Fiche 1 : Algèbre - PCSI

m
(m−1)x+y+z= 0
x+ (m−1)y+z= 0
x+y+ (m−1)z= 0
f:R3→R3
(x, y, z)7→ (x−y−z, −x+y−z, −x−y+z)
fR3
A f
m1∆ = Ker(f−m1Id)m2
P=Ker(f−m2Id)
∆PR3
1∆23P
D f B= (1, 2, 3)R3
n DnAn
ER Rn[X]n∈Nn≥2
(P0, P1,· · · , Pn)E
P0= 1 ; P1=X;Pk=1
k!X(X−k)k−1(2 ≤k≤n).
f E E
∀P∈E f(P) = Q Q(X) = P(X)−P0(X+ 1) .
(P0, P1,· · · , Pn)E
∀k∈ {1, ..., n}Pk
0(X+ 1) = Pk−1(X)
k∈ {0, ..., n}Qk=f(Pk)
Qk0≤k≤n Pj0≤j≤n
f E f∈GL(E)
Pk0≤k≤n Qj
n= 3
X3B= (P0, P1, P2, P3)
PR3[X]
P(X)−P0(X+ 1) = X3.
P