
|
L L
0
(2).
. 1:
,
0
,
A(B (x)) = o
2:
x 2 Im A
B (x) = B (A(z)) = A(B (z)):
A ,
0
3.
B (A(x)) = o:
A
0
0
A:
A
.
,
A L
A (x) = A(x)
A
-
0
e1 ::: ek
A1
A
-
L:
L
L
A
.
,
,
,
,
(4)
-
. VI.
184
-
-
A:
(A ; E) = o
A|
e:
(5)
,
(5)
-
(5)
:
( 11 ; ) 1 + 12 2 + ::: + 1n n = 0
2 1 + ( 2 ; ) 2 + ::: + 2 n = 0
1
2
n
(6)
: : :: :: :: :: :: ::: :: :: :: ::: :: :: :: : :: ::
n 1 + n 2 + ::: + ( n ; ) n = 0:
1
2
n
.
x
A
(
)
: 1) x 6= o 2) A(x) = x:
,
|
.
5.
,
A:
.1:
x
, y
L
x:
y= x
A(y) = A(x) = x:
, A(y)
L:
2:
x|
L:
A(x)
L
: A(x) = x:
x 6= o
.
.
.
0
0
0
0
.
6. i,
i,
i-
= ei
.
5.
A(ei )
A
.
|
i-
,
.
.
A; E
-
.
ei
A(ei ) =
A(ei ) -
,
.
A
A; E
.
-
4.
185
5 x3
a11 ;
,
a12
2;
2
1
:::
n
2
:::
det(A ; E) = det : :: :: : : ::: :: :: :: : :: :: :: ::n:: :
n
n
::: nn ;
1
2
(7),
A
A:
,
,
.
2
1
= 0:
(7)
|
-
-
1.
.
-
.
n:
(10) x 4 . V
,
,
n
,
.
( 11 ; )( 22 ; ):::( nn ; )
:
i (i 6= j)
j
( ii ; ) ( jj ; ):
,
n ; 2:
(8),
(;1)n n + (;1)n 1( 11 + 22 + ::: + nn) n 1:
(8),
;
=0
,
det(A ; E) = (;1)n n + (;1)n
1
;
,
,
n
(8)
,
.
:
.
n
n;1 X i + ::: + det A:
i
i=1
(
(9)
,
n
.
-
;
det(A ; 0E) = det A:
A:
.
-
),
,
-
. VI.
186
, ,
,
.
.
7.
A
.
,
.
A A |
-
0
.
(1)
det(A ; E) = det(S 1 AS ; S 1 S) = det S 1 (A ; E)S =
= det(A ; E) det S 1 det S = det(A ; E):
,
A
A:
,
.
,
.
1 + ::: + n
n 1
(
;
)
1
n
.
tr A
tr A:
(9)
,
,
|
.
6.
.
2.
0
;
;
;
;
;
.
.
5 x2
x1 ::: xs
1 ::: s
i = 1 ::: s
:
.
x1 ::: xs
B i = (A ; i E )
i j
B i (xj ) = A(xj ) ; i xj = ( j ; i )xj :
i 6= j B i(xi ) = o:
, B i (xj ) 6= o
,
,
x1 = 2 x2 + ::: + sxs :
x1
( 1 ; 2):::( 1 ; s )x1
j ( j ; 2 ):::( j ; j ):::( j ; s )xj
-
-
,
(10)
B 2 ::: Bs :
j xj (j = 2
::: s)
4.
. .
0
0
.
|
187
.
.
p( ):
p( ) = ( ; 0 )s p1( )
1
.
3.
0
s:
.
ek+1 ::: en
A
6:
A=
B
C|
,
L
-
0
k:
..0
.
0
0
..
.
0
:::
...
:::
:::
...
:::
A
1 1
0 1
,
.
e1 ::: ek
L:
k
n;k
.
0
:
0
..
. C
0
,
A; E
.
, -
2
.
,
(
:
8.
A
-
s:
0
..
. B
A
)
p1 ( ) |
.
k
,
det(A ; E) = ( 0 ; )k det(C ; E):
k 6 s:
s
,
,
s:
,
.
7.
,
s
A
-
. VI.
188
L
,
q=
-
,
0
.
):
,
A
.
L
L
A2 + pA + qE:
0
0
p = ;( + ) a
B = A2 + pA + qE
3L
.
.
B -
t2 + pt + q (
= Ker B :
|
A
0
,
: B = (A ; E)(A ; E):
det B =
det(A ; E) = 0
,
= det(A ; E) det(A ; E) = 0
B
.
L
.
,
x
A(x) = x
B (x) = 2 x + p x +
+ qx = ( 2 + p + 2q)x:
, + p + q 6= 0
B (x) = o
x = o:
x
.
x|
L:
L |
x A(x):
.
,
y = x + A(x) |
L :2
A(y) = A(x) + A2 (x):
B (x) = o
, A (x) = ;pA(x) ; qx
A(y) = A(x) ; pA(x) ;
; qx:
, A(y)
x A(x) . .
L:
,
x A(x) |
.
,
.
1,
,
,
,
.
.
.
,
,
,
. ,
.
.
,
0
0
00
00
0
8.
.
.
|
9.
.
6
,
.
A
e1 ::: en
-
-
4.
189
(
,
,
10.
.
,
,
L:
|
L
.
L
,
A:
L
L
,
,
::: s:
p(t) |
. 1:
L
,
x
,
-
1,
n:
L:
,
.
,
,
-
S:
A
,
A
|
-
S
A
.
;
p(A) = O
-
.
12.
,
S 1 AS
4.
,
,
(
),
,
,
,
,
n-
,
11
A
,
.
,
.
n
.
11
-
,
,
.
.
L
A
11.
n
. 6).
,
)
L
-
(
.
.
.
,
-
,
x = x1 + ::: + xs
p(t) = (t ; 1 ):::(t ; s )
R = p(A) = (A ; 1 E ):::(A ; s E )
-
.
,
1
:::
-
. VI.
190
(10) (A ; i E )(xj ) = ( j ; i )xj
R (xj ) = ( j ; 1 ):::( j ; j ):::( j ; s )xj = o:
,
R (x) = R (x1 ) + ::: + R (xs ) = o
2:
,
A
p(A) = O
p(t) = (t ; 1):::(t ; s ):
1=p(t)
:
1
p(t) = t ; + ::: + t ; :
,
1
x:
-
s
1
s
1 = q1(t) + ::: + qs(t)
qi(t) = i p(t)=(t ; i) |
,
i
p(t)
t ; i:
A
t : E = q1(A) + ::: + qs(A):
x:
x = x1 + ::: + xs
xi = qi (A)(x):
xi 6= o
,
,
.
,
i |
(A ; iE )(xi ) = i p(A)(x) = o:
,
.
, L
.
9.
.
5.
,
|
.
.
,
-
,
.
A
.
|
-
.
n-
-
(n ; 1)L n 1:
, dim Im(A ; E ) = n ; dim Ker (A ; E ) 6 n ; 1:
(n ; 1)Ln 1
Im(A ; E )
x 2 L n 1 A(x)
(A(x) ; x) + x
A(x) ; x 2 Im(A ; E )
L n 1 x 2 L n 1:
Ln 1
n;1
.
Ln 1
,
n;1
A
.
n;1
,
L n 1:
;
;
;
;
;
;
;
;
4.
Ln
;
1:
,
191
Ln
Ln
2
;
A
n;2 (n ; 1),
1
;
.
,
-
L 1 ::: L n 2 L n 1
e1 2 L 1 e1 e2 2 L 2 ::: e1 ::: en 1 2 L n 1 :
.
;
;
-
;
.
,
.
,
(11)
-
;
-
A
L
A
0
0
.
,
-
A
0
p = ;( + ) q = :
8
x2L
,
(A 2 + pA + qE )x = o:
2
A (x) = A(x)
(A + pA + qE )x = o:
,
B
A2 + pA + qE
. B = A2 + pA +
+ qE = (A ; E)(A ; E).
det B = 0
det(A ; E) =
=0
.
,
.
0
0
0
0
0
1.
2.
.
Im A
Ker A
,
A:
,
L
8, |
.
A : L 2! L :
,
L = Ker A Im A
Ker A = Ker A:
L = Ker A Im A:
e
e1 ::: er 2 Im A er+1 ::: en 2 Ker A?
x y |
n:
,
det(E + xyT ) =
3.
4.
,
5.
A
6.
= 1 + xT y:
7.
,
,
,
,
0
3 ;2 6
;2 6 3 :
6 3 ;2
. VI.
192
8.
.
9.
10.
n
.
,
A B|
A
,
,
A
1 ;2 ;2
4 7 6 :
;1 ;1 1
A=
-
,
x
1.
.
5.
g(x
f
::: n:
f
,
.
-
.
f (x)
L
",
,
,
g
x y
,
,
L
n
n
.
f
1
, ,
n
f
x y
f (x + y) = f (x) + f (y)
f(
L
x) = f (x):
1
::: n
L
,
.
L
.
n:
n
,
\
.
,
y):
x
,
n1
.
,
.
,
A |
0
L
,
2.
(AT = A)
.
11.
L
2L
.
(AT = ;A):
det A 6= 0: AB BA
.
A
.
T
.
x2
-
::: n:
,
x -
,
L
(1)
5.
L
193
,
,
.
,
.
1.
L
,
,
,
f (o) = 0:
= (a x):
.
.
a:
,
n-
4.
x
|
,
,
L
i-
L
0 1] (
C:
-
x
(1)
3.
, i
p:
0
-
,
2.
|
-
.
n
C
1 x 1).
u
C
Z1
e:
,
e:
L:
p1 ::: pn:
.
,
v|
-
= v(t)u(t) dt:
0
,
,
e1 ::: en:
|
u
n-
f
C
u(0):
L
::: n:
f (x) = f ( 1 e1 + ::: + n en ) = 1 f (e1 ) + ::: + n f (en ):
f (e1 ) ::: f(en )
x
f
.
1.
ne
f (x) = '1 1 + ::: + 'n n
1 n . .
,
13
. .
1
.
'1 ::: 'n
f
n-
|
f
k '1 ::: 'n k:
n:
.
e
e:
(2)
.
x
-
(2)
-
-
. VI.
194
:
1
..
.
f (x) = k '1 ::: 'n k
'
, '( + ) = ' + '
(6) x 3
,
'
i:
3.
'0
|
'(
= 'S:
0
i
S:
0
(4).
. x3
.
.
+
.
g
f
g
|
f
g(x) =
:
= (' + ) :
,
,
f+
,
.
x
'+
:
,
f +g
f +g
:
|
nn
.
|
'
'+
|
L
L
3.
,
e:
-
f
x
g
':
n-
0
0
,
L
(4)
e ' |
,
(4)
, 'i = f (ei ) = ' i
ei
i-
x
2.
+g |
.
):
f
= f (x) + g(x):
f (x) = '
'
'(
0
.
g(x) = f (x):
.
)=
(3)
.
e = eS:
.
h
h(x)
(3)
=' :
n
|
5.
L
,
pi
(i = 1 ::: n)
,
,
,L |
n:
:
.
e
L
,
L
(i j = 1 ::: n)
pi
,
.
L:
k '1 ::: 'n k
L
L
::: pn
p1
f2
L
p0 T
L
(7)
L
e1 ::: en
f =
,
e e
(4)
e = eS:
p p:
0
0
;
,
= pT (S 1 )T :
;
(S
1 )T :
;
-
0
(4) x 1
,
T
'T = (S 1 )T ' :
.
-
(6)
(6)
pi :
p
(5),
-
L
f
(7)
L
-
.. = 'p:
.n
= k '1 ::: 'n k
,
-
::: pn
e1 ::: en
L:
= pT 'T :
,
p1
p1
,
p
n-
:
f
(5)
i-
'1 ::: 'n:
k'1 ::: 'nk
f = '1 p1 + ::: + 'npn :
p
i | i-
,
.
-
L:
pi (x) = i
x(
.
-
L
3).
0 i 6= j
pi (ej ) =
1 i=j
.
13*
195
0
,
p
p0
L
-
-
. VI.
196
,
L
, ,
2L
x
,x
:
.
f (x + y) = f (x) + f (y):
L
:
1.
x2L
2.
a
L
:
,
:
|
L
,
L
:
L
x
f (x):
,x
,
,
) f (x) > 0?
,
.
f
L :
L
-
-
dim L = dim L = dim L
.
) f (x) > 0
(8)
-
, (f + g)(x) = f (x) + g(x) ,
,
( f )(x) = f (x)
,
,
L
f
,
L
L
.
.
f:
,
|
4.
L:
x
|
p = S p0 :
L
.
,
x
.
,
,
e1 e2
a = e1 + e2 :
,
(4).
3.
k|
.
n
ka:
,
.
: a) 1 t t2 ::: tn ) 1 (t ; a) (t ; a)2 ::: (t ; a)n :
4.
e1 ::: en 2 L p1 ::: pn 2 L |
.
,
x2L
f 2L
x=
= p1 (x)e1 + ::: + pn (x)en f = f (e1 )p1 + ::: + f (en )pn :
x
0
a
x
1.
.
6.
L
.
b
6.
L
(
|
197
, . .
x y z
)
b (x + y z) = b (x z) + b (y z) b ( x y) = b(x y)
(1)
b (x y + z) = b (x y) + b (x z) b (x y) = b(x y):
1.
.
.
i
j (i j = 1 ::: n) |
e = k e1 ::: enk |
L:
x y
b
(1) :
b (x y) = b
,
,
n2
ij
e:
n
X
n
i ei X j ej
i=1
j =1
X
b (x y) =
i j b (ei ej )
ij
(2)
-
::: 1n
B = : :: :: : : : : ::: ::::::: : :2:n:: :
n1 n2 ::: nn
12
22
,
:
b(x y) = T B :
:
b
b (x y)
(3)
.
ij
C:
.
C
= b(ei ej ) = eTi C ej
4 x2 . V
.
0
-
(2)
C =B . .
(3)
C| -
,
ei ej |
,
ij
cij
,
, .
|
x y
e = eS:
=S
=S :
b (x y) = (S )T B(S ) = (S T BS) :
b
e
,
T
B = S BS:
(4)
0
(3)
B
n:
)
11
21
.
X
ij i j :
ij
= b(ei ej ) (
=
0
0
0
0
0
0
0
0
0
0
. VI.
198
,
X k l
ij =
i j kl (i j = 1 ::: n)
B
0
0
k
i
kl
|
i j . . ij = ji:
.
,
2.
b|
S:
b
b (x y) = b (y x):
,
,
b (ei ej ) = b(ej ei )
B
,
1 1
,
b (x y) = ( T B )T = T B T = T B = b (y x):
1.
,
.
.
x
2.
L
k
.
k(x) = b (x
|
.
k
b:
x y|
.
k(x + y) = b (x + y x + y) = b (x x) + b(x y) + b (y x) + b (y y):
,
b (y x) = b(x y)
b (x y) = 12 (k(x + y) ; k(x) ; k(y))
b
k:
,
.
,
k(x) = T B
k(x) =
.
-
.
.
(3)
(5)
X
ij
ij i j :
:
-
x)
,
(6)
(7)
6.
1
(7) |
::: n: (
ij i j
i 6= j
199
,
12
1 2+
,
.
.
.
1)
C1 |
2)
) 1i = 0
(10).
)
, i-
2 )2 + 2
i=1
,
:
.
-
k
-
.
11
.
B1
"1 0 ::: 0
0
..
.
C1
0
1i 6= 0:
ii 6= 0
-
B
,
( 1i =
i-
.
e
k
.
),
-
i,
(10)
n ; 1:
: 11 = 0:
i = 2 ::: n:
i
(8)
(9)
11 6= 0:
,
1 3 + :::
"i ( i )2
B|
:
13
k
n
X
".)
:
.
B
i-
22(
1.
-
",
.
k(x) =
,
\
\
ji j i
(7)
k(x) = 11 ( 1)2 + 2
.
,
. .
,
.
-
i-
ii
0
11
6= 0:
(10)
=0
,
. VI.
200
k
"1 ::: 0
...
O :
Bk =
0 ::: "k
O
Ck
n;k
Ci
Ck |
.
, (k + 1)n;k
.
,
"1
...
B
"n
k:
-
1
:
Ck
,
,
1
1
;
0
-
-
Ck :
Cn
B =
"1 ::: "k
-
n;k
Bk
Bk+1
(n ; 1)-
(11)
-
.
).
-
,
.
S
ST
( . 4 x 2 . V).
.
e
B = S T BS
0
,
,
k
S:
,
e
S = S1 :::SN |
B
0
0
.
,
B
, . .
S:
-
S
.
.
-
6.
-
.
.
,
201
k(x) = 2( 1)2 + 4 1 2 + 3( 2)2 + 4 2 3 + 5( 3)2 :
,
2(
( 1 )2
,
1:
1)2 + 2 1 2] + 3( 2)2 + 4 2 3 + 5( 3)2 :
2( 2)2 :
2 ( 1)2 + 2 1 2 + ( 2 )2 ] ; 2( 2 )2 + 3( 2 )2 + 4
k(x) = 2 ( 1 + 2)]2 + k (x)
k |
2
3:
2
2
2
3
k (x) = ( ) + 4
+ 5( 3 )2 :
:
2
3
2
k (x) = ( + 2 ) + ( 3 )2 :
0
,
2 3 + 5( 3 )2 :
,
0
0
0
,
k(x) = 2( e1)2 + ( e2)2 + ( e3 )2
e1
= 1+
2
e2
= 2+2
3
e3
= 3:
,
2
;2
12
.
1
= e1 + e2
2 2
12 ( e )
2
1 2:
,
,
= e1 ; e2
.
,
-
i = ei
1
(i > 2):
.
,
2 12( e1 )2 ;
.
.
: -
.
1, ;1
,
0.
"k
-
. VI.
202
1
2.
k-
p
ekp
"k
"k
,
,
),
(5)
k
,
r
( 1 )2 + ::: + ( r )2 :
L:
.
>0
,
x
B B
B = S T BS
det S 6= 0:
3 x 3 . V.
,
,
.
.
k:
.
k
L
L
k(x) < 0
0
,
,
.
,
.
0
0
.
,
,
0
Rg B = Rg BS = Rg B
,
.
.
3.
.
k-
.
.
.
(
k(x)
,
k,
"k 6= 0
.
3.
,
.
,
.
j"k j
k
,
0:
L:
x
0
L:
0
x 6= o
L
k
L:
,
k(x) > 0
k(x) 6 0
0
-
-
6.
,
.
)
203
.
(
,
L(
.
)
;
|
,
dim L (
.
.
)
,
r
,
-
;
.
-
,
.
4.
,
-
k:
-
.
,
k
;1
e1 ::: en
k
s
1
;( )2 ; ::: ; ( j )2 + ( j +1 )2 + ::: + ( r )2 :
L1
e1 ::: ej
L2 |
.
j +1 = ::: = n = 0 k(x) = ;( 1 )2 ; ::: ; ( j )2 < 0
x2L1
x 6= o:
,k
L 1 s > j:
L
k
,
2
1 = ::: = j = 0
x 2 L 2 k(x) = ( j +1 )2 + ::: + ( r )2 : (
,
r < n:)
dim L 2 = n ; j:
L( )
s>j
k
.
,
L2 L( )
n
z
.
k(z) < 0
z 2 L ( ) k(z) > 0
z 2 L 2:
,
j = s:
,
;1
,
.
,
+1
,
r;s
r
s
.
.
.
;
;
;
-
.
0
n
( 1)2 + ::: + ( n )2 :
(12)
n
. VI.
204
| ( 1 )2 ; ::: ; ( n )2 :
-
n
r
( 1 )2 + ::: + ( r )2
;( 1 )2 ; ::: ; ( r )2:
,
.
,
,
.
.
2.
-
,
.
,
,
.
(5): det B = det B(det S)2 .
0
,
5.
.
,
-
11 :::
1k
Mk = : : : ::: :: :: :: ::: > 0 (k = 1 ::: n):
k1 ::: kk
(13)
1:
,
.
,
,
.
|
k
.
.
.
.
1.
-
ii = k(ei ) > 0
,
2:
k
(13)
.
,
.
, M1 = 11 > 0
(10) "1 > 0:
Bk
.
B
,
"1 ::: "k
6.
.
"k+1 = Mk+1=Mk
"k+1 > 0
Ck
.
205
"1 ::: "k+1:
.
L
-
k
.
4.
-
.
-
.
.
b
,
-
-
xy z
b (x + y z) = b (x z) + b (y z) b ( x y) = b(x y)
b (x y + z) = b (x y) + b (x z) b (x y) = b (x y):
:
,
:
.
,
,
L
.
,
.
,
x y
X
b(x y) =
ij i j = T B :
ij
B
.
: ij = b (ei ej ):
S
B
b
B = S T BS:
0
,
b
.
.
ij
B
,
,
.
k
-
b (x y) = b(y x):
,
BT = B
-
= ji :
-
: ij = ji
: ii = ii :
,
-
.
BT = B
k(x) = b(x
,
x)
-
b:
. VI.
206
k
1 ;1
0
.
:
,
,
:
0 1 ;1
.
,
,
.
1.
i
b (x y ) =
)
+
1 2
;2
)
2.
2 2
x y:
+4
2 2
+3
1 3
,
.
: e1 = e1 + e2 e2 = e2 + e3 e3 = e3 ?
( 1 )2 +
0
3.
4.
0
1 2 3
2 4 5
3 5 8
)
)
,
s
x
|
b?
A|
8.
.
m
k<n
AT A
,
B = RT R:
,
10.
,
,
7.
x
k(x) = 0?
11.
,
: )
k:
1 2
?
n
B
b
?
)
:
+ ( 2 )2 :
:
:
b
b (x y) = 0:
s
,
.
y
r:
.
R det R 6= 0
.
?(
.)
3 3
. 1, ),
.
-
n-
6.
+
0
1 2 3
2 4 5
3 5 9
-
5.
9.
1 1
b
j
-
,
?
-
-
7.
x
1.
,
.
1.
A
207
7.
{
.
,
-
p( ) = det(A ; E) |
p(A) = O:
.
(A ; E)
,
(4) x 5 . V.
(A ; E) 1 = det(A1; E ) B( )
A
,
(1)
;
bij ( ) = (;1)i+j dji ( )
n;1
(A ; E) ,
,
n ; 1:
bij ( ) = b0ij + b1ij + ::: + n 1bnij 1:
B( ) |
dji
,
;
;
B( ) = B0 + B1 + ::: + n 1Bn 1
Bk |
bkij (k = 0 ::: n ; 1):
(A ; E)B( ) = det(A ; E)E
(A ; E)(B0 + B1 + ::: + n 1Bn 1 ) = p( )E:
;
;
a0 a1 ::: an:
(2)
,
:
AB0
= a0 E
AB1 ; B0 = a1E
AB2 ; B1 = a2E
: :: : : :: ::: :: : : : : :: ::: :: : :
ABn 1 ; Bn 2 = an 1E
; Bn 1 = anE:
A0 = E
. .,
An
p(A) |
,
|
.
;
;
.
A2
,
(1)
(2)
-
;
,
;
|
.
;
p( )E = a0E + a1 E + ::: + n anE:
(2)
:
,
,
-
,
-
,
;
|
,
A
A
-
. VI.
208
.
L
-
A
p(A) = O :
2.
L
.
n-
-
A:
p(t)
:
p(t) = (;1)n (t ; 1 )k1 (t ; 2)k2 :::(t ; s )ks :
.
-
,
.
qi(t) |
p(t)
-
.
1=p(t)
-
f1 (t)
fs(t)
1
p(t) = (t ; 1 )k1 + ::: + (t ; s )ks :
,
1 = q1(t) + ::: + qs(t)
fi (t)
(t ; i )ki :
qi(t) = (ft ;(t)p()t) (i = 1 ::: s):
, -
i
i
ki
E = Q 1 + ::: + Q s :
Q i = qi(A)
Q iQ j = O
i 6= j:
A
,
t:
(3)
(4)
, p(t)
A
.
(3) Q i
(4),
i = 1 ::: s
Q i = Q iQi:
(5)
L
.
x:
(3)
x = Q 1(x) + ::: + Q s (x)
(6)
. . x = x1 + ::: + xs
xi = Q i (x) 2 Q i (L ):
.
,
,
x = y1 + ::: + ys
yi 2 Q i(L ) (i = 1 ::: s):
,
zi
yi = Q i (zi ):
,
x = Q 1(z1 ) + ::: + Q s (zs )
,
qi(t)qj (t)
7.
(5), . . xi = yi
(6)
209
Qi
L
,
L = Q 1 (L )
3 x4
(i = 1 ::: s):
1.
,
L
Q i (x) = Q i (zi )
.
|
(4)
Q i (L )
,
::: Q s(L ):
Q i (L )
.
.
:
(7)
Ki
A
A:
,
2. K i = Ker (A ; i E )ki
i:
.
(t ; i )ki qi(t)
,
.
{
,
(A ; i E )ki Q i = O :
,
x2L
(A ; i E )ki Q i (x) = o . .
Q i (x) 2 Ker (A ; i E )ki Q i (L ) Ker (A ; i E )ki :
,
x 2 Ker (A ; i E )ki :
Qj
j 6= i
(A ; i E )ki
x
.
(6)
x
x = Q i (x): , x 2 Q i (L )
Ker (A ; i E )ki Q i (L ):
.
.
:
Ker (A ; i E ) K i :
,
(A ; i E )(x) = o
(A ; i E )ki (x) = o:
2
(7)
:
k
k
1
s
L = Ker (A ; 1E ) ::: Ker (A ; sE ) :
(8)
3.
.
Ki
(A ;
; iE )
.
B:
i
.
2
,
B ki = O :
,
,
.
,
K
B:
B k (x) = o
x
,
h
,
B h (x) = o
- x
h<k
B h (x) = o:
14
. .
. VI.
210
B h 1 (x) 6= o
;
x:
B . .
.
Bh
;
1
.
1 m|
A:
B (K ) K
B h (K ) B h 1 (K )
h
m
m
1
fog = B (K ) B (K ) ::: B (K ) K :
Vh
B h (K )
Ker B :
fog = V m V m 1 ::: V 1 Ker B :
Ker B
:
Vm
,
-
m 6 k:
,
-
;
;
-
;
. .
,
e01 ::: e0d
h
V
,
B h (K )
= B h (ehj ):
1
;
V m;2
Vh
e0j
Ker B
V h:
V h+1:
ehj
helj = B h l (ehj ) (l = 1 ::: h)
e0j :
B (elj ) = B h l+1 (ehj ) = elj 1 :
,
e0j :
;
l-
;
,
3.
;
Vm
(9)
l + 1:
e
.
e0j e1j ::: ehj
B (ehj ) = ehj 1 :
B (e2j ) = e1j :::
e0i 62 V 1
|
h
+1
h
0
B (ej ) = B (ej ) = o:
e0j =
,
(9)
l-
.
.
(10)
m:
-
e01 ::: e0d:
K:
e .
,
-
e
.1:
1
;
,
,
,
,
s
,
-
3
;
;
e0j
B (e1j ) = e0j
,
(9)
Vm
,
7.
211
.
s+1
,
.
,
.
B:
(10)
,
,
2:
B h (K ):
,
y
K
x
x
,
B h (x) =
B h (y) = o
1B
,
Cj :
K
,
(A ; i E )
, A(x) = B (x) + i x 2 Cj :
.
14*
, B h (x) 2 V h :
e01 ::: e0p |
.
h-
-
3,
|
K
-
pB
6 h:
.
x 2 Cj
0
B (x) = B ( 0ej + 1e1j + ::: + h ehj ) =
, Cj
B h (x)
h (eh ):
p
h
h
1e1 ; ::: ; p ep
e:
.
6 h:
Ker B
h (eh ) + ::: +
1
pa x:
-
e:
h + 1:
y = x;
. .
x
e
,
.
.
.
.
dim V h = p:
V h B h (x)
.
|
,
,
e:
1
e01 ::: e0d
,
,
,
.
-
,
e0j |
.
Cj
L:
|
(10)
h 1 2 Cj :
0
1ej + ::: + h ej
B:
B|
= Ki
A:
-
;
|
-
. VI.
212
K
4.
B |
K
K = C1
B
d
4.
.
5.
A
::: + ds
A
L
A
Cd
:::
.
|
A
e0j ::: ehj:
L
.
di = dim Ker (A ; i E ):
L|
,
2 x4
.
.
,
.
-
Ker B :
1 4
L
m. e. d1 +
-
A
,
-
i
A(elj ) = B (elj ) + i elj (l = 0 ::: h)
(10)
0
1
0
1
h
h 1
h
j = i ej A(ej ) = ej + i ej ::: A(ej ) = ej + i ej :
|
.
A
A(e0 )
;
-
i 1 0 ::: 0
0 i 1 ::: 0
: :: : : ::: :: :: :: : :: :: : :
0 0 0 ::: 1
0 0 0 ::: i
,
,
i:
-
,
.
h+1
-
.
2.
.
(
),
-
7.
:
213
,
e01 ::: e0d
,
.
,
(
|
,
.
,
,
|
ki
.
i
ki
-
.
,
,
.
i
,
.
.
,
(10)
-
.
6.
5.
)
.
i
,
.
-
,
.
,
,
-
V 1i :::V mi i
,
.
V li = (A ; iE )l (K i) \ Ker (A ; iE )
K i:
,
l
(A ; i E ) (L ) \ Ker (A ; i E ) =
= (A ; i E )l (K i ) \ Ker (A ; i E ): (9)
,
x L
x = x1 + ::: + xs (A ; i E )(x) =
= y1 + ::: + ys
yj = (A ; i E )(xj ) 2 K j
.
xj 6= o i 6= j
yj 6= o
Ker (A ; i E ) K i:
,
Ki
K i:
(9)
V li = (A ; iE )l (L ) \ Ker (A ; iE ):
A
. VI.
214
,
( ; 1)3( ; 2)3 ,
2
0
A = 00
0
0
,
0 1
1 0
0 1
0 0
0 ;1
0 0
3
1 = 1:
1
0
A ; E = 00
0
0
,
a2
,
5,
,
2
.
0 1
0 0
0 0
0 0
0 ;1
0 0
0
0
0
3
0
1
1 0 0 kT
=2
0 0
0 0
1 0 :
0 ;9
2 0
0 ;3
aT1
,
,
Im(A ; E )
A;E
A;E
| 6.
| a2 :
(A ; E) =
0 0 1
0 ;1 0
A ; 2E = 00 00 ;10
0 0 ;1
0 0 0
0
0
0
2
0
1
,
, b
-
1 =k0 1 0 0 0 0k
A ; E:
,
3 = k0 0 0
A
,
,
V1
1
0 0
0 0
1 0 :
0 ;9
3 0
0 ;2
,
T:
2=k00 0301k
.
0
0
0
4
0
1
2
0 0
0 0
1 0 :
0 ;9
1 0
0 ;4
= k 1 0 0 0 0 0 kT :
,
.
2:
,
a3 :
-
.
,
b
-
7.
215
.
1,
|
,
|
2
1
= k 0 0 0 0 1 0 kT :
a2 a3
2
3
0
0
0
1
0
0
)
1
0
0
0
0
0
0
0
1
0
1
0
0 ;1
0 ;1
2 ;1
1 0
0
0
0
0
1
0
:
,
)
,
.
A |
0
1
0
0
A= 0
0
0
0
1
0
0
0
0
0
1
1
0
0
0
0 ;1 0
1 0 1=2
1 1 0 :
0 1 1=2
0 0 1
0
0
0
2
0
0
-
S
,
,
)
1 ;1
1 1
0 1
0 0
0 0
a1
,
0
,
)
2.
)
.
A
1.
3 ;1
1 1
0 0
0 0
2
0
0
0
0
3
0
1
:
b b1 b2 :
A
S
0
1
S = 00
0
0
1
(A ; 2E) = :
T
=
k
0
0
1
0
1
0
k
:
1
(A ; 2E) = 1 :
0
0
0
1
2
0
0
0
0 :
0
1
2
,
?
:
-
.
:
VII
x
1.
1.
.
,
,
,
.
,
.
. I,
,
.
,
.
.
:
,
.
x
(x y)),
(
y
E
,
:
1) (x y) = (y x)
2) (x + y z) = (x z) + (y z)
3) ( x y) = (x y)
4) (x x) > 0
x 6= o:
nE E|
0
(x y + z) = (x y) + (x z):
,
.
.
-
.
E
-
.
-
-
x y
z
E:
,
.
.
(x y) = ( y x) = (y x)
(x y) = (x y):
.
-
(1)
,
(2)
-
1.
217
.
,
1)
4) |
.
|
(2)
,
(1)
2)
,
\
"
.
,
1.
,
.
-
,
,
,
-
.
2. nT
i
,
,
,
=
,
1 1 + ::: +
n n
.
,
(3)
-
,
.
,
( .
)
3.
0 1]
-
.
i
3)
.
-
,
.
.
-
.
,
,
.
,
,
-
Z1
(f g) = f(t)g(t) dt:
2.
p
.
0
1){4)
(x x):
.
-
.
x y
cos ' = (jxxjjyy)j :
x
x4 . I
jxj
'
(4)
. VII.
218
4)
,
.
|
,
3.
= jxj2 + 2(x
,
|
x y
(x y) = 0:
.
-
,
(x x) = 0
1.
.
.
,
-
,
(x y) = 0
(4)
y ,
x = o:
x
(x y) = T ; :
;
(6)
{
:
2
2
y) + jyj 6 jxj + 2jxjjyj + jyj2 = (jxj + jyj)2 :
,
(x y) = jxjjyj . e.
.
(6)
|
,
.
,
=2:
(5)
-
jx + yj 6 jxj + jyj
(x + y x + y)
.
1.
,
,
,
.
(4)
.
(x y)2 6 (x x)(y y)
,
x y
-
,
y
:
y=x
-
e
(3) x 6
(ei ej ) . e.
(e1 e1) ::: (e1 en )
; = : :: : : : : ::: :: :: :: :: ::: : :
(en e1) ::: (e1 en )
e:
.
,
. VI
(7)
gij
1.
219
2.
.
1.
x1 ::: xk |
.
(x1 x1) ::: (x1 xk)
det : ::: :: :: :: ::: :: :: :: :: ::
(xk x1) ::: (xk xk )
,
.
,
.
,
,
,
,
2,
.
,
.
1x1 + ::: + kxk = o
.
,
.
-
1(x1 x1) + ::: + k (x1 xk) = 0
: : :: ::: : : : : :: :: ::: : : : : :: ::: :: : : :
1(xk x1) + ::: + k (xk xk) = 0
1 ::: k:
,
.
{
e
(4) x 6 . VI
,
.
-
(5),
-
e
0
S:
; = S T ;S
(8)
0
4.
.
,
.
(ei ei) = 1 (i j = 1 ::: n):
h1 ::: hn n-
(x y) = T =
3. n
,
-
.
,
,
1 1 + ::: +
: (ei ej ) = 0
(7)
n n:
-
,
-
i 6= j
.
(9)
-
. VII.
220
.
x=
,
h1 ::: hn
n
X
(x
hi )
j hi j 2 h i :
i=1
(hi hj )
.
.
(10)
1
,
x = 1h1 + ::: + nhn:
hi
,
i = (x hi)=jhij2
.
(x x)
i 6= j
(10).
jxj2 =
5.
(10).
(hi hj ) = 0
n
X
(x
i=1
hi ) 2 :
j hi j 2
.
e e = eS:
0
(11)
-
(8) ; = ; = E
0
S T S = E:
(12)
; =E
,
,
.
(12)
-
0
.
,
,
.
(12)
(12),
.
-
ST = S 1:
(13)
;
,
SS T = E:
ST
-
,
i
S
j
,
(12) (14):
n
n
X k k
X
0
i
=
6
j
0 i 6= j
i j
i j = 1 i=j
k k = 1 i = j:
k=1
k=1
,
(9),
,
|
.
.
(14)
(15)
-
1.
.
221
SU
S U|
, (SU)T = U T S T = U 1S 1 = (SU) 1 :
(12),
det S = 1
;
(det S)2 = 1:
det S = ;1:
,
cos
sin
cos
6.
E:
E
.
E
,
0
:::
:::
k
0?
k
X
ia
i
=
k
X
i (x
i=1
ai (i = 1 ::: k)
x:
(17)
1 1 + ::: + n
1
1
n=0
:: : : : : :: :: ::: : : : : :: :::
1 1 + ::: + n n = 0:
k
k
k
a1 ::: ak |
.
0
k(n ; k)-
|
E:
(n ; k)-
0
-
E:
x
0
,
0?
i=1
E
:
,
(x a1) = 0 ::: (x ak) = 0:
x2E
(17),
x
k
E
0?
a1 ::: ak |
E
n
ni |
E
4.
(x a) = x
(16)
E
,
.
,
:
.
0
n-
sin
; cos
n-
.
0?
-
; sin
cos
sin
.
;
,
2
k-
;
(17)
a
ai) = 0:
1
1i
,
n
:::
:::
:
.
|
,
.
. VII.
222
(E ) |
0? ?
; (n ; k) = k:
,
E :
, (E )
E E
E:
E
=E :
0
0?
0? ?
0
E
y2E
00
00
E
:
0?
:
7.
x2E
x2 2 E :
0?
E
E
0
E
0
E
E
|
00
0
0
x
0
E E
, x2E
E=E E
x1
x2 |
x
h1 ::: hk:
E
hk+1 ::: hn
E
0?
0
0?
E
x1 2 E
x E:
0
:
0?
,
:
-
x
,
(10),
k
X
x1 = (xjh hj ) hi :
(18)
i=1
k=1
x1 = ((x h)=jhj2)h
,
(18) |
,
h1 ::: hk:
(10),
,
(11)
.
(x1 x2) = 0
jxj2 = jx1 + x2 j2 = jx1j2 + jx2j2 > jx1j2:
jx 2 j
x E
,
.
6.
x1 |
x E:
y2E
x1
jx2j = jx ; x1 j < jx ; yj:
.
x1 ; y
z
2
2
2
jx ; yj = jx1 + x2 ; yj = jz + x2 j = (z + x2 z + x2) =
= jz j2 + 2(x2 z) + jx2j2:
, jx ; yj2 = jx2j2 + jz j2:
(z x2) = 0
z2E ,
.
,
0
0?
,
E
0
-
.
(x y) = 0
00?
.
,
0? ?
,
n:
5.
0
dim(E ) = n ;
0? ?
0
0?
E
(E ) :
0
i
i
2
0?
0
0
0
1.
8.
223
.
,
(18)
,
E
h1
{
f1 ::: fn:
f2
.
.
h1 = f1 : -
h2
:
h2 = f2 ; (fjh hj ) h1:
,
h2
f1 = h1 f2
h2 6= o
f1 f2
.
.
,
h1 ::: hk
i6k
hi
f1 ::: fi:
k
X
hk+1 = fk+1 ; (f jh j h ) hi:
(19)
i=1
hk+1 |
fk+1
h1 ::: hk
hi
i < k + 1: ,
f1 ::: fk+1
i6k
hi
f1 ::: fi:
,
hk+1 6= o
f1 ::: fk+1
.
fn
n
.
,
h:
e
ei = hi =jhij (i =
= 1 ::: n):
h:
S
h
f:
f1 = h1
(19)
, fj
j
h1 ::: hj
hj
1.
i
,
j
( i > j),
i = j:
,
|
( . 3 x 1 . V)
.
e
h:
h = eD
D|
.
f = hS
f = eDS
,
,
R = DS |
,
S
,
,
.
,
2
1
1
2
i
k+1
i
f
7.
.
2
S
e
h
h f
-
. VII.
224
h
R
.
x6
,
.
. VI,
9. QR-
R
a = gQR:
10.
k-
.
|
.
.
8.
A
.
.
a:
e|
a = gA:
-
.
,
,
, R|
R
e f
a1 ::: an
A
A|
.
a:
,
Q
.
,
A = QR
,
-
.
Q|
-
A
g
g a, . .
7
, e = gQ
QR = A:
.
,
-
,
,
a = eR
e
a = gA
.
k
ff1 ::: fkg
f1 ::: fk
i 0 6 i 6 1 (i = 1 ::: k):
.
,
.
ff1 ::: fk 1g
ff1 ::: fkg
,
,
jhk j
hk
fk
f1 ::: fk 1:
ff g
: V ff g = jf j
kV ff1 ::: fkg
.
,
(20)
,
.
fk
, hk = fk
V ff1 ::: fkg = V ff1 ::: fk 1gjfk j:
,
f1 ::: fk
1
n-
;
-
;
;
-
,
1.
225
(
)
n-
f1 ::: fn
-
.
ff1 ::: fng:
,
-
n-
fh1 ::: hng
.
;
h
h
1 ::: hn |
jh1j2 ::: jhnj2
.
p
V ff1 ::: fng = V fh1 ::: hng = jh1j:::jhnj = det ;h :
S|
7 det S = 1
h1 ::: hn f1 ::: fn:
det ;f = det(S T ;h S) = det ;h :
p
,
V ff1 ::: fng = det ;f :
(20)
e|
, F|
f1 ::: fn
.
|
e f:
;f = F T ;e F:
(20)
p
V ff1 ::: fng = j det F j det ;e = j det F jV fe1 ::: eng:
,
e
V ff1 ::: fng = j det F j:
( . 6 x 1 . VI),
n,
,
.
V ff1 ::: fng = det F
V ff1 ::: fng = det F V fe1 ::: eng:
(21)
n = 2 3 x 4 . I.
1.
62
,
Z1
(p q) = p(t)q(t) dt:
1
;
)
)
)
15
. .
1 t t2 :
1 (t ; 1) (t ; 1)2 :
t2 + 1 t + 1:
. VII.
226
2.
1
dim E = 4
3.
:
)
)
1
E
,
)
)
,
5.
)
-
0
+ 2+ 3=0
2
E
+ 3 + 4 = 0:
k 1 2 3 4 kT :
0
.
1 3 1
2 4 ;3 :
1 1 ;1
)
6.
n = 2k
,
.
QR-
1 1
2 3
E E
0?
4.
-
+ 2 + 3 + 4 = 0:
.
n
?
-
k|
:
-
,
k 1 1 ; 1 0 kT k 1 1 1 ; 1 kT k 1 1 1 1 kT :
.
2.
1.
x
,
,
,
.
x y
e:
e:
e:
,
,
x y
.
.
,
A
.
A
(A(x) y) = (x A (y)):
A
A A
A A
:
(1)
(A )T ; = T ;A
;|
,
T AT ; = T ;A :
,
(1)
, AT ; ;A |
x y
-
(1)
A:
(2)
2.
227
.
,
AT ; = ;A :
A A
,
A
1.
AT :
= AT :
(3)
(3).
,
(4)
e
B
.
(1).
(A )T = T (AT ):
,B
A:
,
A
(4)
.
.
(AT )T = A
(4)
,
(A ) = A:
A B (AB)T = B T AT
(AB ) = B A :
(4)
,
A A
.
,
.
n-
B
Ax = b
n
n
A|
. .b
Ker A :
,
15*
(5)
(6)
-
.
A:
,
x
Im A
,
: b 2 Im A
y 2 Ker A :
2.
.
,
A(x) = b
A:
AT y = o
A (y) = o . .
:
),
A
,
(b y) = 0
:
A
Im A = ( Ker A ) :
e
-
-
?
.V
y 2 ( Im A)
x.
?
,
(x A (y)) = 0,
(
,
.
(A(x) y) = 0
,
. VII.
228
x
,
y 2 Ker A .
2.
A
A=A :
y:
(4)
.
.
(A(x) y) = (x A(y))
,
1.
.
0
?
x
-
.
-
,
:
A
0
.
,
A:
0
.
A
A |
,
3.
,
.
A
, y 2 ( Im A)
A (y) = o.
.
A (x) = A(x)
,
0
-
.
,
8 x 4 . VI
A:
A
0
E
0
E:
A
,
-
0
:
;
+(
2)
,
( +
( ; )2 + 4 2 :
,
;
,
A|
0
E:
.
0
.
,
)2 ; 4(
4.
2.
A|
det(A ; E) = 0
.
+ ) +
-
,
.
.
,
2;(
2 ):
-
-
.
2.
229
.
.
E
A:
(A(x) y) = (x A(y)) = (x y):
( ; )(x y) = 0
3.
A
|
0?
E:
0
, A(y) 2 E
0?
.
4.
.
A|
E:
,
,
L
L
,
x
(
,
|
,
A
L:
.
,
A|
|
?
,
,
.
L
L
L :
A
0
E
5.
-
E:
?
|
,
0?
L
,
?
?
A(x)
0
-
E
A
),
,
E
x
A:
,
A
-
0
y2E :
(A(x) y) = 0
(x A(y)) = 0 ,
.
A
3
E
(x y) = 0
,
.
.
6= :
A(x) = x A(y) = y
(A(x) y) = (x y):
S
,
L
.
,
-
2.
,
S 1 AS |
;
.
E:
|
.
,
.
-
.
-
. VII.
230
,
A
4
,
,
,
(
,
A
.
A
.A=A
n
i ei
6
i
,
4.
)
A
|
.
.
ei
|
.
-
.
(
-
.
,
,
det(A ; E)
(A ; E) = o:
.
,
-
n
.
,
|
,
,
,
.
-
3.
2 x 3 . IV
,
4.
.
)
n-
-
S
6.
.
,
.
.
,
,
.
-
.
,
.
,
,
-
1 0 03
0 03 1 :
2
; 2 + 0 9991 = 0
0
1
=
0
2
= 1:
2;2
+1=0
-
k1 1kT
2.
231
k1 ; 1kT
1
3.
,
= 1 03
2
E E
A:
(A(x) A(y)) = (x y)
E:
x y
.
,
\
,
,
,
.
-
E !E ,
(7)
"
.
-
,
.
5.
,
.
A|
,
-
.
,
.
-
.
E E
E !E
A:
A(x) 2 E
(9) x 1
,
.
.
,
-
= 0 97:
,
.
A
,
,
(7)
,
x2E
.
-
.
-
.
,
.
,
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