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

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