# Линейные операторы: Учебное пособие

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• ..

.

2004.

• 1. 1. .

2. .

3. .

4. . .

5. .

6. .

7. .

2.

1. .

2. .

3. .

4. .

5. () .

6. -.

7. .

8. .

9. .

10. .

3.

1.

2. .

3. .

4. .

5. .

• 1. 1.

-

, -

. -

. -

,

() .

1.1. V (-

) k, ,

) V V V V,

: (v1, v2) v1+ v2, -

:

Ia) v1+ v2= v2+ v1 ()

IIa) (v1+ v2)+v3=v1+ (v2+v3) ()

IIIa) 0, -

, v+0=v, v V;

IV a) v V v V ,

v+v=0. v

- v.

, V -

.

) kV (,v) v V,

,

:

I) (v1+v2)= v1+ v2 II) (+)v=v+v

,, k, v,v1,v2 V; ( )

• III) () v =( v) (), , k, v V;

IV) 1v=v ()

k

C, R.

. , -

-

.

.

1. V= },...,1,),,...,,{( 21 nikin =

n k.

Vn = ),...,,( 21 Vn = ),...,,( 21

Vn = ),...,,( 21 : iii += ni ,...,1= .

, .. ),...,,( 21 n = .

I, II, I-IV -

k, .. . -

(0,0,,0), n -

. ,

= (1, 2,,-n), , -

. -

.

n, kn.

, n-

n.

• 2. R(a,b)

Rba ),( R.

, .. (f+g)(x)= f(x) +g(x), , )())(( xfxf = ,

),(, baRgf , R, x (a,b).

R(a,b)

R.

2.

-

.

2.1. v1,v2,,vm V -

,

mikim ,...,1,),,...,,( 21 = ,

0...2211 =+++ mmvvv (2.1)

( 0 V.)

, m=1 , v, -

, v=0. -

, v=0, , 11 = , 01 = v . , 0= v

0 , , 1 , v=0.

-

.

2.2. v1,v2,,vm, m2,

V , , -

, ..

mmjjjjj vvvvv +++++= ++ ...... 111111 (2.2)

j mjj ,...,,,..., 111 + .

• m>1. -

2.1. .. ),...,( 1 m , -

j , 0j . (2.1) j

, j,

, :

mj

mj

j

jj

j

j

jj vvvvv

= ++

...... 1

11

11

1 . ..

(2.2).

(2.2), , jv

, :

0...... 111111 =+++++ ++ mmjjjjj vvvvv . -

, 2.1 , ..

),...,,1,,...,( 111 mjj + .

.

2.1. -

, .

. , , s

, ..

0...2211 =+++ ss vvv ,

i .

:

00...0... 12211 =++++++ + msss vvvvv .

-

v1,,vs ,,vm.

, 0v=0, -

,

0v=(0+0)v=0v+ 0v.

• 2.2.

.

.

( ).

2.3. , ,

.

2.1, ,

.

2.4. v1,v2,,vk ,

v1,v2,,vk,v , v -

v1,v2,,vk.

.

vvv k ,,...,1 )0,...,0,...,0(),,...,( 1 k ,

0...2211 =++++ vvvv kk . 0 , .

0= , kvv ,...,1 -

, 0...1 === k .

),,...,( 1 k .

3.

-

. , -

.

( ) 3.1. },,{ 1 nuuU K=

},,{ 1 mwwW K= , mn .

.

m. m=1 1122111 ,...,, wuwuwu nn === . j

, .. -

• ,

U, 2.3. , 1=> mn .

01211122112 == wwuu , .. },{ 21 uu -

. U

(. 2.2.).

, W m-1

m .

mnmnn

mm

wwu

wwu

++=

++=

..................................

...

11

11111

niim ,...,1,0 == . -

mmn

• , }',...'{' 11 = nuuU -

},...,{ 11^

= mwwW .

11 mn , .. mn . .

3.2. },...,{ 1 nuuU = },...,{ 1 mwwW = -

V.

n=m.

: W

},...,,{ 1 ms wwu . ,

su U

W. mn .

U W, nm .

, n=m.

.

3.3. -

, -

. ,

.

.

3.2 , -

. -

.

3.4. -

V -

Vkdim ( k ,

).

:

• 1. nk n n. -

, )0,...,1,...0(=ie , i=1,n nk .

=

==n

iiin ev

11 ),...( . 0

1

==

n

iiie , 0),...( 1 =n , ..

0...21 ==== n . , nee ,...,1

.

2. x

},,)({][0=

==n

ii

ii Nnkaxaxfxk

, n

nxx,...,,1 .

4. .

4.1. V n-

nee ,..,1 .

1. V

=

=n

iiiev

1

, ki .

2. v1,v2,,vm,

m

• .. nee ,..,1 , 0...11 === nn ,

.. nn == ,...,11 .

.

nm eevv ,..,,,.., 11 , ,

. siim

eevv ,..,,,..,11

mvv ,..,1 , . , -

, ..

0......112211

=++++++ss iiiimm

eevvv .

ji

-

, , ji

e -

. , siim

eevv ,..,,,..,11

-

.

, V -

nee ,..,1 , -

nm eevv ,..,,,.., 11 . ,

, siim

eevv ,..,,,..,11

.

siim

eevv ,..,,,..,11

V.

.

kn ,..,1 , Vv

: nneev ++= ...11 , v

nee ,..,1 . nneeu ++= ...11 V,

nnn eeuv )(...)( 111 ++++=+ , nn eev )(...)( 11 ++= , k .

, n

, 3- ,

: -

,

• , -

.

n n .

, , -

.

4.2. V U

k . V U -

(.. ) V U ,

:

)()()( 22112211 vvvv +=+ , kVvv 2121 ,,, .

V n , nee ,..,1

V,

V nk : ),...,,()( 21 nv = , n ,...,, 21 -

v nee ,..,1 . , -

V nk . -

, -

:

)()()( 2121 vvvv +=+ , )()( vv = .

, -

-

. ,

.

4.3. k

, .

• . V W

V W .

nvv ,..,1 - V, )(),..,( 1 nvv - W . -

, Ww Vv , ..

==

===n

iii

n

iii vvvw

11

)()()( .

0)(1

==

n

iii v , 0)(

1

==

n

iiiv . ,

: 0)0( = . .. , 01

==

n

iiiv . -

nvv ,..,1 , 0...1 === n . , -

, )(),..,( 1 nvv

W , , VnW kk dimdim == .

, nVW kk == dimdim . nvv ,..,1 ; nww ,..,1

V W . :

niwv ii ,...,1,)( == . .. nvv ,..,1 V, -

V : ==

=n

iii

n

iii wv

11

)( . -

V -

W , .. nww ,..,1 . -

, .. V W .

5.

V n - , nee ,..,1 nee ',..,'1

. ()

nee ',..,'1 ( -

) :

• nnnnnn

nn

etetete

etetete

+++=

+++=

...'......................

...'

2211

12211111

:

=

nnnn

n

n

nn

ttt

tttttt

eeeeee

K

MM

K

K

21

22221

11211

2121 ),...,,()',...,','(

T -

. ,

-

( )

. T - . T

n ,...,1 , 0...11 =++ nnee . -

nee ,...,1 .

)',...,'( 1 nee ),...,( 1 nee ,

n .

.

==

==n

iii

n

iii exexx

11

'' - x

. ie'

ie :

j

n

j

n

ijii

n

i

n

jjjii

n

iii etxetxexx

= == ==

===

1 11 11'''' .

=

=n

iiiexx

1

-

, :

=

=n

ijiij txx

1

' , nj ,...,1= .

• -

:

=

nnnnn

n

n

n x

xx

ttt

tttttt

x

xx

'

''

2

1

21

22221

11211

2

1

M

K

MM

K

K

M,

XTX 1' = , X , 'X x

, [ ]ijtT = . . V 3- , 321 ,, eee

. x : 321 2 eeex += .

: 3211' eeee += ,

3212 32' eeee += , 3213 63' eeee ++= . x -

. .. -

,

X

,

T .

c , -

.

121

631111321

~

231

310410321

~

531

100410321

~

51714

100010021

~

517

20

100010001

, x :

321 '5'17'20 eeex += .

• 6.

U V.

, U V, -

(.. -

1.1)

, V.

, U -

.

() 6.1. U

V . ..,

Uuu 21, k21,

2211 uu + V .

.

-

. mvv ,...,1

V. -

mvv ,...,1 =

m

iiii kv

1

, .

=

=m

iiivv

1

, =

=m

iii vv

1

'' , =

+=+m

iiii vvv

1

)'(' , im

ii vv

=

=1

)( ,

k . , -

.

6.1. . -

>< mvvv ,...,, 21 .

, -

mvvv ,...,, 21 -

>< mvvv ,...,, 21 . -

.

• WV + U W

V wu + , Uu , Ww .

11 wu + , WUwu ++ 22 ,

WUwwuuwuwu ++++=+++ )()()()( 22112211222111 ,

.. Uuu + 2211 , Www + 2211 .

suu ,...,1 U , tww ,...,1 W .

, >=

• Myyxxm ts = ),...,,,...,( 001001

==s

i iiuxx

10 . ( ), -

M UW.

4.3. WUM kk = dimdim -

:

6.2. U W -

V, )(dimdimdimdim WUWUWU kkkk ++= .

UW. , =jm Myyxx jtjjsj ),...,,,...,( )()(1)()(1 ,

j=1, , (s+t)r, (6.1),

M UW,

=

=s

ii

ji

j uxx1

)()( , j=1, , (s+t)r,

UW.

. U: )2,1,1(1 =u , )1,1,0(2 =u . -

W : )1,2,1(1 =w , )1,0,1(2 =w .

,

2121 ,,, wwuu , , 221 ,, wuu

. .. U+W

221 ,, wuu 3)(dim =+WUk .

6.2 , 1)(dim =WUk . -

UW

6.1:

+

=

+

101

121

110

211

2121 yyxx

• , , (1,1,1,0). ,

UV

=+

121

21 uu .

7.

U+W -

, , ,

UW. 0=WU ,

2211 wuwu +=+ 1221 wwuu = . .. Uuu 21 , Www 12 ,

01221 == wwuu .

U+W .

n , -

:

7.1. },{...11 ii

n

i inUuuuUUU ==++= = -

Ui, i=1,,n, i

: 0)......( 1 =++++ nii UUUU) . U

nUUU ...21 ini U1= . (

niini UUUUUUU +++++=++++ + ............ 1111) .)

.

7.2. nUUU ++= ...1 . .., -

Uu

nuuuu +++= ...21 , ii Uu , i=1,,n.

. ini UU 1== . nn uuuuu '...'... 11 ++=++=

u, },...,2,1{ ni , :

= =

n

ijj jjii uuuu 1 )'(' . , ii uu ' ,

• iU , : ni UUU ++++ ......1) . -

, 0' = ii uu , i=1,,n, .. -

u.

, x )......( 1 nii UUUU ++++) . .. x=ui=u1+

+ui-1+ui+1++un. 0...... 111 =+++++ + niii uuuuu . -

0=0++0, -

Ui, i=1,,n.

, : 0...21 ==== nuuu , .. x=0. , -

U .

7.3. nUUU ++= ...1 ...., -

==n

i iUU

1dimdim .

. ini UU 1== . -

n. n=1, , dim U = dim U1. -

dim U 6.2:

dim (U1+ +Un) = dim U1+ dim (U2+ +Un) dim U1 (U2+

+Un).

U1 (U2+ +Un)=0, .. dim U1 (U2+ +Un)=0.

U2+ +Un 7.2. -

==++n

j jnUUU

22dim)...dim( , ..

== =+=n

j jn

j jUUUU

121dimdimdimdim .

, , ==n

j jUU

1 ==

n

j jUU

1dimdim . -

iiki uu ,...,1 Ui, i=1,,n. -

==n

j jUU

1 , },...,,...,,...,{ 1111 1 nnknk uuuu -

U. ,

UUkn

j jn

j jdimdim

11== == . , },...,{ 11 nnkuu -

U. 7.2 =n

j jU

1 .

• .

7.4. U -

V. W V,

WUV = .

. - U: mee ,...,1 . -

V: nm eee ,...,,...,1 .

W >< + nm ee ,...,1 . V=U+W, ,

7.2, 0=WU . .. WUV = .

, -

V.

.

k, -

. WU V -

(u,w), Uu , Ww .

V

)','()','(),( wwuuwuwu ++=+ ; ),(),( wuwu = ; Uuu ', , Www ', , k .

(u,0) V U , U,

(0,w) W , W. -

: (u,0) u,

(0,w) w. , V

U W .

• 2. 1.

-

. -

,

.

-

.

1.1. V W -

k. V W ,

)()()( 22112211 vvvv +=+ , Vvv 21, , k21, .

: .

, W -

, k. -

V -

W Homk(V,W).

.

1) V=W=R2 ,

R2 .

) )()( vv =

• ) )()()( 2121 vvvv +=+

2) V=W=k[x] x, dxd

=

. gdxdf

dxdgf

dxd +=+ )( .

: V W -

}0)(,{ == vVvKer

}),(,{ VvvwWwJm == .

.

1.1. V

k, Homk(V,W), Ker Im

VJmKer kkk dimdimdim =+ .

• . .. VKer ,

• nnvxvxx ++= ...11 V.

= = == =

===m

jj

m

j

n

iijijji

n

i

n

iiii wxawaxvxx

1 1 11 1

)()()( .

.. mmwywyxy ++== ...)( 11

:

=

nm x

xA

y

yMM11

.

,

.

, V W -

A nm , -

.

2.1. >=< nvvV ,...,1 >=< mwwW ,...,1 - -

. -

),( WVHomk

nm - k.

.

1. - 2R .

21, ee . 211 sincos)( eevue +=+= .

• 122 sincos)( eee = . -

cossinsincos .

2. dxd

= -

R[x]. }deg],[)({][ nfxRxfxRn = , -

, n.

][][ xRxRdxd

nn -

][xRn . -

][xRn nxx,...,,1 . 1)( = ii ixx , ni ,...,1= , 0)1( = . -

][xRn

0000000

02000010

KK

MMMMKK

n.

-

, V, V.

• V n- , nee ,...,1 nee ',...,'1

, ),( VVHomk

:

=

=n

jjjii eae

1

=

=n

jjjii eaAe

1

=

=n

jjjii eae

1'''

=

=n

jjjii eaeA

1'''' . ( -

ie (0, , 1, 0)t i- -

)

T nee ,...,1

nee ',...,'1 . ==

==n

iii

n

iii exexx

11

'' Vx

.

AXx

xA

y

yY

nn

=

=

= MM

11, AX

x

xA

y

yY

nn

=

=

= MM

11.

, 2 1 'TXX = , 'TYY = (.. Y 'Y

)(xy = ).

'''' XTATYYAXATX ==== . .. 'X - ,

'TAAT = ATTA 1' = .

A 'A , ,

.

2.9. ,

, .

. 321 ,, eee

:

111122254

.

3211' eeee += , 3212 32' eeee += , 3213 63' eeee ++= .

• AT 1 , 1T

, -

T . , -

A

, T AT 1 .

111122254

631111321

~

365132254

310410321

~

497132254

100410321

~

497153326102217

100010021

~

497153326204435

100010001

.

=

497153326204435

1AT -

T , 321 ',',' eee , -

: 3211 '6'22'29)'( eeee += , 3212 '7'26'34)'( eeee += ,

3213 '6'21'29)'( eeee += .

3.

)(kM n -

k -

n- .

Homk(V, V) Endk(V)

End(V). End(V) ,

k. kVEnd ),(, , :

• )()())(( xxx +=+ , ))(())(( xx = , ))(())(( xx = , Vx .

, -

End(V)

,

)()( = -

+=+ )( , +=+ )( )()()( ==

k . , End(V) -

k k-.

End(V) Mn(k),

.

3.1. , k- A B ,

A B ,

Aaa 21 , , k 21,

)()()( 22112211 aaaa +=+ , )()()( 2121 aaaa = . (1).

3.1. >=< neeV ,...,1 - -

k . A

k- End(V) Mn(k).

. 2.1. A -

k- End(V) Mn(k). -

(1) , n- V -

( ) n . ,

=

=n

iiiexx

1

=

nx

xx M

1

, , xAx =)( ,

= == ==

==

=

n

jji

n

iji

n

ij

n

jjii

n

iii exaeaxexx

1 11 11)( ,

• =

=

nn y

y

x

xAxA MM

11

, =

=n

iijij xay

1

( )( rsaA = ). )()()()())(()( xAxAxxxxA +=+=+=+ ,

Vx , AA ++ . ,

)())(())(()( xAxxxA === , Vx , .. A .

,

=

===

= =

n

jji

n

iji exbxxxA

1 1

))(())(()( == == =

=n

kkkj

n

ji

n

ijij

n

ji

n

iji eaxbexb

11 11 1

)(

( )xAAexba kn

ki

n

i

n

jjikj =

=

= = =1 1 1

, Vx ( ( )rsbA = ), .. AA .

4.

4.1. U V -

( )VEnd ,

UU . U|

U , U

A | .

mee ,...,1 U , ,

nm eee ,...,,...,1 V , , A

:

=

BCA

A u0

,

• ==

mmm

m

u

aa

aaAA

U

.............

...

1

111

| - -

U , BC, - -

k ( )mnm ( ) ( )mnmn .

U -

W , . WUV = WW ,

( )VEnd :

=

w

u

AA

A0

0 ,

=

mmm

m

u

aa

aaA

.............

...

1

111

=

+

+++

nnmn

nmmm

w

aa

aaA

,1,

,11,1

........................

... - -

U W . -

wu = -

wu AAA = , WUV =

.

( ), n -

iU , in

iUV

1== , Vn dim= .

.

4.2. 0, vVv

( )VEndk , k -

vv = .

.

• nee ...1 V .

=

nnn

n

aa

aaA

..............

...

1

111

, in

iiexv

=

=1

- -

v . , -

in

iiexv

=

=1

,

nx

xM1

, ,

kvv = 00 , vvA 0 = ( ) 00 = vEA , E - . -

:

0...00

0...00...0

..............

... 1

0

0

0

1

111

=

nnnn

n

x

x

aa

aaM

( )( )

( ) 0...

0...0...

02211

22022121

12121011

=+++

=+++=+++

nnnnn

nn

nn

xaxaxa

xaxaxaxaxaxa

KKKKKKK