Formes linéaires et hyperplans en dimension finie

(x, y, z)7→ 2x−y+ 3z
P∈Kn[X]7→ P(0)
T r :A7→ Paii
P∈Cn[X]7→ R1
−1
P(t)
1 + t2dt
eiE ei
ei∗E∗E∗
ϕ(Pxiei7→ Pxi)
E
Mn(K)
F
E H
{A|T r(A)=0}
AB −BA Mn(K)
(x, y, z)|2x−3y+z= 0}
{P|P(0) = 0}
J:x7→ (ϕ7→ ϕ(x)) E
E∗∗

(ϕ1, . . . , ϕn)E∗E ϕi=ei∗
(ϕ(i))
(x1, . . . , xn+1)R(y1, . . . , yn+1)
R∀i, P (xi) = yi
(y1, . . . , yn+1) (x1, . . . , xn+1)
Pj(x) = Pyjlj(x)ljRn[X]∗
xi
lj(x) = Q
i6=j
X−xi
xj−xi
Q(xi) = yj
P R R ∈R[X]
x∈E f ∈E∗f(x)X⊥=
{f∈E∗|∀x∈X, f (x)=0}E∗X◦={x∈E|∀f∈Y, f(x)=0}
A1⊂A2⇒A⊥
2⊂Ab 1ot A1⊂A2⇒A2◦⊂A1◦X⊥=¯
X⊥= (vectX)⊥X⊥
E∗(F+G)⊥=F⊥∩G⊥(F∩G)⊥=F⊥+G⊥ ◦ J(F◦) = F⊥
A⊥◦ =A A◦⊥=A dimF +dimF ⊥=E dimF +dimF ◦=E
A=E
∈L(E, F )
tf f ∈E∗7→ fu
F∗E∗
v(y(x)) = y(u(x)).
t(uv) =tvtu(tu)−1=t(u−1)J(u) =tt u Kertu=
Im(u)⊥, Imtu=Ker(u)⊥tu
G⊥tu
E B0B∗B0∗MB,B0(u) =t
(MB0∗,B∗(tu)) MB(u) =t(MB∗(tu))
B C B∗C0∗tP−1MC0∗(u) =tPtMB(u)P
E∗
(e1, e2, e3) (2e∗
1+e∗ 2+e∗ 3,−e∗ 1+ 2e∗ 3, e∗ 1+ 3e∗ 2)
1/13 ∗(6e1−2e2+ 3e3,−3e1+e2+ 5e3,−2e1+ 5e2−e3)
tu
xtu
(Kx)◦
E
∈L(E)P1|P2. . . |P r
Pi

E×E K
(x, y)7→ xy
a(x, .)a(., y)
x7→ a(x, .)y7→ a(., y)
E E∗
a(x, y) = a(y, x)a(x, y) = −a(y, x))
I=−J
Q(x) = a(x, x)
a(x, x) = Q(x
e1, . . . , ena(ei, ej)
(a(ei, .)) B∗a(., ei)
B∗
n2
a(x, y) =tXMY
B B0
tP MP
A=tP BP
E
e e0a a(e, e0)=0
a
a
{y∈E|a(x, y) = 0,∀a∈A}
E∗:{a(x, .)|x∈A}E
⊥A⊥vectA =⊥Abot0 = E⊥(A∪B) =⊥A∩⊥B⊥A+⊥B⊂⊥(A∩B)
dim(H) + dim(⊥H)>E
E
E∗
⊥⊥A=E dim(A) + dim(⊥A) = dimE
⊥A

a(ei, ei)
0−a//a 00
R⊥P DP ∈On(R)D
E∗fidim T
Ker(fi)
=n−rg((fi))
r n −r(fi)T
Ker(fi)
=F
x0n−r
(fi)TKer(fi) + x0=F
ϕ1(x) = b1, ϕ2(x) = b2, ϕ3(x) = b3
ϕi
ϕ3=λϕ1+
µϕ2b36=λb1+µb2
On(R)
On(R)
SOn(R)O−
n(R)

{(ε1,...εn)|εi= 1,−1}
On(R)Mn(R)
C
RnΓM(a)
Γ
y= (∇f(a).x)