.

(-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