Feuille d`exercices 18 - PCSI

F R4
F={(x;y;z;t)∈R4=x+y=z+t= 0}
a= (2; −2; −1; 1)
a, b, c ∈Rx7→ sin(x+a)
x7→ sin(x+b)x7→ sin(x+c)RR
R4
a= (1; 2; 2; 1), b = (5; 6; 6; 5), c = (−1; −3; 4; 0), d = (0; 4; −3; −1)
a= (2; −5; 3; 10), b = (1; −1; 1; 3), c = (3; 3; 1; 1)
a= (1; 2; 5; −1), b = (3; 6; 5; −6), c = (2; 4; 0; −2)
R4F=V ect(u;v;w)G=V ect(a;b)
u= (1; −1; 2; 3) v= (1; 1; 2; 0) w= (3; −1; 6; −6) a= (0; 2; 0; ..3)
b= (1; 0; 1; 0) F G F ∩G F +G
R4F a = (1,2,3,4)
b= (2,2,2,−2) c= (0,2,4,4) G d = (1,0,−1,2)
e= (2,3,0,1)
F, G, F +G F ∩G
E u ∈ L(E)u
∀x∈E, ∃λx∈K, u(x) = λx.x
f g E
g◦f=IdE
f g E
E
g◦f=IdEg◦f y ∈E x ∈E
g◦f(x) = y g(f(x)) = y g
E g
f=f◦g◦g−1f
f g
E=R[X]
f:E→E
P(X)7→ XP (X)
g:E→E
P(X)7→ P0(X)
f g g ◦f=IdE
E{0}
dim(E)=1 E E {0}
•F E dim(F)≤dim(E)
dim(E) = 1 dim(F) = 1 dim(F)=0 F=E
F={0}
•
dim(E)>1E
e1e2vect(e1)vect(e2)
E E {0}

E, F G f g
E F F G
dim(Ker(g◦f)) ≤dim(Ker(f)) + dim(Ker(g))
f Ker(g◦f)
g◦f:E→G Ker(g◦f)
E˜ f f Ker(g◦f)
˜
f
dim(Ker(g◦f)) = dim(Ker(˜ f)) + rg(˜ f)
Ker ˜ f⊂Kerf Im ˜ f⊂Kerg
E u
E
Im(u) = Ker(u)
u2= 0
∃p∈N, dim(E) = 2p
rang(u) = p
•Imu =Keru
x∈E u(x)∈Imu =Keru u(u(x)) = 0
u2= 0 E rg(u)
p∈N rg(u) = p
dim(E) = dim(Keru) + rg(u) = 2rg(u)Imu =Keru
dim(E)=2p
•u2= 0 Imu ⊂Keru
dim(E)=2p rg(u) = p
dim(Keru) = p Imu =Keru
fR
E=Im(f) + Ker(f)⇔Im(f) = Im(f2)
EKn u, v ∈ L(E)
v◦u= 0 u+v rang(u) + rang(v) = n
v◦u= 0 Imu ⊂Kerv rg(u)≤dim(Kerv)
n=rg(v) + dim(Kerv)n≥rg(v) + rg(u)
u+v rg(u+v) = n
Im(u+v)⊂Im(u) + Im(v)
rg(u+v)≤rg(u) + rg(v)−dim(Im(u)∩Im(v)) ≤rg(u) + rg(v)
E
φ Kerφ E
dim(Kerφ) = n−1rg(φ)=1
Imφ ⊂KImφ =Kφ