Réduction (1)

u v KE
u v Im uKer u v
fKE
pIm pKer p f
EKf g E
f◦g=g◦f
Ker fIm f g g(Ker f)⊂Ker f
g(Im f)⊂Im f
p E
p f Im pKer p f
uKE
N=
∞
[
p=0
Ker upI=
∞
\
p=0
Im up
n∈NN= Ker unI= Im un
N I
u u N I
E=F⊕G F G
u u F G
F=N G =I
u∈ L(E) dim E < +∞p∈N∗up= 0
k∈ {1, . . . , p}Fk
E
Ker uk= Ker uk−1⊕Fk
E=F1⊕ ··· ⊕ Fp
u
ERn f ∈ L(E)
f2=−IdE
a∈E(a, f(a))
F(a) = Vecta, f(a)E a1, . . . , ap
E=F(a1)⊕ ··· ⊕ F(ap)
E
E f
ERu E
u3+u= 0
Im u u
x∈Im u u2(x)
v u Im u
v
u
u v Kn∈N∗
u◦v=v◦u v
det(u+v) = det u
n≥1

n= 1 v= 0
n≥2v6= 0 u v
Im v
u v
Im v
E=E1⊕E2K
Γ = u∈ L(E)Ker u=E1Im u=E2
uΓ ˜u=uE2E2
φ: Γ →GL(E2)φ(u) = ˜u
◦Γ
φ(Γ,◦) (GL(E2),◦)
φ
(Γ,◦)
E=C(R,R)f∈E x ∈R
T f(x) = f(x) + Zx
0
f(t) dt
T E
E
T
A= (ai,j )∈ Mn(R)s
∀i∈J1 ; nK,
n
X
j=1
ai,j =s∀j∈J1 ; nK,
n
X
i=1
ai,j =s
U U =t1··· 1∈ Mn,1(R)
A λ
µ
AU =λU tUA =µtU
λ µ
D= Vect(U)H=X∈ Mn,1(R)tUX = 0
AMn(R)
D H
Mn(R)
RE=R[X]
H f
H p ∈N
∀k≥p, Ker fk+1 = Ker fk
F E D
F
DRn[X]n∈N
m∈NF=Rm[X]
F F =R[X]
g∈ L(E)g2=kId + D k ∈Rk
D: P7→ P0R[X]
∆R[X] ∆2= D
A∈ M3(R)A2= 0 A6= 0
A
B=
000
100
000

A∈ Mn(K)
An−16= OnAn= On
A
B=
0 1 (0)
1
(0) 0
A∈ Mn(K)A2= 0 r > 0
A
B=0Ir
0 0 
M∈ M4(R)M2+I= 0
M
0−1 0 0
1000
000−1
0010
A∈ Mn(K)
A n −1
A2= tr(A).A det(In+A) = 1 + tr A
n
diag(1,2, . . . , n)
f:Mn(C)→C
∀(A, B)∈ Mn(C)2, f(AB) = f(A)f(B)
A∈ Mn(C)
A⇐⇒ f(A)6= 0
A∈ M3(R)A2= O3
C=M∈ M3(R)AM −MA = O3
A=
3 6 −5−2
−1−6 5 −2
−1−10 8 −3
0−3 2 0
B=
12621
0 2 2 5
0 0 3 2
0 0 0 5
GMn(R)
(G, ×)r∈N∗
(G, ×) (GLr(R),×)
A∈ M2(C)AtA
A∈ M3(C)A−A
tr(A) = det(A) = 0

fKE
0/∈Sp(f)⇐⇒ f
fKn∈N∗
0∈Sp(fn)
0∈Sp(f)
uKE
Sp u−1=λ−1λ∈Sp u
EKu∈ L(E)a∈GL(E)v=a◦u◦a−1
Sp uSp v Eλ(u)Eλ(v)
uKE
u
u v
λ6= 0 u◦v v ◦u
P∈E=R[X]
u(P) = P0v(P) = ZX
0
P(t) dt
E
Ker(u◦v) Ker(v◦u)
λ= 0
E
f g CE
n≥1
f◦g−g◦f=f
f
fn−16= 0 e E λ ∈C
Matef=
0 1 (0)
1
(0) 0
Mateg= diag(λ, λ + 1, . . . , λ +n−1)
ECu, v L(E)a, b
C
u◦v−v◦u=au +bv
a=b= 0
u v
a6= 0 b= 0
u
un◦v−v◦unu
u v
a, b 6= 0
u v
EC(a, b)∈C2f g
L(E)
f◦g−g◦f=af +bg
f g

E u v
E[u;v] = uv −vu
[u;v]=0 u v
[u;v] = λu λ ∈C∗u u
v
α β [u;v] = αu +βv
u v
f g KE
f◦g−g◦f=I
n≥1fn◦g−g◦fn=nfn−1
f g f ◦g−g◦f=I
E=K[X]f g f(P) = P0
g(P) = XP
E f g
E
f◦g−g◦f=f
fn◦g−g◦fn
P P (f)=0 f◦P0(f)=0
f
A∈ Mn(C)TMn(C)
T(M) = AM −MA
A
T
A, B, C ∈ Mn(R)
AB −BA =C
C A B
A C
C
C
A∈ Mp(R) ∆: M∈ Mp(R)7→ AM −MA
∆Mp(R)
∀n∈N∗,∀(M, N )∈ Mp(R)2,∆n(M N ) =
n
X
k=0 n
k∆k(M)∆n−k(N)
B= ∆(H)A
∆2(H) = 0 ∆n+1(Hn)=0
∆n(Hn) = n!Bn
k · k Mp(R)
Bn 1/n −−−−−→
n→+∞0
B
ECu v
E
u◦v−v◦u=u
Ker(u)v
Ker(u)6={0}
u v
u◦v−v◦u∈Vect(u, v)
E u v