Exercices - Page Personnelle de Jérôme Von Buhren

n∈Nfn:R→Rfn(x) = xnex
(fn)n∈NF(R,R)
n∈N∗fn:R→Rfn(x) = cos(nx)
(m, n)∈N∗×N∗
Im,n =Z2π
0
cos(mt) cos(nt)dt.
(fn)n∈N∗F(R,R)
a∈Rfa:R→Rfa(x) = eax
(fa)a∈RF(R,R)
F= Vect((1,0,0,1)) G= Vect((1,1,1,1))
H1={(x, y, z, t)∈R4|x=z, y =t},
H2={(x, y, z, t)∈R4|x+t= 0, y +z= 0}.
R4=F⊕G⊕H1R4=F⊕G⊕H2
E=C∞(R,R)
F={f∈E|f(0) = f0(0) = 0}, G ={f∈E|f},
H={f∈E|f}.
E=F⊕G⊕H
uR3
1 1 1
0 1 0
1 0 1
.
H={(x, y, z)∈R3|x+y+z= 0}D= Vect(1,0,1).
R3=H⊕D
H D u
u
uR3
2 2 1
1−2 2
2−1−2
.
H= Vect(1,−3,0),(0,1,−1)D= Vect(3,1,1)
R3=H⊕D
H D u
uB=(1,−3,0),(0,1,−1),(3,1,1).
uR2[X]
∀P∈R2[X], u(P) = X2P1
X.
H= Vect(1, X2)D= Vect(X)
R2[X] = H⊕D
H D u
uB= (1, X2, X)

E u ∈L(E) Ker(u)
Im(u)u
H={M∈Mn(R)|Tr(M)=0}
Mn(R)
A, B ∈Mn(C)AB −BA = In
A, B ∈Mn(C)AB −BA =A
p∈N∗Tr(Ap)=0
u∈L(R3)
(i)u(x, y, z)=(x+y+z, x +z, y +z),(ii)u(x, y, z)=(x−y, y −z, z −x).
u∈L(R3[X])
(i)u(P) = P+P0,(ii)u(P) = P(X+1)−P(X),(iii)u(P) = XP 0+P(1).
E u ∈L(E)
1
BEMatB(u)
u2= Tr(u)·u
E p ∈L(E)
E= Ker(p)⊕Im(p)
p E = Ker(p)⊕Im(p)
Tr(p) = rang(p)
Sn(R)An(R)
Mn(R)
Sn(R)An(R)Mn(R)
Mn(R) = Sn(R)⊕An(R)
1 1 1
0 1 1
0 0 1
.
Sn(R)An(R)
A, B ∈Mn(R)
AB A B
M∈Mn(R)MMTMTM
M∈Mn(R)X=x1· · · xnT∈Mn,1(R)
XTX x1, . . . , xnXTX= 0
X= 0
Ker(M) = Ker(MTM)