Post on 13-Apr-2018
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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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