.
(-119)
5: &..
2009
&
: 1 2, , , kv v vG G G
m\ 1 2, , , k \ 0 , :
1 1 2 2 0k kv v v + + + =
GG G G" ( ): 1 2, , , kv v v
G G G m\
.
{ } { 0i .. 1 1 2 2 0i i k kv v v v + + + + =
GG G G G" " } 1 21 2 ki ki i i
v v v v = G G G G"
(. ivG ).
: 1 2, , , kv v vG G G m\ . ,
: 1 1 2 2 0k kv v v + + + =GG G G" 1 2 0k = = = =" .
. 0i .. 1 1 2 2 0k kv v v + + + =
GG G G" : 1 1 2 2 0k kv v v + + + =
GG G G" :
[ ] 121 2
| | | 0kk
v v v
=
GG G G" #
0Ax = GG . , 1 2, , , kv v v
G G G , [ ]1 2| | | kA v v v= G G G" 0Ax = GG 0x = GG . : ) ( )r A k= ( 0Ax = GG
0x = GG ) ) ( )r A k< ( 0Ax = GG
)
: 4\ .
1 (1, 2,0,1)v = G , 2 ( 2,0,3, 1)v = G , 3 ( 1, 2,3,0)v = G
1 2 31 3
1 1 2 2 3 3 1 2 32 31 2
21 2 1 0 02 22 0 2 0 00 0 3 3 0 03 3
1 1 0 0 0v v v
+ + = + + = = +
GG G G
N N
123
0
01 2 12 0 2 0
0 3 3 01 1 0 0
xA
= G G
2 ( 1)1 2 1
2 0 20 3 31 1 0
A =
(+) (+)
3/ 4 1/ 41 2 10 4 40 3 30 1 1
(+)
(+)
1 2 10 4 40 0 00 0 0
U =
( ) { - }=2r A U= , . ( ) 3r A k< = , 0Ax = GG , . 1 2 3( , , ) (0,0,0) ..
1 1 2 2 3 3 0v v v + + =GG G G . , 1 2 3, ,v v vG G G .
: ,
1 1 2 2 3 3 0v v v + + =GG G G , .
1 2 3, ,v v vG G G .
: 4\ .
1 (1,7,6,3)v =G , 2 (2, 1,5,4)v = G , 3 ( 3, 3,0, 1)v = G , 4 (0,4,2,1)v =G
1 2 3 4, , ,v v v v
G G G G :
1 2 3 07 1 3 46 5 0 23 4 1 1
A
=
, ( )r A U, . , . ( ): det 56A = , det 0A 0Ax = GG 0x = GG . , 1 2 3 4, , ,v v v vG G G G .
1) 1 0v
GG , , 1 1 0v =GG
1 0 = .
.. 1
1 02 0
0 01 0
=
1 0 = .
2) , 0
G , ,
10 0 =G G
1 0 . 3) , 1 2, , , kv v v
G G G 0G . .. 1,0, , kv v
GG G , : 10 10 0 0kv v+ + + =G GG G" , 2 1 0 =
4) 1 2, , , kv v v
G G G 1 2 1, , , , , ,k kv v v v v+G G G G G . (, , ).
5)
.
.. : 1 (1,7,6,3)v =G , 2 (2, 1,5,4)v = G , 3 ( 3, 3,0, 1)v = G , 4 (0,4,2,1)v =G
, : 1 2 3, ,v v vG G G .
1 2 4, ,v v vG G G .
1 3 4, ,v v vG G G .
2 3 4, ,v v vG G G .
1 2,v vG G .
1 3,v vG G , ...
6) k m> 1 2, , , mkv v v G G G \ .
: [ ]1 2| | | kA v v v= G G G" m k , ( ) min( , )r A m k m k = < , 0Ax = GG .
: 3 2\ 4 3\ 5 4\
... 7) 1 2,
mv v G G \ , \ .. 2 1v v=G G . .. 1 ( 2,1,0, 3)v = G , 2 ( 4,2,0, 6)v = G , ,
2 12v v=G G , 1 ( 2,1,0, 3)v = G , 2 ( 4,3,0, 6)v = G ,
\ .. 2 1v v=G G . & 2\
2\ (0,0) (x,y) , 7, 2\ .. 1 ( 2,1)v = G 2 (4, 2)v = G , 2 12v v= G G
-2 -1 1 2 3 4
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
1vG
2vG
, 2 - 2\ . .. 1 ( 2,1)v = G 2 (4,1)v =G , \ .. 2 1v v=G G .
-2 -1 1 2 3 4
-0.5
0
0.5
1
1.5
1vG
2vG
, 6, 3 2\ .
, 3 21 2 3, ,v v v G G G \ , (.. 3 1 1 2 2v v v = +G G G ) 1 2 3, ,v v vG G G .
: 1 ( 2,1)v = G , 2 (4,1)v =G , 3 32, 2v =
G . 1 2, \ 3 1 1 2 2v v v = +G G G . , .
13 1 1 2 2 1 2
2
2 2 4 2 2 43/ 2 1 1 3/ 2 1 1
v v v
= + = + = G G G
. : 12
2 4 21 1 3/ 2
= . :
(+)
2 4| 20 3| 5/ 2
2 4| 2 1/ 2|
1 1| 3/ 2A b
= G
:
2 : 2 25 532 6
= = 1 : 1 2 1 1
5 22 4 2 2 4 26 3
+ = + = = : 3 1 2
2 53 6
v v v= +G G G
-2 -1 1 2 3 4
-0.5
0
0.5
1
1.5
1vG
2vG
3vG
256
vG1
23
vG
& 3\ 3\ (0,0,0) (x,y,z) , 7, 3\ .. 1 ( 1,3, 5)v = G 2 (3, 9,15)v = G , 2 13v v= G G 0 (0,0,0)=G , - 3\ . .. 1 ( 1,3, 5)v = G 2 (3,6,15)v =G , \ .. 2 1v v=G G . : 3\ . , 0 (0,0,0)=G .
, 6, 4 3\ .
, 4 31 2 3 4, , ,v v v v G G G G \ , (..
4 1 1 2 2 3 3v v v v = + +G G G G ) 1 2 3 4, , ,v v v vG G G G . : 1 2, , , kv v v
G G G m\ . , : { }1 1 2 2 1 2.. , , ,m kV v v v v v = = + + + G G G G G\ " \ m\ .
: V m\ . 1 2, , , kv v vG G G m\ V : 1) 1 2, , , kv v v VG G G 2) ,v V G 1 2, , , \ , .. 1 1 2 2 kv v v v = + + +G G G G" (.
V 1 2, , , kv v vG G G )
: 1 2, , , kV v v v= G G G 1 2, , , kv v vG G G V
& : V V . , { }1 2, , , kv v v= G G GB V : 1) 1 2, , , kv v v VG G G 2) 1 2, , , kv v v
G G G 3) ,v V G 1 2, , , k \ , .. 1 1 2 2 k kv v v v = + + +G G G G" (.
V 1 2, , , kv v vG G G )
.: , k- ( )1 2, , , k ,v VG 1 2, , , k vG B .
: { }1 2 3, ,e e e= G G GB 1 (1,0,0)e =G , 2 (0,1,0)e =G ,
3 (0,0,1)e =G 3\ . (1, 2,5)v = G B .
1) 1 2 3, ,e e e
G G G 3 . : 31 2 3, ,e e e G G G \ 2) 1 2 3, ,e e e
G G G : 1 0 00 1 00 0 1
A I = =
det 1A = , det 0A 0Ax = GG 0x = GG . , 1 2 3, ,e e eG G G .
3) 3( , , )v x y z= G \ :
0 0 1 0 00 0 0 1 00 0 0 0 1
x xy y x y zz z
= + + = + + : 1 2 3v xe ye ze= + +G G G G
, (1, 2,5)v = G 1 2 31 2 5v e e e= +G G G G , vG { }1 2 3, ,e e e= G G GB 1, 2,5 : . , vG { }1 2 3, ,e e e= G G GB 1, 2,5 , vG { }2 1 3, ,e e e= G G GB 2,1,5 : { }1 2 3, ,v v v= G G GB 1 (0,1,2)v =G , 2 (2,2,6)v =G , 3 ( 1, 2, 8)= vG 3\ . (1, 2,5)v = G B .
3 1 2 3( , , ) .. 1 1 2 2 3 3v v v v = + +G G G G . :
11 1 2 2 3 3 1 2 3 2
3
1 0 2 1 1 0 2 12 1 2 2 2 1 2 2
5 2 6 8 5 2 6 8
= + + = + + =
v v v vG G G G
, 123
10 2 11 2 2 22 6 8 5
=
.
Gauss: ( 2)1 2 2 | 2
0 2 1 | 12 6 8 | 5
(+)
0 2 1 | 1| 1 2 2 | 2
2 6 8 | 5A b
= G
( 1)1 2 2 | 20 2 1 | 10 2 4 | 9
(+)
1 2 2 | 20 2 1 | 1 |0 0 3 | 8
U d =
G
:
3 : 3 383 83
= = 2 : 2 3 2 2
8 52 1 2 13 6
= + = = 1 : 1 2 3 1 1
5 8 172 2 2 2 2 26 3 3
+ = + = =
, 1 2 317 5 83 6 3
v v v v= G G G G , (1, 2,5)v = G
{ }1 2 3, ,v v v= G G GB 17 5 8, ,3 6 3
: . , V . ( { }0G 0) : V . : V V dim( )V . : dim( )V = [ V ] .. dim( )V =3, : 1 ( - ) 2 3 : 4 5 6 ... : dim( )m m=\ : 2dim( ) 2=\ , 3dim( ) 3=\ , 4dim( ) 4=\ , ... : V 2\ (0,0) 3\ (0,0,0), : dim( ) 1V = .. { }2( , ) \ 2 0L x y x y= =\ dim( ) 1L = { }3( , , ) \ 2 0 & 2 0W x y z x y z x y z= + = + =\ dim( ) 1W = : V 3\ (0,0,0), : dim( ) 2V = .. { }3( , , ) \ 2 0E x y z x y z= + =\ dim( ) 2E = : , { }1 2 1 1 2 2( , , , ) \ 0nn n nV x x x x x x = + + + = \ " ( 1 2, , , n 0 ) dim( ) 1V n= .. { }41 2 3 4 1 2 3 4( , , , ) \ 2 5 3 0W x x x x x x x x= + + =\ dim( ) 3W = : { }0W = G n\ 0
G. , dim 0W =
: dim( )V k= , k V . .. -- 2 2\ (. 2 -
2\ ) . -- 3 3\ (. 3 -
3\ ) . -- 4 4\ . -- 5 5\ . ... , -- - 2\ (0,0)
3\ (0,0,0), . -- 2 3\
(0,0,0) . : V dim( )V k= , { }1 2, , , kv v v= G G GB ( k ) V , : 1) 1 2, , , kv v v VG G G 2) 1 2, , , kv v v
G G G : W V, : ) dim dimW V ) dim dimW V= , W V=
.. W 3V =\ : 1) 3 dim 3W W= =\ 2) W 0 (0,0,0)=G dim 2 dimW V= < 3) W 0 (0,0,0)=G dim 1 dimW V= < 4) { }0W = G dim 0 dimW V= <
: n n A , n\ : { n nA \ } {det 0A } { 0Ax = GG 0x = GG } { n A }. , n n\ . , A n\ . : { }1 2 3, ,v v v= G G GB 1 (0,1,2)v =G , 2 (2,2,6)v =G ,
3 ( 1, 4, 8)v = G 3\ .
: 0 2 11 2 42 6 8
A =
:
2 1 2 1det 2 ( 16 6) 2( 8 2) 2
6 8 2 4 = + = + + + = A
. det 0A { }1 2 3, ,v v v= G G GB 3\ : det 1 0nI = . , nI n\ : n\ nI .
.. 2\ { }1 0,0 1 3\
1 0 00 , 1 , 00 0 1
4\ 1 0 0 00 1 0 0, , ,0 0 1 00 0 0 1
... : { }3( , , ) \ 2 0V x y z x y z= + =\ 3\ . ) ; ) . ) ( 1,1,3)b = G V . , .
) V 0 (0,0,0)=G . ) ( , , )v x y z V= G , ,x y z :
2 0 2x y z x y z + = = . v VG : 2 y zv y
z
= G , ,y z\ .
. 2 11 00 1
v y z = +
G , ,y z\ (*)
. V
1210
v =
G 21
01
v =
G , V
2 0x y z + = , 1 2,V v v= G G (. V 1 2,v vG G ) 1 2,v v
G G . :
( 1/ 2)2 11 00 1
A =
(+) 2
2 10 1/ 20 1
(+)
2 10 1/ 20 0
U =
, ( ) 2r A = ( )r A n= 1 2,v vG G . , : 1) 1 2,v v VG G , 2) 1 2,v v
G G , 3) V 1 2,v v
G G , { }1 2,v vG G V . , V 2 . , dim( ) 2V = : 2 0x y z + = 1 3
[ ]1 2 1A U= = x ,y z . 0Ax = GG (*). , - V ( ).
) ( 1,1,3)b = G V 1 2 1 3 0 + = ,
2 0x y z + = . b VG { }1 2,v vG G V ,
2 1 2( , ) .. 1 1 2 2b v v = +G G G . :
11 1 2 2 1 2
2
1 2 1 1 2 11 1 0 1 1 0
0 13 0 1 3b v v
= + = + = G G G
, 12
12 11 0 10 1 3
=
.
,
A U , Gauss
| |A b U d G G
. bG
A U . :
( 1/ 2)113
b =
G (+)
21
3/ 23
(+)
13/ 2
0d
= G
: 1 12 2
12 1 2 1 11 0 1 0 1/ 2 3/ 20 1 0 0 03
= =
: 3 : 1 20 0 0 + = , 21 2( , ) \ 2 : 2 2
1 3 32 2 = =
1 : 1 2 1 12 1 2 3 1 1 = = =
, 1 23b v v= +G G G , ( 1,1,3)b = G
{ }1 2,v vG G 1,3. : 3dim( ) 3=\ , ( 1,1,3)b = G 3\ , 3 3\ . , 3\ 1,1,3 . , bG { }3( , , ) \ 2 0V x y z x y z= + =\ 3\ dim( ) 2V = , bG 2 V .
: V { }1 2, ,..., kv v vG G G V. u VG 1 1 ... ...i i k ku v v v = + + + +G G G G 0i . { }1 1 1,..., , , ,...,i i kv v u v v +G G G G G V. :
{ }3( , , ) \ 2 0V x y z x y z= + =\ { }1 2,v vG G , 1 210
v =
G 21
01
v =
G .
V 1vG
uG : 1 1 2 2u v v = +G G G , 1 0 ,
.. 1 2
2 1 72 3 2 1 3 0 2
0 1 3u v v
= = = G G G , V { }2,u vG G
, 2v
G wG : 1 1 2 2u v v = +G G G , 2 0 .
.. 1 2
2 1 11 0 10 1 1
w v v = + = + =
G G G , V { }1,v wG G . , V , . : { }3( , , ) \ 2 0V x y z x y z= + =\
{ }3( , , ) \ 0W x y z x y= =\ 3\ . ) V W . 3\ ; ) V W ) V W ;
) { }3( , , ) \ 2 0 & 0V W x y z x y z x y = + = =\ .
. V W 3\ , V W 3\ .
) ( , , )v x y z V W= G , ,x y z :
}N N0
1 2 12 0 00 01 1 0
A x
xx y z yx y z
+ = = = GG
: ( 1)1 2 1
1 1 0A = (+)
1 2 10 1 1 U
=
, 1 2 1 00 0 00 1 1
xAx Ux y
z
= = = G GG G
: ,x y : z
: 2 : 0y z y z = = 1 : 2 0 2 0x y z x z z x z + = + = =
. v V W G : zv zz
= G , z\ .
. 111
v z =
G , z\
, V W 111
.
, V W 1 . , dim( ) 1V W =
) V W 111
v z =
G z\ . , V W
(1,1,1) 3\ [. (0,0,0) (1,1,1)] 1: U, - .
..
0 3 1 4 7 60 0 2 1 4 50 0 0 0 2 30 0 0 0 0 0
U
= , 1 2 3
0 0 03 0 01 2 0, ,4 1 07 4 26 5 3
r r r
= = =
G G G
.
: 1 2 3, ,r r rG G G
0 0 03 0 01 2 04 1 07 4 26 5 3
A
=
.
. 0 3 1 4 7 60 0 2 1 4 50 0 0 0 2 3
TA =
3 - . ,
( ) 3Tr A = . , A , ( ) ( )Tr A r A= . , ( ) 3r A = ( ) 1 2 3, ,r r r
G G G . 2: U,
..
0 3 1 4 7 60 0 2 1 4 50 0 0 0 2 30 0 0 0 0 0
U
= , 1 2 3
3 1 70 2 4, ,0 0 20 0 0
v v v = = =
G G G
.
: 1 2 3, ,v v vG G G
3 1 70 2 40 0 20 0 0
A
= ,
. , ( ) 3r A = ( ) 1 2 3, ,v v v
G G G .
m nA \ . n A m\ . m\ A , ( )AR . . 1 2, , , nv v v
G G G A , 1 2( ) , , ,= nA v v vG G GR . ( )AR mbG \ 1 2, , , nv v v
G G G .
.. 1 3 3 22 6 9 51 3 3 0
= A
1 3 3 2( ) 2 , 6 , 9 , 5
1 3 3 0
= AR
: b
G ( )AR ,
1 2, , , n \ .. 1 1 2 2 = + + + n nb v v vG G G G" :
[ ]N
1 12 2
1 2
| | |
= = = n
n nx
b v v v b A Ax b
G
G G GG G G G" # #
, ( )b AG R { =Ax bGG ( )} : ( 1,0,4)= bG
1 3 3 22 6 9 51 3 3 0
= A .
( 1,0,4)= bG ( )AR , 1 2 3 4, , , \
.. N N
12
1 2 3 434
1 1 3 3 2 1 1 3 3 20 2 6 9 5 0 2 6 9 54 1 3 3 0 4 1 3 3 0
= + + + = Ab x
G G
, =Ax bGG ( ). :
( 2) 11 3 3 2 | 1| 2 6 9 5 | 0
| 41 3 3 0
= A bG
(+) (+)
( 2)1 3 3 2| 10 0 3 1| 2
| 30 0 6 2
(+)
1 3 3 2| 10 0 3 1| 2 |
| 10 0 0 0
= U dG
= =Ax b Ux dG GG G . 3 0 1= =Ax bGG . , b
G ( )AR .
: ( )r A m= (. U ) =Ax bGG ( m n= m n< )
mbG \ . , : ( ) mA \R U
( )UR . : dim ( ) ( )U r U=R
A U :
{ A } { U } :
A U A
U ( )AR . : dim ( ) ( )A r A=R , ( ) ( )A UR R
: ( ) ( )A UR R , 1 3 3 22 6 9 51 3 3 0
= A
1 3 3 20 0 3 10 0 0 0
U =
.
: { ( )UR } = {1 & 3 U } = 1 30 , 30 0
, ( )UR :
1 2
1 30 30 0
v = + G . : 1 22
330
v + =
G .
, .. 3 A , 393
.
, 39 ( )3
U
R ( ) ( )A UR R . : V 1 (1, 2,4)v = G , 2 (3,0, 1)v = G , 3 (0,6, 13)v = G , 4 (1,4, 9)v = G . V ;
: 1 2 3 4, , ,V v v v v= G G G G V 3\ A 1 2 3 4, , ,v v v v
G G G G . : 1 3 0 12 0 6 4
4 1 13 9A
= .
: ( )V AR , :
(13/ 6)1 3 0 10 6 6 60 13 13 13
2 41 3 0 1
2 0 6 44 1 13 9
A =
(+) (+) (+)
1 3 0 10 6 6 60 0 0 0
U =
, ( ) 2r A = (= - U ) , dim ( ) 2 dim 2A V= =R { V } = { ( )AR } = { }1 2,v vG G (. A U ) , V 3\ 10,v
G G 2vG , : (0,0,0) , (1, 2,4) (3,0, 1) . , : 3\ (0,0,0) :
0x y z + + = , , 0 . , (1, 2,4) (3,0, 1) :
} { }1 ( 2) 4 0 2 4 0 1 2 4 03 0 13 0 ( 1) 0 3 0 0 + + = + = = + + = = ,
( 3)1 2 43 0 1
(+) 1 2 40 6 13
: , :
:
2 : 136 13 06
= = 1 : 13 12 4 0 2 4 0
6 3 + = + = =
, : 1 13 1 130 0 2 13 6 03 6 3 6
x y z x y z x y z + + = + + = + + =
, { }3( , , ) \ 2 13 6 0V x y z x y z= + + =\
m nA \ . m A n\ . n\ A . ( )TAR , TA
.. 1 3 3 22 6 9 51 3 3 0
= A
1 2 13 6 3( ) , ,3 9 32 5 0
TA =
R
. ( )TAR 4bG \
1 2 13 6 3, ,3 9 32 5 0
, . 1 2 3
1 2 13 6 33 9 32 5 0
b = + +
G
: ( )b AG R { Ax b= GG ( )}. : ( )Tb AG R { TA x b= GG ( )} U , -
( )TUR . : dim ( ) ( )TU r U=R
A U : ( ) ( )T TA UR R
( A U , )
:
A U - U ( )TAR . : dim ( ) ( )TA r A=R
.. 1 3 3 22 6 9 51 3 3 0
= A .
1 3 3 20 0 3 10 0 0 0
U =
.
: dim ( ) 2TA =R (= - U )
{ ( )TAR } = { - U } = 1 03 0,3 32 1
m nA \ . 0Ax = GG n\ ( )AN A . : ( )A N , 0Ax = GG 0 nx = GG \ . , ( ) nA \N xG 0Ax = GG n\ . , 1x
G 2xG 0Ax =GG . , 1 2, ( )x x AG G N .
: 1 2 1 2( ) 0 0 0A x x Ax Ax+ = + = + =G G GG G G G . , 1 2( ) ( )x x A+ G G N . ( )AN
. , 1 1( ) ( ) 0 0A x Ax = = =
G GG G . , 1( ) ( )x A G N , 1 ( )x A G N & \ . ( )AN . , ( )AN n\
A U 0 0Ax Ux= =G GG G . : ( ) ( )A UN N
dim ( ) ( )A n r A= N = ( ) ( )r A n= (. U , . )
0Ax = GG 0x = GG . : { }( ) 0A = GN (, n\ ) dim ( ) 0A =N ( )AN
:
1 3 3 22 6 9 51 3 3 0
= A .
1 3 3 20 0 3 10 0 0 0
U =
.
: dim ( ) ( ) dim ( ) 4 2 dim ( ) 2A n r A A A= = =N N N ( )AN 0Ax = GG ,
01 3 3 2
0 0 0 3 1 00 0 0 0 0
xyUx zw
= =
GG .
: ,x z : ,y w : 3 : 4( , , , )x y z w \ 2 : 13 0
3z w z w+ = =
1 : 13 3 2 0 3 3 2 0 33
x y z w x y w w x y w + + + = + + + = =
, ( )AN :
3
13
y wy
x w
w
=
G , :
3 3 11 01 0 1/ 3
3 0 1
y wy
x y ww
w
= = +
G , ,y w\
, ( )AN 3 1
1 0,0 1/ 30 1
.
: - Ax b= GG , 1x
G 2xG , : 1 2 1 2( ) 2A x x Ax Ax b b b b+ = + = + =
G G G GG G G G , . 1 2( )x x+G G Ax b=GG .
, Ax b= GG . , .
: m nA \ . TA , ( )TAN A . : dim ( ) ( )A n r A= N dim ( ) ( )TA m r A= N [ TA m ( ) ( )Tr A r A= ] :
1 3 3 22 6 9 51 3 3 0
= A . ;
, ( ) 2r A = . , ( )TAN : dim ( ) ( ) dim ( ) 3 2 dim ( ) 1T T TA m r A A A= = =N N N ( )TAN 0TA x = GG . :
( 3) ( 3) ( 2)1 2 13 6 33 9 32 5 0
TA =
(+) (+)
(+)
1 2 10 0 00 3 60 1 2
( 3)1 2 10 1 20 3 60 0 0
(+)
1 2 10 1 20 0 00 0 0
U =
U TA , U A .
01 2 10 1 2 00 0 0 0 0 00 0 0 0
Tx
A x U x yz
= = =
G GG G
: ,x y : z : 4 & 3 : 3( , , )x y z \ 2 : 2 0 2y z y z+ = = 1 : 2 0 2( 2 ) 0 5x y z x z z x z+ = + = =
, ( )TAN : 52z
x zz
= G , :
52
1x z
= G , z\ . , ( )TAN
52
1
.
( )TAN 3\ (0,0,0) (5, 2,1) . : m nA \ .
( ), ( ), ( )TA A AR R N ( )TAN : 1)
A U
2) ( )r A = ( - U ) 3) ( ), ( ), ( )TA A AR R N ( )TAN : dim ( ) ( )A r A=R , dim ( ) ( )TA r A=R , dim ( ) ( )A n r A= N , dim ( ) ( )TA m r A= N 4) { ( )AR } = { A U
} 5) { ( )TAR } = { - U } 6) dim ( ) 0A =N , { }( ) 0A = GN ( )AN . dim ( ) 0A >N , 0Ux = GG
( U ) . n\ . ( )AN .
7) dim ( ) 0TA =N , { }( ) 0TA = GN ( )TAN . dim ( ) 0TA >N ,
TA U , 0U x = GG ( U ) . m\ . ( )TAN .
: ( )AR ( )AN A . : V
1 2, , , kv v vG G G A=[ 1 2, , , kv v vG G G ]
, ( )V AR . ( )1 2, , , nx x x V
1 1 2 2 0n nx x x + + + =" , A=[ ] ,
( )V AN .
.. 1 2 3 4, , ,V v v v v= G G G G 1 (1, 2,4)v = G , 2 (3,0, 1)v = G , 3 (0,6, 13)v = G , 4 (1,4, 9)v = G , ( )V AR , 1 3 0 12 0 6 4
4 1 13 9A
= 1 2 3 4, , ,v v v v
G G G G .
.. { }3( , , ) \ 2 6 0V x y z x y z= + =\ V ( , , )x y z 2 6 0x y z + = [ ]2 1 6 [0]xy
z
= . , ( )V A N [ ]2 1 6A =
.. { }3( , , ) \ 2V x y z x y z= = = \ V { } { }2 2 0 1 2 0 00 1 0 1 0xx y x y yx z x z z = = = = + = . ,
( )V A N 1 2 01 0 1A = .
1: m nA \ . ( )AR ( )TAN m\ . , 3 : ( ) mA \R & ( )T mA \N dim ( ) dim ( ) dimT mA A+ = \R N { }( ) ( ) 0TA A = GR N 2: m nA \ . ( )TAR ( )AN n\ . , 3 : ( )T nA \R & ( ) nA \N dim ( ) dim ( ) dimT nA A+ = \R N { }( ) ( ) 0TA A = GR N : ( ) mA \R dim ( ) 0TA =N [. ( )r A m= ] ( ): { }3( , , ) \ 2 0V x y z x y= + =\ { }3( , , ) \ 0W x y z x y z= + =\ 3\ . ) V W ; ) V W .
; ) 3\ V W .
) 3 ( , , )x y z
0x y z + + = . , V W 0 (0,0,0)=G .
, dim 2V = dim 2W = ) { }3( , , ) \ 2 0 & 0V W x y z x y x y z = + = + =\ V W :
}N N0
2 0 1 2 0 00 1 1 1 0
+ = = + = A x
xx y yx y z z GG
, ( ) V W AN , 1 2 01 1 1 = A :
( 1)1 2 01 1 1
= A (+) 1 2 00 3 1
= U , 0 0= =Ax UxG GG G : ,x y : z :
2 : 13 03
+ = =y z y z
1 : 1 22 0 2 03 3
+ = + = = x y x z x z
, V W :
23
13
=
z
x z
z
G , :
2 /31/31
= x zG , z\
, V W 2 /3
1/31
.
1 , dim( ) 1 =V W V W 3\ (0,0,0) ( 2 /3,1/3,1) . , V W .
) ( ) V W AN , V W A , ( )TAR . ( )TAR - U ,
{ ( )TAR }= 1 02 , 30 1
COVER5Binder1Dianysm_xwroi_p1Dianysm_xwroi_p2Dianysm_xwroi_p3Dianysm_xwroi_p4Dianysm_xwroi_p5Dianysm_xwroi_p6
Aneksarthsia_Baseis_klp