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R
f:R3→Rf(x, y, z) = x+y+ 2z
f:R2→Rf(x, y) = x+y+ 1
f:R2→Rf(x, y) = xy
f:R3→Rf(x, y, z) = x−z
f:R2→R2f(x, y)=(x+y, x −y)
fR2
J:C([0 ; 1],R)→RJ(f) = R1
0f(t) dt
J
ϕ:C∞(R,R)→ C∞(R,R)ϕ(f) = f00 −3f0+ 2f
ϕ
a X E K
Ea:F(X, E)→E Ea(f) = f(a)
Ea
:K(X)→K[X]
E, F Kf∈ L(E, F )A, B
E
f(A)⊂f(B)⇐⇒ A+ Ker f⊂B+ Ker f
uKE F
E
u−1u(F)FKer u
uu−1(F)FIm u
uu−1(F)=u−1u(F)
E F Kf∈ L(E, F )
(Ei)1≤i≤nE(Fj)1≤j≤p
F
f(
n
X
i=1
Ei) =
n
X
i=1
f(Ei)
f Ei
f(Ei)
f−1(
p
X
j=1
Fj)⊃
p
X
j=1
f−1(Fj)
f∈ L(E)x∈E x f(x)
f

f, g ∈ L(E, F )
∀x∈E, ∃λx∈K, g(x) = λxf(x)
λ∈K
g=λf
ERn≥2a∈E Fa
f E x ∈E
x, f(x), aFaa= 0 n= 2
Faa∈E
H
v H h ∈Hh, v(h)
Fa
EKf∈ L(E)
f2−3f+ 2Id = 0
f f
Ker(f−Id) Ker(f−2Id)
E
f g KE f ◦g= Id
Ker f= Ker(g◦f) Im g= Im(g◦f) Ker fIm g
f g E R C
f◦g= Id
Ker(g◦f) = Ker fIm(g◦f) = Img
E= Ker f⊕Img
g=f−1
(g◦f)◦(g◦f)g◦f
f, g ∈ L(E)
g◦f◦g=g f ◦g◦f=f
Im fKer g E
f(Im g) = Im f
EKf E
n∈N∗fn= 0 Id −f
f
K
E
EKF
L(E) IdE
F∩GL(E) (GL(E),◦)

EKp∈ L(E)
pId −p
Im(Id −p) Ker(Id −p) Im pKer p
p, q ∈ L(E)
p◦q=p q ◦p=q
p q
EKp, q E
p◦q E
EK
s E s2= Id
F= Ker(s−Id) G= Ker(s+ Id)
F G E
s F
G
α∈K\ {1}f E
f2−(α+ 1)f+αId = 0
F= Ker(f−Id) G= Ker(f−αId)
F G E
f F G
α
f∈ L(E)f2−4f+ 3Id = ˜ 0
Ker(f−Id) ⊕Ker(f−3Id) = E.
f
EKp E q = Id −p
L={f∈ L(E)| ∃u∈ L(E), f =u◦p}M={g∈ L(E)| ∃v∈ L(E), g =v◦q}
L M L(E)
p q KE p ◦q= 0
r=p+q−q◦p
ECu∈ L(E)
p E u =p◦u−u◦p
u(Ker p)⊂Im pIm p⊂Ker u
u2= 0
E F Kn p
n>p
u∈ L(E, F )v∈ L(F, E)
u◦v= IdF
v◦u
p q RE
Im p⊂Ker q
p+q−p◦q

f g E R C
f◦g= Id
Ker(g◦f) = Ker fIm(g◦f) = Img
E= Ker f⊕Img
g=f−1
(g◦f)◦(g◦f)g◦f
HKE
a E H
H⊕Vect(a) = E
HKE D
H
D H E
HKE
F E H
F=H F =E
f, g ∈E∗Ker f= Ker g α ∈K
f=αg
e= (e1, . . . , en)KE
n∈N∗
∀f∈E∗, f(e1) = . . . =f(en) = 0 =⇒f= 0
e E
EKV E
f∈ L(E)
V⊂f(V) =⇒f(V) = V
f∈ L(E, F ) (x1, . . . , xp)
E
rg(f(x1), . . . , f(xp)) = rg(x1, . . . , xp)
f:R3→R3f(x, y, z)=(y−z, z −x, x −y)
f:R4→R3f(x, y, z, t) = (2x+y+z, x +y+t, x +z−t)
f:C→Cf(z) = z+i¯zC R
EKn≥1f
E p fp=˜ 0
x∈E
x, f(x), f2(x), . . . , fp−1(x)
fn=˜ 0
f n
(I, f, f2, . . . , fn2)
f
E

f
Ker f= Im f
f=u◦v
u v
a0, a1, . . . , anK
ϕ:Kn[X]→Kn+1
ϕ(P) = (P(a0), P (a1), . . . , P(an))
K
a0, . . . , anϕ:R2n+1[X]→R2n+2
ϕ(P) = (P(a0), P 0(a0), . . . , P(an), P 0(an))
ϕ
EKf, g ∈ L(E)
rg(f+g)≤rg(f) + rg(g)
|rg(f)−rg(g)| ≤ rg(f−g)
E F K
f∈ L(E, F ), g ∈ L(F, E)f◦g◦f=f g ◦f◦g=g
f, g, f ◦g g ◦f
f, g ∈ L(E)EK
|rg(f)−rg(g)| ≤ rg(f+g)≤rg(f) + rg(g)
u v E
|rg(u)−rg(v)| ≤ rg(u+v)≤rg(u) + rg(v)
u v L(R2)
rg(u+v)<rg(u) + rg(v)
u v R2
rg(u+v) = rg(u) + rg(v)
E, F Kf, g ∈ L(E, F )
rg(f+g) = rg(f) + rg(g)⇐⇒ Im(f)∩Im(g) = {0}
Ker(f) + Ker(g) = E
f g E
rg(f◦g)≤min(rg f, rg g)
rg(f◦g)≥rg f+ rg g−dim E
EKf, g ∈ L(E)f+g
g◦f=˜ 0
rg f+ rg g= dim E
EKn∈N∗u E
u3=˜ 0
rg u+ rg u2≤n