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 ++++=+++ )()()()( 22112