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R
R
x∈R\Q(p, q)∈Z×N∗
x−p
q
<1
q2
un=1
nsin n
A B Mn(R)A
P∈R[X]P(0) = 1 B=AP (A)Q∈R[X]
Q(0) = 1 A=BQ(B)
E p q L(E)
p◦q=q q ◦p=p p q
EC(a, b)∈C2f g
L(E)
f◦g−g◦f=af +bg
f g
n∈N∗A B Mn(C)λ1, . . . , λn, λn+1
C1≤i≤n+ 1 A+λiB
A B
ϕM2(C)C
∀A, B ∈ M2(C), ϕ(AB) = ϕ(A)ϕ(B)ϕλ0
0 1=λ
ϕ= det
EC
f∈ L(E)E f
ECu∈ L(E)
I1={P∈C[X]/P (u)=0} I2={P∈C[X]/P (u)}
I1I2C[X]
P1P2
P1P2
Q∈ I2u−Q(u)
Φ: M2(R)→R
∀A, B ∈ M2(R),Φ(AB) = Φ(A)Φ(B) Φ 0 1
1 06= Φ(I2)
Φ(O2) = 0
AΦ(A)=0
A∈ M2(R)B A
A
Φ(B) = −Φ(A)
AΦ(A)6= 0

0··· 0 1
0··· 0 1
1··· 1 1
∈ Mn(R)
Mn=
0 (b)
(a) 0
∈ Mn(C)
Mn
k
eiθ+ eikθ = 1
θ∈R
Sku
∀n∈N, un+k=un+un+k−1
k Sk
A∈ Mn(C) tr(Am)→0m→+∞
A < 1
KA1, A2, . . . , AnMn(K)
A1A2. . . An=On
f:Mn(C)→C
∀(A, B)∈ Mn(C)2, f(AB) = f(A)f(B)
A∈ Mn(C)
A⇐⇒ f(A)6= 0
E u E
λ λ u −λId
u
u
mL(E)v u ◦v
m
u
EKu∈ L(E)P∈K[X]
P(u)=0 Q∈K[X]R∈K[X]R(Q(u)) = 0
A∈ Mn(R)
A3=A2A2A2−A
Ak+1 =Akk > 0
p > 0ApAp−A

n∈N∗
En={A∈ Mn(Z)| ∃m∈N∗, Am=In}
A∈En
ω(A) = min {m∈N∗|Am=In}
ω(En)
R[X]
hP, Qi=Z1
0
P(t)Q(t) dt
A∈R[X]
∀P∈R[X], P (0) = hA, P i?
Qn(X) = 1
2nn!(X2−1)n(n)
n≥1Qnn]−1 ; 1[
Qn=Xn+ (X2−1)Rn(X)
Rn∈R[X]Qn(1) Qn(−1)
(P, Q)∈R[X]2
hP, Qi=Z1
−1
P(t)Q(t) dt
QnRn−1[X]
kQnk2
(E, h., .i)
ϕ∈ O(E)M(ϕ) = Im(ϕ−IdE)F(ϕ) = ker(ϕ−IdE)
u∈E\ {0}suu⊥
ϕ∈ O(E)M(ϕ)⊕⊥F(ϕ) = E
(u1, . . . , uk)
M(su1◦ ··· ◦ suk) = Vect(u1, . . . , uk)
(u1, . . . , uk)v1, . . . , vk∈E\ {0}
su1◦ ··· ◦ suk=sv1◦ ··· ◦ svk
(v1, . . . , vk)
m, n ∈N∗M∈ Mm,n(R)
U∈ Om(R)V∈ On(R)N=UMV
∀(i, j)∈ {1, . . . , m}×{1, . . . , n}, i 6=j=⇒Ni,j = 0

H H 6={0}
a= inf {h > 0|h∈H}
a6= 0 a∈H x ∈H
a H =aZa= 0
ε > 0α∈H∩]0 ; ε]αZ⊂H
x∈Rh∈αZ⊂H|x−h| ≤ α≤ε H
R
x∈R\QN∈N∗
f:{0, . . . , N} → [0 ; 1[
f(k) = kx − bkxcN+ 1 f
N[i/N ; (i+ 1)/N[i∈ {0, . . . , N −1}
k < k0∈ {0, . . . , N}
|f(k0)−f(k)|<1
N
p=bk0xc−bkxc ∈ Zq=k0−k∈ {1, . . . , N}
|qx −p|<1/N
x−p
q
<1
Nq <1 q2
N(p, q)
(p, q)∈Z×N∗
x−p
q
<1
q2
π pn/qn
π−pn
qn
<1
q2
n
qn→+∞
|upn|=
1
pnsin pn
=
1
pnsin (pn−qnπ)
≥1
|pn|
1
|pn−qnπ|≥qn pn→1 π
(un)
{|sin n| | n∈N}={|sin(n+ 2kπ)| | n∈Z, k ∈Z}=|sin (Z+ 2πZ)|
H=Z+ 2πZπ
HR|sin(.)|
R[0 ; 1] {|sin n| | n∈N}[0 ; 1]
n|sin n| ≥ 1/2
|un| ≤ 2/n
(un)
(un)
p∈N∗Ap=On
P B =AP (A)
B=A+a2A2+··· +ap−1Ap−1
B2=A2+a3,2A3+···+ap−1,2Ap−1Bp−2=Ap−2+ap−1,p−2Ap−1, Bp−1=Ap−1
Ap−1=Bp−1, Ap−2=Bp−2+bp−1,p−2Ap−1A2=B2+b3,2B3+···+bp−1,2Bp−1
A=B+b2,1B2+··· +bp−1,1Bp−1
Q∈R[X]Q(0) = 1 A=BQ(B)
p◦p=p◦(q◦p) = (p◦q)◦p=q◦p=p p q
p q p q
p q p =q◦p=p◦q=q
a=b= 0
f g
Cf
λ Eλ(f)6={0}g

g Eλ(f)
g
f Eλ(f)f g
a= 0 b6= 0
f◦gn−gn◦f=nbgnn∈N
u∈ L(E)7→ f◦u−u◦fL(E)
dim L(E)<+∞n∈Ngn6=˜
0nb
n∈Ngn=˜ 0
ker g6={0}ker g f
fker g f g
b= 0 a6= 0
a6= 0 b6= 0
f◦(af +bg)−(af +bg)◦f=b(f◦g−g◦f) = b(af +bg)
f af +bg
f g
M∈ Mn(C)Mn=On
(A+xB)n
n λ1, . . . , λn, λn+1
x∈C(A+xB)n=On
x= 0 An=Ony6= 0 y= 1/x
(yA +B)n=Ony→0Bn=On
ϕ(I2)=1 P ϕ(P−1) = ϕ(P)−1A B
ϕ(A) = ϕ(B)
µ0
0 1 1 0 0µϕ1 0 0µ=µ
ϕλ0
0µ=λµ A ϕ(A) = det A
A A M2(C)
λ α
0λ
λ= 0 A2= 0 ϕ(A) = 0 = det A
λ6= 0 λ α
0λ1 0 0 2=λ α 0 2λ λ α 0 2λ
2ϕ(A) = 2λ2= 2 det A
f F f
fFF
f
f F
f
f∈ L(E)
F=⊕
λ∈Sp fEλ(f)f
f
{0}CF=E f
I1u P1=πu
u
I2
(K[X],+)
I2K[X]I1
I1⊂ I2P1∈P2K[X]P2|P1
n E
v E vn=˜ 0P2(u)
(P2)n(u) = ˜ 0P1|Pn
2
P2|P1P1|Pn
2P2P1
u
P2=Y
λ∈Sp u
(X−λ)αλ
P2(u)
Qλ∈Sp u(u−λIdE)