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    K E K

    E

    E

    E

    K.

    E = L (E, K) E. K

    F K

    L (E, F) K

    E

    = 0 x0 E = (x0) = 0.

    y,

    y=y

    =

    y

    (x0)

    y = (x)

    x=

    y

    x0 E ,

    Im ()

    K

    {0} ,

    1

    K,

    E n B= (ej)1jn E, B

    pj :x =ni=1

    xieixj

    1, 2, , n,

    =ni=1

    ipi: x =ni=1

    xieini=1

    ixi

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    E

    n

    B= (ej)1jn E.

    x

    E

    x=

    nj=1

    xjej E,

    (x) =

    nj=1

    xjej

    =

    nj=1

    xj (ej) =n

    j=1

    jxj

    j = (ej) j 1 n.

    L= (1, 2, , n)

    B E

    x E , (x) =LX= (1, 2, , n)

    x1

    x2

    xn

    =n

    j=1

    jxj.

    (E, , )

    E (x) =, x x E.

    x E, (x) =n

    j=1

    jpj(x) =

    nj=1

    jpj

    (x)

    pj j 1 n, xxj.

    B E

    =n

    j=1

    jpj

    j j = (ej) j

    1

    n.

    E n, E

    E

    n, B = (pi)1in , pi

    B

    pi=ei , i 1 n B

    = (ei )1in B.

    ei(ej) =ij =

    1

    i= j

    0 i=j

    (1 i, jn)

    E

    =

    nj=1 (ej) e

    j .

    E, F

    L (E, F) dim(E)dim(F) .

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    E

    B E (v1, , vp) E V1, , Vp B Kn.

    (V1, , Vp) Kn,

    (V1, , Vp) p

    p n

    (V1, , Vp) K

    n

    (V1, , Vp) p.

    tV1

    tVp

    .

    1, , p R

    n

    xRn

    , i(x) =

    nj=1

    i,jxj (1 i p)

    A=

    L1

    Lp

    =

    11 12 1n21 22 2n

    p1 p2 pn

    p

    Li i B K

    n

    p

    E

    E.

    B = (i)1in E,

    B= (fi)1in

    E B B.

    B0 = (ej)1jn E.

    i

    1

    n

    x=

    nj=1

    xjej E

    i(x) =i1x1+ + inxn

    Q= ((ij))1i,jn i E

    i

    Q

    i B0 F1, , Fn Q1,

    QQ1 =In

    Q (F1, , Fn) = (QF1, , QFn) = (E1, , En)

    (Ei)1in Kn.

    j

    1

    n

    QFj =Ej

    X Kn,

    QX=

    11x1+ + 1nxn

    n1x1+ + nnxn

    = 1(x)

    n(x)

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    QFj =Ej

    1(fj)

    n(fj)

    =Ej =

    1j

    nj

    j

    1 n, fj E Fj B,

    i(fj) =ij i, j 1 n. B B.

    E, F

    E

    u L (E, F)\ {0} . u r

    1, , r E

    y1, , yr

    F

    x E, u (x) =

    ri=1

    i(x) yi.

    ker(u) =

    ri=1

    ker(i) .

    rg (u) =r 1, Im (u) r (yi)1ir

    Im (u) ,

    x E , 1(x) , , r(x)

    u (x) =r

    i=1 i(x) yi. i (i)1ir r,

    r =

    r1i=1

    ii x E

    u (x) =r1i=1

    i(x) (yi+iyr)

    r 1 (yi+ iyr)1ir1 Im (u) ,

    (i)1ir

    x ker (u)

    u (x) =

    ri=1 i(x) yi = 0,

    i(x) (yi)1ir

    E

    1, , r

    E

    y1, , yr F

    x E , u (x) =r

    i=1

    i(x) yi

    E

    n.

    B= (ej)1jn

    E,

    i

    i(x) =i1x1+ + inxn

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    E

    u

    B E B = (yj)1jr Im (u) A= ((i,j))1ir1jn

    rg (u) = rg (A) = rg( tA)

    = rg (1, , r) =r

    (i,j)

    1jn

    i B.

    E

    K

    2,

    E

    u= 0,

    u= I d Id.

    u

    u=

    1

    2u +

    1

    2u

    K

    2

    u

    r

    1

    n 1.

    1, , r E y1, , yr

    F

    x E , u (x) =r

    i=1

    i(x) yi

    (y1, , yr) E, (1, , r) E

    v, w

    xE ,

    v (x) =1

    2

    r

    i=1

    i(x) yi+n

    i=r+1

    i(x) yi

    w (x) =1

    2

    ri=1

    i(x) yin

    i=r+1

    i(x) yi

    E

    n

    u= v + w.

    K

    u Id ,

    v=u Id u= v + w w= Id.

    E

    1, , p,

    E

    pi=1

    ker(i) ker () ,

    i.

    i

    pi=1

    ker(i) =E

    r

    (i)1ip

    (i)1ir

    pi=1

    ker(i) =r

    i=1

    ker(i) .

    (1, , r, )

    E

    (ej)1jn E.

    i(er+1) =ei(er+1) = 0 1 i r

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    er+1r

    i=1

    ker(i) .

    (er+1) =er+1(er+1) = 1

    er+1 /ker()

    pi=1

    ker(i)ker() (1, , r, )

    i.

    E

    E , F , G

    u

    E

    F

    v

    E

    G.

    ker(u) ker(v)

    w

    F

    G

    v= w u.

    Im (u)

    F,

    H

    F

    F = I m (u) H.

    w L (F, G) y = u (x) Im (u) , w (y) =v (x)

    y H, w (y) = 0.

    y= u (x1) =u (x2) , x1 x2 ker(u)

    ker(v) v (x1) =v (x2) . x E, w (u (x)) =v (x) , v= w u.

    u L (E, Kp)

    x E, u (x) = (1(x) , , p(x))

    ker(u) =

    pi=1

    ker(i) ker() .

    Kp

    K

    = u x E

    (x) = (u (x)) = (1(x) , , p(x)) =p

    j=1

    jj(x)

    =

    pj=1

    jj.

    j

    Kn[x]

    E= Kn[x] .

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    Kn[x]

    B= (ej)0jn E, ej(X) =Xj.

    PE

    P =n

    j=0

    ajXj =

    nj=0

    ajej aj = P(j) (0)

    j!

    B

    PE, ej(P) =aj = P(j) (0)j!

    (0 j n)

    E= Kn[x] n+ 1 x0, x1, , xn K.

    L= (Li)0in

    Li(x) =

    nj=0j=i

    x xjxi xj

    (1 i n)

    E.

    L.

    K = R

    xi [a, b] .

    0, 1, , n

    P Rn[x] ,

    ba

    P(t) dt=n

    j=0

    jP(xj)

    n= 2, x0=a, x1 =a +b

    2

    x2=b.

    L n+ 1 = dim (E) n

    j=0

    jLj = 0, xi, i 0 n, i = 0.

    PE P =

    nj=0

    jLj xi,

    i

    0 n,

    Li (P) =i=P(xi) .

    : P

    ba

    P(t) dt

    =n

    j=0

    (Lj) Lj ,

    P Rn[x] , (P) =

    ba

    P(t) dt=n

    j=0

    (Lj) Lj(P) =

    nj=0

    jP(xj)

    j = (Lj) =

    ba

    Lj(t) dt

    P

    ba

    P(t) dt P

    P(xj) j 0 n Rn[x] .

    n= 2, x0=a, x1=c =

    a + b

    2

    x2 = b,

    L0(x) = 2

    (b a)2(x c) (x b) , 0 =

    b a

    6

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    L1(x) = 4

    (b a)2(x a) (x b) , 1 = 4

    b a

    6

    L2(x) = 2

    (b a)2(x a) (x c) , 2=

    b a

    6

    P R2[x] ,

    ba

    P(t) dt=b a

    6

    P(a) + 4P

    a + b

    2

    + P(b)

    .

    E

    E

    E

    E= K [x]

    B= (ej)jN , ej(X) =Xj.

    B =

    (ej

    )jN

    ei(ej) = ij i, j N,

    E.

    E

    KN

    K.

    B B

    P =n

    k=0

    akxk R [x] , (P) =

    nk=0

    ak

    =

    m

    j=0

    jej , m j

    1 = (em+1) =

    mj=0

    jej(em+1) = 0

    KN

    E

    u= (uk)kN , (u) :P =n

    k=0

    akxk

    nk=0

    akuk

    ( (u) = 0) (k N, (u) (ek) =uk = 0) (u= 0)

    E

    u= (uk)kN

    k N, uk = (ek)

    P=n

    k=0

    akxk,

    (u) (P) =n

    k=0

    ak (ek) = (P) , = (u)

    KN

    E.

    E

    dim(E) = card(E) .

    E E

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    Mn(K)

    Mn(K)

    Mn(K) , (ei)1in E= Kn

    (Eij)1i,jn Mn(K) .

    Eij

    k {1, , n} , Eijek=

    0

    k =j,

    ei k= j.

    k =j Eij j i

    1

    Eij = (0, , 0, ei, 0, , 0)

    ei j.

    Ei (p, q)

    (Eij)pq=ipqj

    rs

    Mn(K) .

    i =j {1, , n}

    EijEji =Eii.

    EijEjj =Eij EjjEij = 0.

    Mn(K) (AB) = (BA)

    A, B Mn(K) .

    (Eii) = (Ejj ) i, j 1 n.

    (Eij) = 0 i=j {1, , n} .

    (A) = Tr (A)

    A

    Mn(K) .

    u

    Mn(K) u (In) =In u (AB) =u (BA)

    A, B Mn(K) . u

    Mn(K) L (E) , E n.

    i =j {1, , n}

    k

    1

    n,

    EijEjiek=Eij(Ejiek) =

    Eij(0) = 0 k =iEij(ej) =ei k= i

    =Eiiek

    EijEji = Eii.

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    EijEjjek =Eij(Ejjek) =

    Eij(0) = 0 k =jEij(ej) =ei k= j

    =Eijek

    EjjEijek=Ejj (Eijek) =

    Ejj (0) = 0

    k =jEjj (ei) = 0 k= j = 0

    EijEjj =Eij EjjEij = 0.

    A = ((aij))1i,jn B =

    ((bij))1i,jn Mn(K) , C=AB D= BA,

    cii=n

    k=1aikbki dkk =

    n

    k=1bkiaik

    i, k

    1

    n,

    Tr (AB) =ni=1

    cii=ni=1

    nk=1

    aikbki =

    1i,kn

    aikbki

    Tr (BA) =n

    k=1

    dkk =n

    k=1

    ni=1

    bkiaik =

    1i,kn

    aikbki = Tr (AB)

    AB

    BA

    i, j

    1

    n,

    (Eii) = (EijEji) = (EjiEij) = (Ejj ) .

    = (Eii) i 1 n.

    i =j 1 n,

    (Eij) = (EijEjj) = (EjjEij) = (0) = 0.

    A= ((aij))1i,jn Mn(K) A=

    1i,jn

    aijEij,

    (A) =

    1i,jn

    aij (Eij) =ni=1

    aii (Eii) =

    ni=1

    aii= Tr (A) .

    (In) = Tr (A) =n,

    Mn(K) (In) =n (AB) = (BA) A,B.

    Mn(K)

    A Mn(K) , (A) = Tr (u (A)) .

    (AB) = (BA)

    A, B

    Mn(K)

    (A) = Tr (A) ,

    Tr (u (A)) = Tr (A)

    A Mn(K) .

    u (In) =In = 1.

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    Mn(K)

    Mn(K) A Mn(K)

    B Mn(K)

    (B) Mn(K)

    A Mn(K) , (B) (A) = Tr (BA) .

    Mn(K) ,

    B Mn(K)

    A Mn(K) , (A) = Tr (BA) .

    Mn(K) L (E) , E n

    Mn(K)

    (AB) = (BA)

    A, B Mn(K) ,

    (A) = Tr (A)

    A Mn(K)

    Mn(K) .

    A= ((aij))1i,jn Mn(K) , AEij =EijA,

    i, j

    1 n

    AEijej =Aei =n

    k=1

    akiek =EijAej =Eij

    nk=1

    akjek

    =ajjei,

    aki = 0 k =i aii=ajj , A= In.

    A

    Mn(K) , A

    px Kx. A

    A

    B

    Tr (BA) = 0

    A Mn(K) , Tr (BEij) = 0 i, j 1

    n.

    BEijek=

    0

    k =j

    Bei = b1ie1+ b2ie2+ + bnien k=j

    k

    1 n,

    Tr (BEij) =bji bji B = 0. ker() ={0}

    Mn(K) Mn(K) (B) B

    Tr (BA) = Tr(AB) ,

    Mn(K)

    ATr (AB) .

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    B0 Mn(K) (A) = Tr(AB0) A Mn(K) ,

    A, B

    Mn(K)

    (AB) = Tr (ABB0) = (BA) = Tr (BAB0)

    = Tr (B (AB0)) = Tr ((AB0) B) = Tr (AB0B)

    A Mn(K) , Tr (A (BB0 B0B)) = 0

    (BB0 B0B) = 0, BB0 B0B = 0 B Mn(K) ,

    B0 B0=In

    (A) = Tr (AB0) = Tr (A) A Mn(K) .

    GLn(K) , K

    A

    GLn(K) . B Mn(K) ,

    det(B In) n K K B In

    A (B In) = (B In) A, AB A = BA A AB=BA.

    A

    Mn(K) A= In = 0.

    Z(GLn(K)) = K.

    GLn(R) GLn(C)

    : GLn(R) GLn(C)

    GLn(R) GLn(C) .

    A

    GLn(R) AM = MA M GLn(R) ,

    MGLn(R) , (A) (M) = (M) (A) ,

    M GLn(C) , (A) M =M (A)

    (A)

    GLn(C) .

    R

    C,

    i

    4 C

    4

    R.

    E= Kn.

    x E E, x

    x= ( (e1) x, , (en) x) = (( (ej) xi))1i,jn

    ( ei) z z E , E i

    1

    n.

    ej x x E j 1 n.

    ( ei) A

    (ej y

    )

    y E, E,

    A Mn(K) i, j 1 n.

    Mn(K) {0} Mn(K) Mn(K) L (E) , E n

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    Mn(K)

    ei= ( (e1) ei, , (en) ei)

    =

    0 0 0

    0 0 0 (e1) (e2) (en)

    0 0 0

    0 0 0

    (ej) i

    ( ei) z=

    0

    0n

    j=1

    (ej) zj

    0

    0

    =

    0

    0 (z)

    0

    0

    = (z) ei.

    ej x=(

    ej(e1) x, , ej(en) x

    )= (0, , 0, x, 0, , 0)

    x

    j.

    ( ei) A(

    ej y)

    = ( ei) A (0, , 0, y, 0, , 0)

    = ( ei) (0, , 0,Ay, 0, , 0)

    = (0, , 0, ( ei) (Ay) , 0, , 0)

    = (0, , 0, (Ay) ei, 0, , 0)

    I

    Mn(K) {0} , A = 0

    y

    Ay = 0. E

    (Ay) = 1 = 0 i, j 1 n

    ( ei) A (ej y) = (0, , 0, ei, 0, , 0) =Eij I

    I=Mn(K) In =ni=1

    Eii.

    E= K [X]

    I={u L (E)| rg (u) }

    L (E) I ={0}

    P P(0)

    I =L (E) I, u I v L (E) u v I Im (u v) Im (u) vu I (u (xk))1kr Im (u) ,(v (u (xk)))1kr

    Im (v u) u= 0, v u= 0

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    E,

    E.

    H= ker ()

    E,

    (x) = 0

    H.

    H

    E,

    D

    E=H D.

    H= ker()

    E.

    a

    E

    (a) = 0. D= Ka

    a,

    E = HD. x H D, x = a

    (x) = (a) = 0

    = 0.

    H D ={0} .

    x E , y =x (x)

    (a)

    a

    H= ker ()

    x= y + (x)

    (a)

    a,

    x H+ D. E=H+ D E=H D.

    H E

    D

    x

    E

    D

    H,

    E

    E

    E

    , E

    ker() ker () .

    = 0, ker() = ker() .

    = 0,

    E= ker () ker() , ker() =E = 0.

    = 0. a E\ker() E= ker()Ka x E

    x= y +z

    y =x

    (x)

    (a)a ker () z=

    (x)

    (a)a Ka.

    ker() ker() , x E ,

    (x) = (z) = (x)

    (a) (a) = (x)

    =

    (a)

    (a) K. = .

    = 0,

    = 0,

    = 1

    , ker()ker()

    ker() = ker() .

    E

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    E

    n 1 B= (ej)1jn E,

    E

    x=n

    j=1

    xjej E

    1x1+ 2x2+ + nxn= 0

    j = (ej)

    dim (ker ()) =n 1.

    H

    n 1 E n,

    (ei)1in1 (ei)1in E H

    n

    pn: x =n

    j=1

    xjej xn.

    E

    n

    E

    n 1.

    E

    H

    Mn(K) , n 2, HGLn(K) =.

    H= ker() ,

    : A Tr (AB) =n

    i=1

    n

    k=1

    aikbki

    Mn(K)

    B = diag (1, , n)

    AB=

    1a1,1 2a1,2 na1,n1a2,1 2a2,2 na2,n

    1an,1 2an,2 nan,n

    A GLn(K) aii = 0 i 1 n H.

    A=

    0 0 0 11 0 0 0

    0 1

    0 00 0 0 1 0

    = (e2, e3, , en, e1)

    B

    p =q bpq = 0 A= In+Eqp,

    A GLn(K)

    (A) = (In) + (Eqp) = Tr (B) + Tr (EqpB) = Tr (B) +bpq

    =

    1

    bpqTr (B) ,

    AH GLn(K) .

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    E x E (x) = 0.

    (E, | ) : x a| x

    E

    X

    E

    X ={ E | x X, (x) = 0} .

    E

    Y

    E

    Y ={x E| Y, (x) = 0} .

    X E Y

    E.

    X=,

    X =E

    Y =, Y =E .

    Y

    E

    Y =

    Y

    ker() .

    (x Y) ( Y, (x) = 0) (Y, x ker )

    A, B E U, V E.

    A B, B A.

    UV, V U.

    A

    (A)

    ,

    U(U) ,

    A = (Vect (A)) .

    U = (Vect (U)) .

    {0} =E, E ={0} , {0} =E (E) ={0} .

    AB, B A, B A.

    A

    U V, x V V,

    U

    xU. U

    A

    (A)

    A

    A

    0

    (A

    )

    E.

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    A Vect (A) , (Vect (A)) A. A, x Vect (A)

    x =

    pj=1

    jxj, xj A, (x) =p

    j=1

    j (xj) = 0

    (Vect (A)) .

    A= , Vect (A) ={0} A ={0} =E.

    0,

    {0} =E. {0} =E

    E (x) = 0 x E, = 0. E ={0} .

    x (E) (x) = 0 E.

    E B = (ej)1jn E

    B = (ej)1jn

    .

    x=n

    j=1

    xjej, xj =ej(x) = 0 j

    1

    n

    x= 0.

    (E) ={0}

    x (E) B = (ej)jJ E

    x= ek k J. E

    jJ, (ej) =

    0

    j =k

    1 j =k

    0 = (x) = (ek) = 1

    x= 0.

    H

    E,

    H={0}

    H =E.

    H={0} , H =E.

    H =E.

    H= {0} , E= H F F = E E

    (x) = 1

    x H (x) = 0 x F H,

    H =E. H={0} .

    E

    n 1.

    F

    E,

    dim(F) + dim(

    F)

    = dim (E)

    G

    E,

    dim(G) + dim (G) = dim (E)

    F E G E,

    F =(

    F)

    G= (G)

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    X

    E,

    (X)

    = Vect (X) .

    F1 F2 E, (F1+ F2) =F1 F

    2 .

    F1 F2 E, (F1 F2) =F1 +F

    2 .

    G1 G2 E

    ,

    (G1+ G2) =G1 G

    2.

    G1 G2 E

    ,

    (G1 G2) =G1+ G

    2.

    F = {0} F = E. F

    E.

    B1 = (ej)1jp F B= (ej)1jn E B =

    (ej)1jn B. B2 = (ej)p+1jn F. F ej(ei) = 0 1 i p p+ 1 j n, B

    2 F

    . F,

    =

    nj=1

    (ej) ej (ej) = 0 1 j p F

    ej F.

    =

    nj=p+1

    (ej) ej B

    2 F

    .

    dim(F) + dim(

    F)

    =p + (n p) = dim (E) .

    G = {0} G = E. G

    E

    . B1 = (j)1jp G B

    = (j)1jn E

    B= (ej)1jn B.

    B2 = (ej)p+1jn

    G.

    G

    j(ei) = 0 1 j p p+ 1 i n, B2 G. x G

    x =n

    j=1

    xjej xj = j(x) = 0 1 j p x G ej G.

    x=n

    j=p+1

    xjej B2 G.

    dim(G) + dim (G) =p + (n p) = dim(E) .

    F

    (F)

    G (G)

    x

    (F)

    ,

    (x) = 0

    F,

    xj = ej(x) = 0 p+ 1 j n

    x=n

    j=1

    xjej F. (

    F)

    F

    (X

    )

    =

    (Vect (X))

    = Vect (X)

    FkF1+F2 k= 1 k= 2, (F1+ F2) Fk (F1+ F2) F1 F2 .

    F1 F2 F1 F2, F1+ F2 (F1+ F2)

    .

    (F1+F2)

    =F1 F2 .

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    F1F2 Fk k= 1 k= 2, Fk (F1 F2)

    F1 +F2 (F1 F2)

    dim(F1 F2) =n dim(F1 F2)

    dim(

    F1 + F2

    )= dim

    (F1)

    + dim(

    F2)

    dim(

    F1 F2

    )= 2n dim(F1) dim(F2) dim

    (F1+ F2)

    = 2n dim(F1) dim(F2) n + dim (F1+ F2)

    =n dim(F1 F2) = dim

    (F1 F2)

    Gk G1+G2

    k= 1

    k= 2,

    (G1+ G2)

    G

    k

    (G1+ G2)

    G

    1G

    2.

    x G1 G2 G1 G2,

    G1+ G2 x (G1+ G2)

    .

    (G1+ G2) =G1 G

    2.

    F =

    (F)

    (F

    )

    F, x / F, x /

    (F

    )

    .

    B1 F, B1 {x}

    B= B1 {x} B2 E.

    e B, (e) =

    0

    e B1 B2

    1 e= x

    F

    x,

    x /

    (F)

    .

    (X)

    = Vect (X) ,

    X E.

    (F1+ F2)

    =F1 F2

    (F1 F2)

    =F

    1 + F

    2

    B1= (ej)jJ1 F1F2,L1= (ei)iI1 B1 F1 L2= (ei)iI2 B1 F2. B1L1L2

    k1K1

    xk1ek1 +k2K2

    xk2ek2 +k3K3

    xk3ek3 = 0

    K1 J1, K2 I1 K3 I2,

    k1K1

    xk1ek1+k2K2

    xk2ek2 =k3K3

    xk3ek3 F1 F2

    xk2

    k1K1

    xk1ek1+k3K3

    xk3ek3 = 0 xk1 xk3

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    B1 L1 L2 B1 L1 L2 L3 E.

    (F1 F2) ,

    1 2 E

    e B, 1(e) = 0

    e B1 L1

    (e) e L2 L3

    e B, 2(e) =

    0

    e B1 L2 L3

    (e) e L1

    x F1, 1(x) = 0x F2, 2(x) = 0x E, 1(x) +2(x) = (x)

    = 1+ 2 F1 +F

    2 . (F1 F2)

    F1 + F2

    F1, F2

    E,

    F1 F2 E

    .

    E=F1 F2, {0}= E = (F1 F2)

    =F1 F2 .

    dim(

    F1)

    + dim(

    F2)

    = 2n (dim(F1) + dim (F2)) =n = dim (E)

    E =F1 F

    2 .

    B1 F1 B2 F2 B= B1 B2

    E

    E, 1 2

    1(e) =

    0

    e B1

    (e) e B2

    2(e) =

    0

    e B2

    (e) e B1

    = 1 + 2 1 F

    1 2 F

    2 . E

    =F1 + F2 E

    =F1 F2 .

    E

    n.

    (i)1in E

    ni=1

    ker(i) ={0} .

    (i)1in

    E

    E

    = Vect {1, , n} ,

    {0}= (E) = ({1, , n}) =

    ni=1

    ker(i)

    ni=1

    ker(i) ={0} ,

    Vect {1, , n}= ((Vect {1, , n}))

    =

    n

    i=1ker(i)

    ={0} =E

    (i)1in E

    .

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    E

    n2.

    (1, 2, , p) E r,

    F =

    pi=1

    ker(i) E n r.

    F

    E

    m,

    (1, 2, , r) E r= n m, F =r

    i=1

    ker(i) .

    (i)1ip E

    r

    G

    E

    (i)1ip . (i)1ip r G r.

    r= 0, i F =E n.

    (xF) (i {1, , p} , i(x) = 0) x G

    F =G

    dim(F) =n dim(G) =n r.

    F

    E

    m.

    m= 0,

    F ={0}=

    ni=1

    ker(i) (i)1in E.

    m= n,

    F =E= ker (0) .

    1 m n 1, (ei)1im F (ei)1in

    E

    (e

    i )m+1in

    F

    (x F)

    x=

    mi=1

    xiei

    (i {m+ 1, , n} , ei(x) = 0)

    F =

    ri=1

    ker(i) r= n m i=em+i 1 i r.

    E, F K

    u L (E, F) tu F

    E

    F, tu () = u

    tu

    u tu L (E, F)

    L (F, E) .

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    u, v

    L (E, F) , K,

    t (u +v) = tu + tv

    tu= 0

    u F,

    F, x E, (u (x)) = 0

    x E, u (x) (F) (F) = {0} .

    u= 0

    E

    F

    L (E, F) L (F, E)

    u

    L (E, F) v L (F, G)

    t (v u) = tu tv

    F =E , tIdE=I dE

    u

    E

    F,

    tu

    F

    E

    ( tu)1 = tu1

    ker( tu) = (Im (u))

    u

    tu

    Im ( tu) = (ker (u))

    u

    tu

    E

    F

    u

    tu

    G x E,

    t (v u) () (x) = (v (u (x))) = tv () (u (x)) = tu(

    tv ())

    (x)

    t (v u) = tu tv.

    G,

    tIdE() = IdE=

    tIdE=I dE .

    u u1 =I dF u1 u= I dE,

    IdF = tIdF =

    t(

    u u1)

    = tu1 tu

    IdE = tIdE=

    t(

    u1 u)

    = tu tu1

    tu

    tu

    F

    E

    ( tu)1 = tu1.

    F.

    (ker (tu)) ( u= 0) (x E, (u (x)) = 0)

    (y Im (u) , (y) = 0)

    (Im (u))

    ker( tu) = (Im (u)) .

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    u

    Im (u) =F,

    (Im(u)) =

    F,

    ker( tu) =F ={0} tu

    E.

    ( Im ( tu)) ( F | = u)(xker(u) , (x) = (u (x)) = (0) = 0)

    (ker (u))

    Im ( tu) (ker (u)) .

    E

    F

    n

    m,

    dim(

    Im(

    tu))

    =m dim(

    ker(

    tu))

    =m dim

    (Im(u))

    = dim(Im (u)) =n dim(ker(u)) = dim(ker(u))

    Im ( tu) = (ker (u)) .

    (ker (u)) (x) = 0 x ker(u) ,

    ker(u) ker() .

    H

    ker(u) ,

    v u

    H

    H

    v (H) = u (H) .

    L

    u (H)

    F

    F

    y=u (x) u (H) , (y) = (x)

    y L, (y) = 0

    y u (H) y =u (x) x H u H

    u (H)

    xE x= x1+ x2 x1 ker(u) x2 H,

    (u (x)) = (u (x2)) = (x2) = (x1+ x2) = (x)

    ker(u) ker() , x1 ker() =

    tu () Im ( tu) .

    Im ( tu) = (ker (u)) .

    u

    ker(u) ={0} , (ker(u)) =

    {0} , Im ( tu) ={0} =E tu

    E

    n

    F

    m.

    rg (u) = dim (Im (u)) =m dim

    (Im(u))

    =m dim(

    ker(

    tu))

    = rg(tu)

    u

    tu rg (tu) = rg (u) .

    E

    n, F

    m

    B = (ei)1in E B = (fj)1jm F.

    B B.

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    A Mm,n(K) u L (E, F) B B,

    tu B B tA.

    u (ej

    ) =m

    i=1

    aij

    fi (1

    j

    n)

    j

    1

    m, x=

    nk=1

    xkek E,

    tu(

    fj)

    (x) =fj (u (x)) =fj

    nk=1

    xku (ek)

    =

    nk=1

    xkfj (u (ek))

    =n

    k=1

    xkfj

    mi=1

    aikfi

    =

    nk=1

    xk

    mi=1

    aikfj (fi)

    =n

    k=1

    xkajk =n

    k=1

    ajkek(x)

    tu(

    fj)

    =n

    k=1

    ajkek (1 j m)

    tu B B tA.

    C

    u L (E) u

    E.

    E

    n.

    u L (E)

    r

    E

    r

    {1, 2, , n 1}

    uL (E)

    E.

    xE,

    (x)

    u (x) = (x) x.

    x, y

    E,

    x

    y

    y= x

    = 0,

    (y) y=u (y) =u (x) = (x) x= (x) y

    (y) = (x)

    x

    y

    (x + y) (x + y) =u (x + y) =u (x) +u (y) = (x) x + (y) y,

    (y) = (x + y) = (x) .

    (y) = (x) .

    K,

    u

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    r= 1,

    u

    r

    2

    n 1.

    u

    E.

    H

    (ei)1in1 . H {e1, , er1, ek} , k

    r

    n 1, u, u (ej) H j 1

    n 1. H u.

    tu

    E.

    E \ {0} ,

    E= ker() Ka (a) = 0 x E x= x1+ a x1 ker()

    =

    (x)

    (a),

    tu () (x) = (u (x)) = (u (x1)) + (u (a))

    u (x1) ker() ker() u,

    t

    u () (x) = (u (a)) = (u (a))

    (a) (x) ,

    tu () =

    tu

    E.

    tu

    u

    E

    E,

    E.

    : EE x E,

    (x)

    E

    E, (x) () = (x)

    x

    ,

    (x) = 0

    E, x (E) =

    {0} .

    E

    E

    E

    E

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