exercices

R[X]
(P, Q)7→
∞
P
n=0
P(n)Q(n)−nR[X]
P, Q ∈E=R[X]<P |Q>=Z1
0
P(t)Q(t)
√1−tdt
E
E
A∈Rn+1[X]P∈E RA(P)
P A
RA
RA
P, Q ∈E=R2[X]
<P |Q>=P(−1)Q(−1) + P(0)Q(0) + P(1)Q(1)
(P0, P1, P2)E
A, B ∈ Mn(R)<A|B>= (t AB)
<·|·>
M∈ Sn(R)N∈ An(R)<M|N>= 0
A∈ Mn(R)
inf
S∈Sn(R)X
16i,j6n
(ai,j −si,j )2
f, g ∈E=C2([0,1],R)ϕ(f, g) = Z1
0
(fg +f0g0)
H1={f∈E;f(0) = f(1) = 0}H2={f∈E;f00 =f}
ϕ
OT QR
E1E2E3EPas
E1→E3
Pas
E1→E2
Pas
E2→E3

P∈GLn(R)O∈On(R)T P =OT
P=3 1
−1 2P=1
9
8 25 −22
−4−8−7
1 11 −14
E=C([0,1],R)f(t) = (tln tsi t6= 0
0 si t= 0
Z1
0
(f(t)−(at +b))2dt a b R
n∈NP∈Rn[X]Q∈Rn[X]
Q(945) −1515Q0(42) = Z2048
2016
P(t)Q(t) sin2tdt.
E=R3
A
−→
n=
1
1
1
A
R3F= (f1, f2, f3)f1= (1,1,1) f2= (1,−1,1) f3=
(1,0,0)
R4
R4H= Ker ϕ
ϕ(x1, x2, x3, x4) = x1+x3
R2[X]
<P |Q>=Z1
−1
P(t)Q(t)dt.
Pn=1
n!(X2−1)n(n)
R[X]
Pnn
]−1,1[
kPnk2
R2[X]
<P |Q>=Z]−1,1[
P(t)Q(t)
√1−t2dt.

Tn(cos θ) = cos(nθ)
R[X]
Tn
R2[X]
<P |Q>=Z1
−1
P(t)Q(t)p1−t2dt.
sin θ Un(cos θ) = sin ((n+ 1)θ)
R[X]
Un
w
I
E=R[X]
∀P, Q ∈E, <P |Q>=ZI P(t)Q(t)ϕ(t)dt.
(Pn)n∈NE
Pnn I
M∈ Mn(R)uRnH
a1x1+···+anxn= 0 H u 
a1
an
t
M
F G E (x1, ..., xn)
(y1, ..., yn)
∀(i, j)∈[[1, n]]2, <xi|xj>=<yi|yj>
(x1, ..., xn) (y1, ..., yn)
F G
p E
p
p
p
x∈Ekp(x)k6kxk
(e1, ..., en)E f ∈ L(E)
∀x, y ∈E, <x|y>= 0 =⇒<f(x)|f(y)>= 0

(f(e1), ..., f(en))
<f(ei) + f(ej)|f(ei)−f(ej)> f(ei)
f
n∈N∗M∈On(R)
X
i,j
mi,j  6n6X
i,j |mi,j |6n3/2.
f∈O(E)E(>2n∈N∗
gn=1
nE+f+f2+··· +fn−1
Ker (f−E) Im (f−E)
x∈Ker (f−E)gn(x)
x∈Im (f−E)
g= lim
n→∞ gn
T E
∀u, v ∈E, <u|T(v)−v>=<T −1(u)−u|v>
S=T−(Im (S))⊥= Ker (S)
n∈N∗Tn=1
n+T+T2+··· +Tn−1x∈E
(Tn(x))n∈N∗P(x) 0
P
E
1
53−4
4 3 1
53 4
4−3
a, b ∈E x ∈E a ∧x=b
a b
R3(2e1+e2)⊥
(e1, e2, e3)R3
e1+e3e1−e3
π
2
π
3
R3
M=1
3
2−2−1
212
−1−2 2 

R3
M=1
3
2 2 1
1−2 2
2−1−2
(a, b, c)∈R3
a c b
b a c
c b a
SO3(R)t∈[0,4/27]
a, b c X3−X2+t
v E n
S=
n
P
i=1
<v(ei)|ei>(e1, ..., en)
T=
n
P
i=1
n
P
j=1
<v(ej)|fi>2E F
T v r
E n n X= (x1, ..., xn)
Y= (y1, ..., yn) (i, j)<xi|xj>=<yy|yj>
n= 2 X Y ψ i ∈[1, n]] ψ(xi) = yi
(fxk)16k6n
∀k∈[[1, n]], <xk|fxk>= 0 Vect(x1, ..., xk) = Vect(fx1, ..., fxk)
E v1, ..., vp∈E
G(v1, ..., vp) = ((<vi|vj>))16i,j6p∈ Mp(R)
(v1, ..., vp) det (G(v1, ..., vp)) = 0
x∈E F = Vect(v1, ..., vp)
d(x, F )2=det (G(v1, ..., vp, x))
det (G(v1, ..., vp))
a E α ∈Ruα∈ L(E)
∀x∈E, uα(x) = x+α <x|a> a
α uα
∈ Mn(R)t M M =Mt M M2=−InM∈On(R)