énoncés et corrections

L(E, F )
F∪G E F ∪G∈ {F, G}
Jr
Mn(k)
E u v w
E w
u w u w−1
u◦v+u+v= 0 u v
E
d∈NX
dX(X−1)···(X−d+1) d!∆
R[X]P P (X+ 1) −P(X)
Ker ∆ ∆X
dd
P∈Rd[X]P(Z)⊂Z(d+ 1) (cd, . . . , c0)
P=cdX
d+· · · +c0X 0P∈R[X]P(Z)⊂Z
(Z,+) (Z3 ,+)
(U,×) (R,+)
(Q,+) (R,+)
(Q,+) (Z,+)
G n g ∈G
gn= 1 G
E A B E A ∆B= (A∪B)\
(A∩B) ∆ P(E)
∆∩ P(E)

E F, G E
ϕ:u∈ L(E)7→ (u|F, u|G)∈ L(F, E)× L(G, E)
ϕ
ϕ
ϕ
EL(F, E)×
L(G, E) Ker ϕ ϕ
P=X3+ 2X2+X+ 1
P x1, x2, x3
x1+x2+x3x1x2x31
x1+1 x2+1 x3
1
2−x1+1 2−x2+1 2−x3
E F (u, v)∈ L(E, F )×L(F, E)
(∗)uvu =u vuv =v
(u, v) (∗)E= Ker(u)⊕Im(v)F= Ker(v)⊕Im(u)
u∈ L(E, F )E1F1Ker(u) Im(u)E
F v ∈ L(F, E) (u, v) (∗) Ker(v) = F1
Im(v) = E1
E u E
Ker u= Ker u2
Im u= Im u2
E= Ker u⊕Im u
E
A∈ Mn(R)B C Mn(R)A[B, C] = BC −CB
Tr(A)=0
ϕMn(K)A
M ϕ(M) = Tr(AM)
Mn(K)ATr(A·)
Mn(K)
ϕMn(K)A B
ϕ(AB) = ϕ(BA)ϕ
V=Mn(R)A∈V V
X XA
W V
A∈W W X AX −XA
E
a c b
b a c
c b a
(a, b, c)∈C3E
E

A∈ Mn(R)tAA =AtAtAA
A
A∈ M3(R)A2= 0
A
000
001
000
100
000
000
A=
1−1 1
2−2 2
1−1 1
A B Mn(C)M∈ Mn(C)
M+ Tr(M)A=B
A∈ M3,2(R)B∈ M2,3(R)AB =
000
010
001
BA
w−1
(u+ 1)(v+ 1) = 1 v+ 1 u+ 1
X k[X]
Ker ∆ = RX
0=R∆X d=X d−1d>1
d d = 0 ∆(P)
cdX
d−1+· · · c1X 0= ∆(cdX d+· · · c1X 1)
P(0) ∈Z
Pd∈NZX
d
Q=Sn,m{x|nx =m1}
g∈G h 7→ gh G Qh∈Gh=
Qh∈Ggh =gnQh∈GhQh∈Gh

E=F+G ϕ F +
G E H (F+G)⊕H=E u H
F+G
F∩G= 0 ϕ F ⊕G E
x∈F∩G u1(x)6=u2(x) (u1, u2)
ϕ
dim(L(F, E)×L(G, E)) = dim(E)(dim(F)+dim(G)) H
Ker ϕ' L(H, E) dim Ker ϕ= dim(E)(dim(E)−dim(F+G)) ϕ
dim(E)(dim(F)+dim(G)−dim(F∩G))
P
x1+x2+x3=−2x1x2x3=−1P0
P=1 X−x1+1 X−x2+1 X−x3
1
x1+1 x2+1 x3=
P0(0)
P(0) = 1 1 2−x1+1 2−x2+1 2−x3=P0(2) P(2) =21 19
u(vu −1) = 0 Im(vu −1) ⊂Ker(u)x∈E
x= (x−vu(x)) + (vu(x)) ∈Ker(u) + Im(v)vx ∈Ker(u)∩Im(v)
vuvx = 0 = vx Ker(u)∩Im(v)=0
v E1u|E1
Im u uv =πIm(u)vu =πE1
(∗)uv vu
v u E1
⇔
x∈E u(x) = u2(y)y
x−u(y)∈Ker u E = Ker u+ Im u x ∈Ker u∩Im u x =u(y) 0 = u(x) =
u2(y)y∈Ker u2= Ker u x = 0
x∈E x =x0+u(y)x0∈Ker u
u(x) = u2(y) Im u= Im u2u2(x)=0 u(x)∈Ker u∩Im u= 0
∧ ⇔
⇔X
k[X]
(Ker u= Ker u2)⇔(Ker u∩Im u= 0)
(Im u= Im u2)⇔(E= Ker u+ Im u)
(Ker u= Ker u2)⇔(Im u= Im u2)
A n x
(x, Ax)A x
(x, Ax)A
0∗ · · · ∗
1
0A0

A0n−1
P P A0P−1Q=1 0
0P
QAQ−1=
0∗ · · · ∗
∗
P A0P−1
∗
A
A
A[B, D]D= diag(d1, . . . , dn)
[B, D]ij =bij (dj−di)dibij =aij
dj−dii6=j
Tr(AEij ) =
aji
A
ϕ
A∈ Mn(K)ϕ= Tr(A·)A=P JrQ
r∈[[1, n]] P Q B Tr(AB)
C0 = Tr(AQ−1CP −1) =
Tr(P JrCP −1) = Tr(JrC)C
D
In−rD=
1
1
1
ϕ(Eij Ek`) =
ϕ(Ek`Eij )δjk ϕ(Ei`) = δi` ϕ(Ekj )ϕ(Ekj )=0 k6=j
ϕ(Eii)
Φ
Φ(Eij ) = Eij A=X
k,`
ak,`Eij Ek` =X
`
aj`Ei`
Tr(Φ) = Pi,j aji
A
C(ei)CnAei=λieiet
iej−et jeii < j
W et
iej−et jeiλi+λjPi<j (λi+λj) =
1
2Pi6=j(λi+λj)=(n−1) Tr A
E=C[Ω] Ω = 
1
1
1
RA= 0