Applications linéaires

f:R[X]→R[X]
P7→ XP
R[X]
a)f:R2→R2
(x, y)7→ (y, 2x)b)g:R3→R2 (x, y, z)7→ (x−z, y +z)
c)h:R2→R2
(x, y)7→ (x−y, 3x−3y)d)i:R2→R3 (x, y)7→ (x−y, y−x, x + 2y)
e)j:R2→R2
(x, y)7→ (xy, 0) f)k:R3→R2 (x, y, z)7→ (x−z, 1 + y)
E=D2(R,R)R
u:E→RR
f7→ f00 + 4f
E f g E
f◦g=g◦f(f) (f)g
u v E
v◦u= 0L(E)(u)⊂(v)
u E
(u) = (u2)⇐⇒ (u)∩(u) = {0E}.
E=Rn[X]f:Rn[X]→Rn[X]
P7→ P+ (1 −X)P0
(f) (f) (f) (f)
E
u∈L(E)E k ≥0
(uk)⊂(uk+1)
(uk) = (uk+1)⇒(uk+1) = (uk+2).
E=R3[X]ϕ E P
(X4−1)P X4−X
ϕR3[X]
(ϕ) (ϕ)
F1F2p F1
F2s F1F2
s p IdE
p f p ◦f=f◦p
(p) (p)f
f:P7→ X3P(1
X)R3[X]
RR

E n ∈N∗u∈L(E)
u u
p up= 0
x6∈ (up−1)
(x, u(x), u2(x), . . . , up−1(x))
un= 0
E u E u
∀x∈E∃p∈Nup(x)=0E.
E
E=K[X]
f:R2→R2
(x, y)7→ (x+y, x −y)
f
(a1, . . . , an)∈Rn
ϕ:Rn−1[X]→Rn
P7→ (P(a1), . . . , P (an)) .
ϕ
n(b1, . . . , bn)∈Rn
PRn−1[X]
∀i∈J1, nKP(ai) = bi.
EKu E
u2−3u+ 2IdE= 0L(E).
u E
E= (u−IdE)⊕(u−2IdE)
E
(x1, . . . , xn)E∀i∈J1, nKu(xi) = xiu(xi) = 2xi
G={P∈R3[X] : P(0) = P(1)}
F
G
G
f E n
E= (f)⊕(f) =⇒(f) = (f2)
(f) = (f2)⇐⇒ (f) = (f2)
(f) = (f2) =⇒E= (f)⊕(f)