Devoir maison 7

n1,2, ..., n
A=
3−3 2
−1 5 −2
−1 3 0
R∈ M3(R)R2=A
R
fa b
c d ∈ M2(R)d2b 2c a 
f
φMn(R)φ(M) = M−tr(M)Id
M∈GLk(C)M2M
(A, B)∈GLn(C)2N=0A
B0N2
A B
A∈ Mn(C)
M∈GLn(R)M2+tM=In
M M −In
φ φ(P)(X)=(X−a)(X−b)P0(X)−nXP (X)a6=b
Rn[X]
E(u, v)∈ L(E)2λ6= 0 v◦u
λ u ◦v
E λ = 0
E=R[X]u(P) = P0v(P) = Q Q P
Ker(u◦v)Ker(v◦u)
f E n n
g f f g
f g
P∈Rn−1[X]g=P(f)