SL(E) est engendré par les transvections

SL(E)
SL(E)
E k
u∈SL(E)u(t) = t+f(t)a f E
a∈ker(f)a6= 0 H= ker(f)u
SL(E)
x∈E− {0}H1, H2x6∈ H1∪H2
u u(x) = x u(H1) = H2
u x H1∩H2
H=H1∩H2+kx ky
kz H1∩H2H1H2u(y) = z
x H +H1=H+H2=E z ∈H2−H
z=a+y a ∈H y ∈H1.
f H f(y) = 1 y6∈ H u
u(t) = t+f(t)a u(y) = y+a=z
dim(E)≥2x, y ∈E− {0}u
u(x) = y
x y H y −x
x a =y−x u(t) = t+f(t)a f H f(x)=1
u(x) = x+y−x=y
x y z
u1, u2u1(x) = z u2(z) = y u =u2◦u1
dim(E)≥2
n= dim(E)n= 1
n≥2v∈SL(E)x∈E− {0}
v(x) = x H ⊂E
x6∈ H x =v(x)6∈ v(H)
v(H) = H v|H
v|Hui,H
uiE x v(x) = x v
ui