Algèbre linéaire - rappels et compléments

E E x ∈E
(x, u(x))
E
E1E2E E1∪E2E
E1⊂E2E2⊂E1
E u ∈ L(E) 2u3+ 5u2−3u= 0 n>3un
u u2
EKf, g ∈ L(E)f◦g=E
Ker (g◦f) = Ker (f)
Im (g◦f) = Im g
E= Im (g)⊕Ker (f)
n>2p E =Rnr∈[[1, n−1]]
pL(E)E p
E u ∈ L(E) Ker (u) = Im (u)
u◦u= 0 dim(E) = 2 (u)
E f ∈ L(E)
Im (f)⊕Ker (f) = EIm (f) = Im (f2)
Im (f) = Im (f2) Ker (f) = Ker (f2)
E n f g L(E)f+g=E(f)+ (g)6n
Im (f)⊕Im (g) = E f g
E f, g ∈ L(E)E= Im (f) + Im (g) = Ker (f) +
Ker (g)
H1H2E H1⊂H2H1=H2

a∈Rfa:x∈R7−→ cos(x+a)
a1, a2, a3∈R(fa1, fa2, fa3)
E H E u ∈ L(E)H
u λ ∈KIm (u−λE)⊂H
E n >1 (ek)16k6n∈En
Φ
∀u∈ L(E),Φ(u)=(u(e1), u(e2), ..., u(en)
(ek)16k6nEΦ
f∈ L(E)E n >2f n
x0∈Ex0, f(x0), ..., fn−1(x0)E
Mat(f, B)
g∈ L(E)g◦f=f◦g g ∈Vect( E, f, ..., fn−1)
E3f E f2= 0
E f 
001
000
000
p>3g∈ L(E)gp=f
g2=f g f
g∈ L(E)g2=f
RE f E
Im (f) = Im (f2)E= Im (f) + Ker (f)
RE n ∈N∗σ∈ L(E)σ2=−E
n
p=n/2E u
(0) −Ip
Ip(0) 
E f, g ∈ L(E)f2=g2=E
f◦g+g◦f= 0
dim(E)
E f g
In0
0−In 0In In0
E3f1, f2, f3E f2
1=f2 2=f2 3= 0
fi◦fj= 0 i6=j f1◦f2◦f3= 0

n∈N∗P=X2n−2 cos(nθ)Xn+ 1 PC[X]
R[X]
(X+i)2m+1 −(X−i)2m+1 m∈N∗
m
Y
k=1
cotan kπ
2m+ 1·
H0= 1 H1=X n ∈N∗
Hn=X(X−1) . . . (X−n+ 1)
n!·
H15(20)
H15(−10)
n∈Np∈ZHn(p)∈Z
P∈R[X]
p∈ZP(p)∈Z
λ0, ..., λn∈ZP=λ0H0+λ1H1+· · · +λnHn
n∈NTn, Un∈R[X]θ∈Rcos(nθ) = Tn(cos θ)
sin ((n+ 1)θ) = sin θUn(cos θ)
TnUnTnUnn∈N
TnUn
(Bn)n∈NB0= 1 n∈N
B0
n+1 = (n+ 1)BnZ1
0
Bn= 0
B1B2
Bn
Bn(1 −X)=(−1)nBn(X); Bn(X+ 1) −Bn(X) = nXn−1;
Bn(0) = Bn(1) B2n+1(0) = 0.
E=Rn[X]f∈ L(E)f(P) = P−P0
f
Q∈R[X]P∈R[X]f(P) = Q
ϕ E =Rn[X]ϕ(P) = P(X)−P(X−1)
ϕ E
ϕ

P∈R[X]RP0R
r∈Rn>2
P∈Rn−1[X]P(k) = rkk∈[[1, n]]
P(n+ 1)
P=X3−X2−1
P
P
P=a0+· · · +anXnan6= 0 k∈[[0, n]] ak=an−k
P n P (X) = XnP(1/X)
1P1 2
−1
P P
P=α
p
Y
k=0
(X2+bkX+ 1)
P0, ..., Pn∈Kn[X] (P0, ..., Pn) (a0, ..., an)∈Kn+1
((Pi(aj))06i,j6n
(a0, ..., an)∈Kn+1 (a0, ..., an)
n∈N∗Φn:P∈Rn[X]7−→ Q−XR Q R P (X2)
An= 1 + X+· · · +Xn
ΦnRn[X]
Φ3R3[X]
n∈N∗∆∈ L(Rn[X]) ∆(P) = P(X+ 1) −P(X)
∆
∀P∈Rn−1[X],
n
X
j=0
(−1)n−jn
jP(X+j)=0.
B=−8−14
3 5 
Bn
(un) (vn)un+1 =−16un−28vn
vn+1 = 6un+ 10vn
unvnu0v0

E
E
A∈ Mn(C) (A) = (A)=1 A2=A
X, Y ∈ Mn,1(R)
M=Xt Y
M−λIn
A, B ∈ Mn(R) (AB −BA) = 1 (AB −BA)2
A, B ∈ Mn(C)M∈ Mn(C)M+ ( M)A=B
T
T
T
T∈ T
ψ:M7→ MT T
ψ ψ
T−1∈ T
A∈ M3,2(R)B∈ M2,3(R)
AB =
1 1 2
−2x1
1−2 1
x
A B
a, b ∈C
a+b ab
1a+b ab (0)
1
(0) ab
1a+b
A=Diag(1,2,1) M∈ M3(R)7→ AM +MA
u
u(e1)u(e1+e2)