Exercice - Alain Camanes

2015/2016
EK K =R C
Exercice 1. (-)
1. E=R?
+×RE(a, b)⊕(a0, b0) = (aa0, b +b0)
R×E λ (a, b)=(aλ, λb) (E, ⊕,)R
2. A⊂RE={f∈F(R,R) ; ∀x∈A, f(x)=0}ER
Exercice 2. (-)
1. E1={(xi)∈J1,nK∈Rn;x1= 0 x2= 0}
2. E2={(x1, . . . , xn)∈Rn;x1+x2= 0}
3. E3={(x1, . . . , xn)∈Rn;x16= 0}
4. E4={(x1, . . . , xn)∈Rn;x1=x2}
5. E5={(x1, . . . , xn)∈Rn;x1x2= 0}
6. E6={u∈S(R) ; lim
n→+∞(un+1 −un)=0}
7. E7={u∈S(R) ; lim
n→+∞(un+1 −un)=1}
8. E8={f∈C(R,R) ; f(0) = 0}
9. E9={f∈C(R,R) ; |f(0)|= 2}
Exercice 3. ( ! )A, B E
(i)A∪B E
(ii)A∪B=A+B
(iii)A⊂B B ⊂A
Exercice 4. (-)L, M, N E
1. L∩(M+N)=(L∩M)+(L∩N)
2. L∩(M+ (L∩N)) = (L∩M)+(L∩N)
Exercice 5. (♥)F, G, H E
1. F∩G=F∩H F +G=F+H G ⊂H G =H
2. G H F
G⊂H G =H
3.
Exercice 6. (-)u= (0,1,2,3) v= (3,2,1,0) w= (1,1,1,1) R4
(x, y, z, t) Vect{u, v, w}
Exercice 7. (-)R3E1=
{(0, y, z), y, z ∈R}E2= Vect{u, v}u= (1,2,3) v= (1,3,4) E1∩E2
E1+E2
Exercice 8. n>2E1, . . . , EnE E1+··· +En
i∈J1, nK, Ei∩ P
j6=i
Ej!={0E}
Exercice 9. ( ! )EC∞
2π D E
E= Ker(D)⊕Im(D)

Exercice 10. (-)p∈N?
1. (fk)k∈[|1,p|]x∈Rfk(x) = δxk δ
2. (f1, f2, f3, f4)x∈Rf1(x) = cos x, f2(x) = sin(x), f3(x) = xcos x, f4(x) =
xsin x
3. (fk)k∈[|1,p|]x∈Rfk(x) = ekx
4. (fk)k∈[|1,p|]x∈Rfk(x) = |x−k|
Exercice 11. (-)
1. E1={(x, y, z)∈R3;x+ 2y+z= 0 2x+y+ 3z= 0}
2. E2={(x, y, z, t)∈R4;x+y= 0 2x−z+t= 0}
Exercice 12. (-)
1. B1=(1,0,0),(1,1,0),(1,1,1)R3
2. B2=(1,3,5),(2,5,−2)R3
3. B3=(X−1)2,(X−1)(X+ 1),(X+ 1)2R2[X]
4. B4=(X−1)(X+ 1), X2,1R2[X]
Exercice 13. (Pi)i∈N
i<j 06deg Pi<deg Pj(Pi)i∈N
Exercice 14. n>2E1, . . . , EnE{0E}
E1+···+En
E1× ··· × En
Exercice 15. ( ! )P(ln(p))p∈P
Q R
Exercice 16.
f∈F(R,R) ; ∃(A, θ)∈R2;∀t∈R, f(t) = Acos(t+θ).
Exercice 17. (-)
1. f1:R3→R2,(x, y, z)7→ (−x+ 2y, 2x−3y+z)
2. f2:R3→R3,(x, y, z)7→ (2x+y−z, x −y+ 3z, 4x+y−z)
3. f3:R3→R3,(x, y, z)7→ (y+z, x +y+z, x)
4. f4:R3→R2,(x, y, z)7→ 2(x+y, x −y)
5. f5:R3→R2,(x, y, z)7→ z(x+y, x −y)
6. f6:R3→R2,(x, y, z)7→ 2(x+y+z, x −y)
Exercice 18. ( ! )a∈Cfa:C→C, z 7→ z+az faR
a fa

Exercice 19. (Polynômes d’endomorphismes, ♥)EKu∈L(E)
P(u) = a0IdE+
n
P
k=1
akuk∈L(E)
1. P, Q ∈K[X] (P+Q)(u) = P(u) + Q(u) (P Q)(u) = P(u)◦Q(u)
2. Ker P(u) Im P(u)u
Exercice 20. (-)TR[X]P
T(P)
T(P) = (8 + 3X)P−(5X−X2)P0+ (X2−X3)P00.
T
Exercice 21. (-)f E f
Exercice 22. (♥)u∈L(E, F )v∈L(F, G)v◦u= 0
Im(u)⊂Ker(v)
Exercice 23. (♥)u∈L(E)u3=uIm(u2) Ker(u)
E
Exercice 24. (♥)f, g ∈L(E)f◦g=g◦fKer(f) Im(f)
g
Exercice 25. ( ! )E=C∞(R,R) +,·
ϕ:E→E, y 7→ y0−xy E1={y∈E;y(0) = 0}
1. (E1,+,·)
2. ϕ E1E
Exercice 26. (♥,
! )f∈L(E)x∈E(x, f(x)) f
Exercice 27. E=C1([0,1],R)
F={f∈E;f(0) = f0(0) = 0}, G ={g: [0,1] →R, x 7→ ax +b, (a, b)∈R2}.
1. F G E
2. F G G F
Exercice 28. (-)u∈L(R3) (x, y, z)∈R3u(x, y, z) = 1
3(x+2y+2z, 2x+
y−2z, 2x−2y+z)u
Exercice 29. (♥)p q E
1. p◦q=p⇔Ker(q)⊂Ker(p)
2. p+q p ◦q=q◦p= 0
3. p◦q p ◦q=q◦p
Exercice 30. (-)p E IdE+p
E p

Exercice 31. (-)p1, p2p1◦p2= 0 q=p1+p2−p2◦p1
qIm(p1) + Im(p2) Ker(p1)∩Ker(p2)
Exercice 32. ( ! )n, N ∈N?u∈L(CN)un= Id E
CNu p E
q=1
n+ 1
n
X
k=0
uk◦p◦un−k.
1. q
2. CN=E⊕Ker q
Exercice 33. E=
p
L
k=1
Eki∈J1, pKpi:E→E,
p
P
k=1
xk7→ xi
(i, j)∈J1, pK2
1. pi
2. i6=j⇒pipj= 0L(E)
3. IdE=
p
P
i=1
pi