Algèbre linéaire: généralités 1. E est un K

2015 −2016
EKn n > 0f∈ L(E)
a∈E ϕ E
K
∀x∈E, f(x) = ϕ(x).a
g=IdE+tf t ∈K g
E=C(R,R)n fn:x7→ cos(2x+n)
N(f0, ..., fN)
F1F2KE F1∪F2
F1⊂F2F2⊂F1F1∩F2=F1+F2F1=F2
E, F, G Ku∈ L(E, F )v∈ L(F, G)
rg(v◦u)≤inf(rg(u), rg(v))
v0=v/u(E)rg(u) + rg(v)−dim(F)≤rg(v◦u)
E=F(R,C)
ep:x7→ exp(ipx)p∈Z
f, g E
E=Im(f) + Im(g) = Ker(f) + Ker(g)
E1={(x, y, z, t)∈R4/−x−2y+ 2z+t= 0}D=V ect((1,2,1,1)) E1
D
F1={(x, y, z, t)∈R4/2x−y+z=y−2z+t= 0}
F2={(x, y, z, t)∈R4/x + 2y−t=x+ 2z−t= 0}
F1F2
R[X]B0= 1
Bp=X(X−1)...(X−p+ 1)
(Bp)p∈NR[X]
Z
E, F KW E
A={u∈ L(E, F )/W ⊂Ker(u)}AL(E, F )
E=C([0,2],R)F E f E
F
EKn n > 0 (e1, ..., en)E
(λ1, λ2, ..., λn)Kni i = 1..n ui=u+ei
u u =
n
X
i=1
λiei
(u1, ..., un)E
n
X
i=1
λ16=−1

2015 −2016
E, F Kf∈ L(E, F )
g∈ L(F, E)f◦g=IdF
E=C([0,1],R)f∈E T (f) [0,1]
R∀x∈[0,1] T(f)(x) = Zx
0
f(4(t−t2))dt
T E T
EKf∈L(E)Im(f)
f Im(f)∩Ker(f) = {−→ 0E}
Kerf Imf
f
p, q E p +q p ◦q=
q◦p=O
p+q Im(p+q)Ker(p+q)
EKf∈L(E)
(i) : f2=O
(ii) : ∃(g, h)∈(L(E))2/g ◦h=f h ◦g=O
E=R4BE f ∈ L(E)AB
A=
2 3 −1 0
−3 1 0 2
−4 5 −1 4
−5 9 −2 6
Im(f)Ker(f)ϕ
Ker(ϕ)⊂Im(f)
E∗=L(E, R)
n∈N∗a1, . . . , an, b1, . . . , bn2n a1< b1< a2< b2<· · · <
an< bn
a, b a < b P ∈R[X]Zb
a
P(t)dt = 0
P
]a, b[
ϕRn−1[X]RnP
ϕ(P) = Zb1
a1
P(t)dt, . . . , Zbn
an
P(t)dtϕ
∀(α1, . . . , αn)∈RnPRn−1[X]
∀i∈ {1, . . . , n},Zbi
ai
P(t)dt =αi.