Algebre lineaire - les classiques

f2
E n
f∈L(E)
(f2) = f−dim(ker f∩f)
dim(ker f2)≤2 dim(ker f)
e
f f f e f: (f)−→ E
x7→ f(x)
(e f) = f( (f)) = (f2) ker( e f) = ker f∩f
f
dim( f) = dim( ( e f)) + dim(ker( e f))
(f) = (f2) + dim(ker f∩f)
n−dim(ker f) = n−dim(ker f2) + dim(ker f∩f)
dim(ker f2) = dim(ker f) + dim(ker f∩f)
dim(ker f∩f)≤dim(ker f) ker f∩f⊂ker f
dim(ker f2)≤2 dim(ker f)
∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
E F
f∈L(E, F )G E
dim(f(G)) = dim(G)−dim(ker f∩G)
e
f f G e f:G−→ E
x7→ f(x)
(e f) = f(G) ker( e f) = ker f∩G
f
dim(G) = dim( ( e f)) + dim(ker( e f))
dim(G) = dim(f(G)) + dim(ker f∩G)
∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
E F f ∈L(E, F )G F
f−1(G)E
(f−1(G)) = (G(f) + (ker f)
f(0E) = 0F∈G0E∈f−1(G)f−1(G)
∀x, y ∈f−1(G),∀λ∈, f(x+λy) = f(x)
|{z}
∈G
+λ f(y)
|{z}
∈G
∈G x +λy ∈f−1(G)
f−1(G)E
e
f f f−1(G)f−1(G)f −−→ F

•ker(f)⊂f−1(G) ker( e f) = f−1(G) ker(f) = ker(f)
•y∈(e f),∃x∈f−1(G), y =e f(x) = f(x)y∈G(f)
(e f)⊂G(f)
y∈G(f)y∈G∃x∈E, y =f(x)
y=f(x)∈G, x ∈f−1(G)y=f(x) = e f(x)∈(e f)
(e f) = G(f)
e
f
(f−1(G)) = (G(f) + (ker f)
∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
p q E
p+q
poq+qop= 0
poq=qop= 0
p+q
(p)∩(q) = {0}ker p+ ker q=E
p+q
p q pop=p qoq=q
•p+q⇐⇒ (p+q)2=p+q
⇐⇒ pop
|{z}
=p
+poq+qop+qoq
|{z}
=q
=p+q
⇐⇒ poq+qop= 0
a)⇐⇒ b)
•c) =⇒b)
b) =⇒poq+qop= 0
=⇒popoq+poqop= 0 p
=⇒poq+poqop= 0
p, poqop+qopop= 0
=⇒poqop+qop= 0
poq−qop= 0 poq=qop= 0
b)⇐⇒ c)
p+q
•x∈(p)∩(q).∃t∈E, ∃z∈E, x =p(t) = q(z)
p(x) = pop(t) = p(t) = x=poq(z) = 0 poq= 0
(p)∩(q) = {0}
• ∀x∈E, x =p(x)+(x−p(x))
q(p(x)) = qop(x) = 0 qop= 0 p(x)∈ker q
p(x−p(x)) = p(x)−pop(x) = 0 p x −p(x)∈ker p
E⊂ker p+ ker q
•(p+q)⊂(p)+ (q)
(p+q)) = (p+q) = (p+q)p+q
= (p)+ (q)
= (p)+ (q)p q
(p)+ (q)) = (p))+ (q)) −p∩q)
(p)∩(q) = {0}
(p)+ (q)) = (p))+ (q))
(p+q) = (p)+ (q)
•ker p∩ker q⊂ker(p+q)
x∈ker(p+q)p(x) = −q(x)
p(x)∈p q(x)∈q p(x)∈p∩q={0}p(x) = 0 q(x) = 0
x∈ker p∩ker q
ker(p+q) = ker p∩ker q
p+q(p)+ (q) ker p∩ker q
∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗

f n
(f) + (IdE−f)≤n f E
x∈E, x =f(x)+(IdE−f)(x)
E⊂(f) + (IdE−f)
dim (f) + (IdE−f)
| {z }
=n
= dim( f) + dim( (IdE−f))
| {z }
≤n par hypothese
−dim (f)∩(IdE−f)
| {z }
≥0
dim( f) + dim( (IdE−f)) = n(f)∩(IdE−f) = {0}
x∈E, f2(x)−f(x) = ff(x)−x= (IdE−f)−f(x)∈(f)∩(IdE−f) = {0}
f2(x) = f(x)f
∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
f f2
f n
(f) = (f2)⇐⇒ ker(f)⊕(f) = E
f2⊂fker f⊂ker f2
•ker(f)⊕(f) = E
f f ˜ f f f
ker ˜ f=f∩ker f={0}˜ f
∀x∈f2,∃t∈E, x =f2(t) = f(f(t)
|{z}
∈Im(f)
) = ˜ f(f(t)) ˜ f
˜
f f f2f f2(f) =
(f2)
•(f) = (f2)
dim(ker f) = dim(ker f2)
ker f⊂ker f2ker f= ker f2
x∈ker(f)∩(f)
∃t∈E, x =f(t)f(x) = 0 f2(t) = f(x) = 0
t∈ker f2t∈ker fker f= ker f2f(t) = 0 = x
ker(f)∩(f) = {0}ker(f)⊕(f)
dim(ker(f)⊕(f)) = dim(ker(f)) + dim( (f)) = dim E
ker(f)⊕(f)⊂Eker(f)⊕(f) = E
∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
f g E n
(f+g) = (f) + (g)⇐⇒ f∩g={0}
ker f+ ker g=E
∀y∈(f+g),∃x∈E, y = (f+g)(x) = f(x)
|{z}
∈f
+g(x)
|{z}
∈g
(f+g)⊂
|{z}
(1)
f+g
(f+g) = dim( (f+g)) ≤
|{z}
(10)
dim( f+g) = (f) + (g)−dim( f∩g)
(f+g) (f) + (g)
dim( f∩g) = 0
(f+g) = (f) + (g)⇐⇒ (f+g) = f+g
f∩g={0}
♣(f+g) = (f) + (g)
f∩g={0}
dim(ker f+ ker g) = dim(ker f) + dim(ker g)−dim(ker f∩ker g)

=n−f+n−g−dim(ker f∩ker g)
= 2n−(f+g)−dim(ker f∩ker g)
ker f∩ker g= ker(f+g)
ker f∩ker g⊂ker(f+g)
x∈ker(f+g)
(f+g)(x) = 0 f(x)
|{z}
∈f
=g(−x)
|{z}
∈g
f(x) = g(x) = 0 f∩g={0}
x∈ker f∩ker g
dim(ker f+ ker g) = 2n−(f+g)−dim(ker(f+g)) = 2n−n=n
f+g
ker f+ ker g⊂E
ker f+ ker g=E
♣f∩g={0}ker f+ ker g=E
(f+g) = f+g
(f+g)⊂f+g f +g⊂(f+g)
z∈f+g∃x, y ∈E z =f(x) + g(y)
ker f+ ker g=E a ∈ker f b ∈ker g x =a+b
c∈ker f d ∈ker g y =c+d
(f+g)(b+c) = f(b) + f(c)
|{z}
0
+g(b)
|{z}
0
+g(c)
z=f(x) + g(y) = f(a+b) + g(c+d) = f(a)
|{z}
0
+f(b) + g(c) + g(d)
|{z}
0
=f(b) + g(c)
z= (f+g)(b+c)∈(f+g)
(f+g) = f+g
∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
E, F G f ∈L(E, F )g∈L(F, G)
Ef −→ Fg −→ G
−−−−−−−−−−−→
gof
(gof) = (f)−(ker g f)
(gof) = (g) + (ker g+f)−F
(gof) = (f) (gof) = (g)
eg g (f) (f)g −−−→ G
x7→ g(x)
(gof) = g( (f)) = (eg)
ker(eg) = (f)∩ker g
egdim( f) = dim( eg) + dim(ker eg)
f= (gof) + dim( (f)∩ker g)
( (f) + ker(g)) = ( f) + (ker g)−(fker g)
=f+ (F)−g−(fker g)
(gof) = f−dim( (f)∩ker g)
=f+ ( (f) + ker(g)) −f−(F) + g
(gof) = g(f+ ker g)−(F)
•f= (gof) + dim( f∩ker g)
(gof) = f⇐⇒ f∩ker g={0}
•(gof) = g+ ( f+ ker g)−(F)
(gof) = g⇐⇒ (f+ ker g) = (F)
⇐⇒ f+ ker g=F f + ker g⊂F
∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗

E, F G f ∈L(E, F )g∈L(F, G)
(gof) = (g)⇐⇒ ker g+ (f) = F
ker(gof) = ker(f)⇐⇒ (f)∩ker g={0}
Ef −→ Fg −→ G
(gof)⊂(g)
•(gof) = (g)
x∈F g(x)∈(g) = (gof)∃t∈E, g(x) = gof(t)
x=f(t)+(x−f(t)) f(t)∈(f)x−f(t)∈ker g
g(x−f(t)) = g(x)−gof(t) = 0
F⊂ker g+ (f) ker g(f)F
•ker g+ (f) = F
y∈(g)∃x∈F, y =g(x)
ker g+ (f) = F∃a∈ker g, ∃b∈(f), x =a+b∃c∈E, b =f(c)
y=g(x) = g(a+b) = g(a)
|{z}
0
+g(b) = g(f(c)) ∈(gof)
(g)⊂(gof)
ker(f)⊂ker(gof)
•ker(f) = ker(gof)
y∈(f)∩ker g:∃x∈E, y =f(x)g(y) = 0 g(f(x)) = 0 x∈ker(gof) = ker(f)
f(x) = 0 y=f(x) = 0
(f)∩ker g={0}
•(f)∩ker g={0}
x∈ker(gof)gof(x) = 0 f(x)∈(f)∩ker g={0}f(x) = 0 x∈ker f
ker(f)⊂ker(gof)
∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
L(E, F )
E F G E
W={u∈L(E, F ), G ⊂ker u}
WL(E, F )
E, F G
ω E F
•ker(ω) = E G ⊂ker ω ω ∈G W
•u v ∈W λ ∈
u v E F G ⊂ker u G ⊂ker v
∀x∈G, u(x) = v(x) = 0 ∀x∈G, (u+λv)(x) = 0 G⊂ker(u+λv)
u+λv ∈W
WL(E, F )
n= dim(E), p = dim(F)m= dim(G)
G H E G ⊕H=E
•u∈L(E, F )
u|Gu|Hu G H
∀x∈E, ∃x1∈G, ∃x2∈H, x =x1+x2
u(x) = u(x1+x2) = u(x1) + u(x2) = u|G(x1) + u|H(x2)
u∈W⇐⇒ G⊂ker u⇐⇒ ∀x1∈G, u(x1) = 0 ⇐⇒ ∀x1∈G, u|G(x1) = 0 ⇐⇒ u|G=ω
•ΦL(E, F )L(G, F )u∈L(E, F )u|G
Φ : L(E, F )−→ L(G, F )
u7→ u|G
u∈W⇐⇒ u|G=ω⇐⇒ Φ(u) = ω W = ker Φ
dim(L(E, F )) = dim(ker Φ
|{z}
W
) + dim( Φ)
dim(W) = dim(L(E, F )) −dim( Φ)