Espaces de dimension finie 1 Dimension d`un espace vectoriel

n∈N
∀k∈J1, nK, Hk=X(X−1) · · · (X−k+ 1)
k!.
(H0, . . . , Hn)Rn[X]
∀k∈J1, nK, , ∀x∈Z, Hk(x)∈Z.
P∈Rn[X]P(x)∈Zx∈Z
n∈NPk=Xk(1 −X)n−kk∈J0, nK
(P0, . . . , Pn)Kn[X]
n∈N
∀P∈Rn[X],∃Q∈Rn[X], P =
n
X
k=0
Q(k).
a0, . . . , an∈R
ϕ:Rn[X]→Rn+1, P 7→ (P(a0), . . . , P (an)) .
ϕ
a0, . . . , an∈R
ϕ:R2n+1[X]→R2n+2, P 7→ P(a0), P 0(a0), . . . , P (an), P 0(an).
ϕ
p∈N∗
E=n(un)n∈N∈RN
∀n∈N, un+p=uno.
E
n∈Nn>2
f:Rn[X]→Rn[X], P 7→ P(X+ 1) + P(X−1) −2P(X).
fRn[X]
f(Xk)k∈J0, nK
Ker(f) Im(f)
Q∈Rn−2[X]
∃!P∈Rn[X], f(P) = Q P (0) = P0(0) = 0.
f∈L(E)
E
fn6= 0 x∈E(x, f(x),...fn(x))
E
fdim E= 0
E f ∈L(E)
∀x∈E, ∃n∈Nfn(x)=0.
f
E

E n f ∈L(E)
(Id, f, f2, . . . , fn2)
P∈K[X]P(f) = 0
f∈GL(E)P∈K[X]f−1=P(f)
u∈L(E)E
n>1
∀p∈N, Ip= Im(up), Np= Ker(up).
(Ip)p>0(Np)p>0
s∈NIs+1 =IsNs+1 =Ns
r s
∀s>r, Is=Ir, Ns=Nr.
IrNrE
f∈L(E)E
n∈N∗a∈E
(a, f(a), . . . , fn−1(a)) E
C={g∈L(E)|g◦f=f◦g}.
CL(E)
C= Vect(Id, f, . . . , fn−1)
C
u1, . . . , un∈L(E)
E n
∀k∈J0, n −1K, Fk= Im(uk+1 ◦ · · · ◦ un).
Fk6={0}Fk+1 Fk
u1◦ · · · ◦ un= 0
f∈L(E, F )E F
G E
dim f(G) = dim G−dim(G∩Kerf).
H F
dim f−1(H) = dim(H∩Im(f)) + dim(Kerf).
f, g ∈L(E)
E
dim Ker(g◦f)6dim Ker(f) + dim Ker(g).
f, g ∈L(E, F )E F
dim Ker(f+g)6dim(Ker(f)∩Ker(g)) + dim(Im(f)∩Im(g)).
f∈L(E)E
rang(f) = 1
λ∈Kf2=λf
λ= 1 Id −f
f, g ∈L(E)
E
rang(f) + rang(g)−dim E6rang(f◦g)6min(rang(f),rang(g)).

f, g ∈L(E)
E
|rang(f)−rang(g)|6rang(f+g)6rang(f) + rang(g).
f◦g= 0 f+g
rang(f) + rang(g) = dim E.
f, g ∈L(E)
E
rang(f+g) = rang(f) + rang(g)⇔Im(f)∩Im(g) = {0}
Ker(f) + Ker(g) = E.
u∈L(E)E
u3= 0
rang(u) + rang(u2)6dim E
2·rang(u2)6rang(u)
u, v ∈L(E)
E
Ker(u) + Ker(v) = Im(u) + Im(v) = E.
u, v ∈L(E)
E
u+v= Id rang(u) + rang(v)6dim(E).
u v
E u ∈L(E)
ur= Id r∈N∗
p=1
r
r−1
X
k=0
uk.
pKer(u−Id)
dim Ker(u−Id) = 1
r
r−1
X
k=0
Tr(uk).
E G
GL(E)
F=\
g∈G
Ker(g−Id) p=1
|G|X
g∈G
g.
p F
|G| · dim(F) = X
g∈G
Tr(g).
E G
GL(E)
p=1
|G|X
g∈G
g.
p
X
g∈G
Tr(g)=0 ⇒X
g∈G
g= 0.

p1, . . . , pr∈L(E)E
p1+· · · +pr= IdE∀i∈J1, rK, p2
i=pi.
pi◦pj= 0 (i, j)∈J1, rK2i6=j
E n F G
E p < n F G
E n F
E F
F1, . . . , Fr
E E =F1+· · ·+Fr
G1⊂F1, . . . , Gr⊂FrE=G1⊕ · · · ⊕ Gr
f∈L(E)E
F1={g∈L(E)|f◦g= 0}F2={g∈L(E)|g◦f= 0}.
F1F2L(E)
F1L(E, Ker(f))
F1
SIm(f)E F2
L(S, E)F2
f∈L(E)E
F={g∈L(E)|f◦g=g◦f= 0}.
FL(E)
SIm(f)E F
L(S, Kerf)
F
u, v ∈L(E)
E
ϕ:L(E)→L(E)f7→ u◦f◦v
SIm(v)EKer(ϕ)
L(Im(v),Ker(u)) ×L(S, E)
ϕ
E F
(u1, . . . , un)E
ϕ:L(E, F )→Fn, f 7→ (f(u1), . . . , f (un))
dim L(E, F )
u∈L(E, F )v∈L(E, G)E F G
Ker(u)⊂Ker(v)⇔ ∃f∈L(F, G), v =f◦u.

u∈L(E, G)v∈L(F, G)E F G
Im(u)⊂Im(v)⇔ ∃f∈L(E, F ), u =v◦f.
f∈L(E)E
(p, g)∈L(E)2p g
f=g◦p
(p, g)∈L(E)2p g
f=p◦g
f∈L(E, F )E F
f
(∀g∈L(E), f ◦g= 0 ⇒g= 0).
f
(∀g∈L(F), g ◦f= 0 ⇒g= 0).
F G
E
f∈L(E)f(F) = G
f∈GL(E)f(F) = G
f∈L(E) Kerf=FImf=G
E
f∈L(E) Kerf= Imfdim(E)
f∈L(E)f2= 0 E
∃g∈L(E), f ◦g+g◦f= Id ⇔Im(f) = Ker(f).
f1, . . . , fp, f ∈L(E, K)
E
f∈Vect(f1, . . . , fp)⇔
p
\
i=1
Ker(fi)⊂Ker(f).
x1, . . . , xp∈E E
ϕ:L(E, K)→Kp, f 7→ (f(x1), . . . , f (xp)).
ϕ(x1, . . . , xp)
ϕ(x1, . . . , xp)
f1, . . . , fp∈L(E, K)
E
ϕ:E→Kp, x 7→ (f1(x), . . . , fp(x)).
ϕ(f1, . . . , fp)
ϕ(f1, . . . , fp)
E
ϕ∈L(E)E