poly de cours

K[X]
±

T M
...................
0n−r,r 0n−r,p−r
,
T∈ Mr(K)r
AX =Y
InA−1
A In
P Q P AQ =Jrr A
K
K+
K×E→E(E, +)
−→
0E
E λ1−→ x1+···+λn−→ xn
λiK−→ xiE
E
KnK CnC R
Kn[X]K[X]
C∞(I)C1(I)D1(I)
Sn(K) (n, n)Mn(K)
fn+2 =fn+fn+1
CN
u∈ L(E)E u L(E)
2

x1, ..., xn∈E xiE
E xixi
(xi)i∈I
F E
{(x, y, z)∈R3;x+y+z= 0}={(−α−β, α, β)|α, β ∈R}={α−→ u+β−→ v|α, β ∈R}= Vect(−→ u , −→ v),
−→
u= (−1,1,0) −→ v= (−1,0,1)
(−→ vi)i∈IE−→
f E −→ vi
λ1, ..., λn∈Ki1, ..., in∈I−→ f=λ1−→ vi1+··· +λn−→ vin
−→
viλ1−→ vi1+···+λn−→ vin=−→ 0
λi
E
−→
vi
I={1,2, ..., n}
Φ
Kn−→ E
(λ1, ..., λn)7−→ λ1−→ v1+···+λn−→ vn
(v1, v2)
λ v1=λv2v2=λv1
v16= 0 v2= 0
n
KnE= (e1, ..., en)ei= (0, ..., 0,1,0, ..., 0)
Kn[X] (1, X, ..., Xn)
Mn,p(K)Ei,j (n, p) (k, `) 1 (i, j) =
(k, `) 0 (Ei,j )k,` =δi,kδj,`
(ϕn
1)n∈N(ϕn 2)n∈Nϕ1ϕ2X2−X−1
un+2 =un+un+1
n∈N
n n
n
K[X] (Pn)n∈NPnn n ∈N
C0(R)fn:x7→ |x−n|
C∞(R)en:x7→ inx
K(X)1
X−ααK

R3v1=
1
2
3
(1,2,3) v2=
2
1
1
v3=
−1
4
7
F= (v1, v2, v3) Vect(F)
R3
f E F
∀x, y ∈E, ∀α, β ∈K, f (αx +βy) = αf(x) + βf(y).
E F L(E, F )K
(x, y, z)∈R37→ (x+y+z, 2x−z)∈R2(x1, ..., xn)∈Rn7→
n
P
i=1
ixi∈R
P∈K[X]7→ XP 0∈K[X]P∈R[X]7→ Z999
945
P(t)dt ∈R
M∈ Mn(K)7→ M+ (M)In∈ Mn(K)M∈ Mn(K)7→ (AM)∈K
f∈ C∞(I)7→ f00 + 2f0+f∈ C∞(I)
u∈ L(E, F )E1E F1F u(E1)
F u<−1>(F1)E
u u<−1>(F1)
E1={0}F1=F
u∈ L(E, F ) 0Fu
Ker (u) := u<−1>({0F}) = {x∈E;u(x)=0F}
Im (u) := u(E) = {u(x)|x∈E}.
E F
Ker (u) Im (u)
{(α, β, 2β−α)|α, β ∈R}(x, y, z)7→ x−2y+z
(α, β)7→ (α, β, 2β−α) (1,0,−1) (0,1,2)

u v vn=un+2 −un−un+1 n∈N
{0}
u∈ L(E, F ){0E}
(a, b)∈R27→ (a+b, a + 2b, 2a+b, a + 3b)∈R4
(a, b, c, d)∈R47→ (a+b+c+d, a −b+c−d)∈R2
P∈Rn[X]7→ (P(1), P (2), ..., P (n+ 1)) ∈Rn+1
EE u ∈ L(E)7→ Mat(u, E)∈ Mn(K)
u∈ L(E, F )
u u E
F
(u(v1), ..., u(vn)) F(v1, ..., vn)E
u(v1, ..., vn)E(u(v1), ..., u(vn))
F
u u
u∈ L(E, F )
u−1
EL(E, E)L(E)E
E(L(E),+, ., ◦)
◦
u∈ L(E, F )v∈ L(F, G)
Ker (u)⊂Ker (v◦u) Im (v◦u)⊂Im (v)
v◦u= 0 Im (u)⊂Ker (v)
E=F u ◦v=v◦u u v