corrigé officiel

u:R2→R3v:R3→R2
u
udim R3= 3
dim(ker u) + rg u= dim(R2) = 2 dim(ker u)≥0
rg u= dim(R2)−dim(ker u)≤2
v
v{0}
dim(ker v) + rg v= dim(R3) = 3 rg v≤dim R2= 2
dim(ker v) = dim(R3)−rg v≥3−2=1
ker v⊂ker(u◦v)uker(u◦v) = ker v
x∈ker v v(x) = 0 (u◦v)(x) = u(v(x)) = u(0) = 0 u(0) = 0
u x ∈ker(u◦v) ker v⊂ker(u◦v)
u x ∈ker(u◦v)u(v(x)) = 0
u v(x)=0 x∈ker v
uker(u◦v)⊂ker v
ker(u◦v) = ker v u
BR2CR3u
R2R3
A= MatB,C(u) = 
1 2
2−1
2 3
vR3R2
B= MatC,B(v) = 1−2 1
2 1 −3.
A B
Arg A= 2
Brg B≤2
Brg B≥2 rg B= 2
u v
u X =x1
x2∈R2AX = 0
x1+ 2x2= 0
2x1−x2= 0
2x1+ 3x2= 0

x1=x2= 0 ker u={0}
rg A= 2
v BX = 0 X=
x1
x2
x3
∈R3
x1−2x2+x3= 0
2x1+x2−3x3= 0 .
x1−2x2+x3= 0
5x2−5x3= 0
x2=x3x1= 2x2−x3=
x3
x1
x2
x3
∈ker vssi 
x1
x2
x3
=x3
1
1
1
, x3∈R.
ker vR3
1
1
1
AB BA
u v
AB =
5 0 −5
0−5 5
8−1−7
et BA =−1 7
−2−6.
AB u ◦v:R3→R3u◦v
R3R3BA
v◦u:R2→R2v◦uR2
R2
u◦v
u◦v AB
R3AB
rg (AB)≥2
rg (u◦v) = rg (AB) = 2
u
ker(u◦v) = ker v
rg(u◦v) = dim R3−dim(ker(u◦v)) = 3−1=2
v◦u
rg (v◦u) = rg (BA)
BA rg (BA) = 2
n n
v◦uR2
ERf E
E E f ◦f=−IdE
x∈E

(x, f(x)) f x
f(x)
λ, µ ∈Rλx +µf(x)=0 f
f(λx +µf(x)) = λf (x) + µf ◦f(x) = λf (x)−µx =f(0) = 0
λ µ λx +µf(x)=0 λf (x)−µx = 0
f(x)λ−µ
(λ2+µ2)x= 0 x λ2+µ2= 0
λ=µ= 0 (x, f(x))
E(x, f(x)) E f
(x, f(x))
(x, f(x))
dim E= 2 E e1=x e2=f(x)
f(e1) = f(x) = e2f(e2) = f(f(x)) = −x=−e1f
(e1, e2)
0−1
1 0 .
E y E
Vect {x, f(x)}x f(x)
(x, f(x), y, f(y))
α β γ δ αx +βf (x) + γy +δf (y)=0
f αf(x)−βx +γf (y)−δy = 0
f(y)γ−δ
(αγ +βδ)x+ (βγ −αδ)f(x)+(γ2+δ2)y= 0
γ2+δ2y x f(x)
y6∈ Vect {x, f(x)}γ2+δ2= 0 γ=δ= 0
(x, f(x)) α=β= 0 (x, f(x), y, f (y))
E
E
Edim E≥4
E E f
0−1 0 0
1000
000−1
0010
.
E(x, f(x), y, f (y))
f