Université F. Rabelais 2012-2013 L3S2 Algèbre Feuille d

EK R C
e1, ..., enE(e1, ..., en)
(x→cosn(x))n≥0C0(R)
F G H E F +H=G+H F =G
F G H E F ⊕H=G⊕H F G
F G E E F ∪G6=E
u∈ L(E)E= Keru⊕Imu u
u v E Im(u+v) = Imu+ Imv
u∈L(E)G H E u(G+H) = u(G)+u(H)
u v E u ◦v= 0 Imv⊂Kerv
u E F dimE > dimF u
u E F dimE > dimF u
u v E u ◦v u
v
F E u ∈L(F, E)v v(x) = u(x)
x∈F v(x) = 0 x /∈F E
F E u ∈L(F, E)v E
F u
Mn(K)Mn(K)
A B C ∈ Mn(K)M=A C
0BM2n(K)
A B
Mn(C)
A B Mn(C) rg(AB) = rg(BA)
Mn(K)K
A∈ Mn(C)λ∈CA−λIn
A B Mn(C)AB =BA = (detA)InB
A
x∈Edim(E)−1
l∈E∗l(x)=1
(e1, ..., en) (f1, ..., fn)E M ∈ Mn(R) (ei) (fi)
∀i∈J1, nK, fi=Pn
j=1 Mi,j ej(e∗
i) (f∗ i)TM
b:Rn×Rn7→ RB∈ Mn(R)
∀x, y ∈Rn, b(x, y) = TxBy.

ER(e1, e2, e3)E f∗
1, f∗ 2, f∗ 3∈E∗
f∗
1= 2e∗ 1+e∗ 2+e∗ 3f∗ 2=−e∗ 1+ 2e∗ 3f∗ 3=e∗ 1+ 3e∗ 2.
(f∗
1, f∗ 2, f∗ 3)E∗(f1, f2, f3)E
ER R I P
I P E
E=IMP.
f∈EI P
n≥2φR[X]
φ(P)(X)=(X+ 2)P(X)−XP (X+ 1).
φRn[X]
Ker(φ)
n≥1U=
1
1
∈ Mn,1(R)
l:(Mn,1(R)2→ Mn(R),
(X, Y )7→ XtU+UtY.
l