Applications linéaires I Généralités sur les applications linéaires

K=R K =C
E F K
f:E−→ F E F f
E F
∀(x , y)∈E2f(x+y) =
∀x∈E, ∀λ∈Kf(λx) =
L(E, F )E F
E
EL(E)
x E −→ 0F
OE,F
E:x7−→ x E E
f∈L(E, F )fKEKF
λ0K
f:E−→ E
x7−→ λ0·x
P∈R[X]
P0
f:R[X]−→ R[X]
P7−→ P0

f
fR[X]
f E F
f(0E) =
∀x∈E, f(−x) =
∀(x, y)∈E2,∀(λ , µ)∈K2, f(λ x +µ y) =
∀(x1, x2, . . . , xp)∈Ep,∀(λ1, ..., λp)∈Kp, f p
X
i=1
λi·xi!=
E F G K
f∈L(E, F )g∈L(F, G)g◦f
f∈L(E, F )g∈L(F, G)
E F K
L(E, F )F(E, F )

E F f :E−→ F
f∀(x, x0)∈E2,[ (x6=x0)⇒( ) ]
f∀y∈F , ∃x∈E;
f
f:R\ {1} −→ R
x7−→ x
x−1
f:R−→ [−1,1]
x7−→ cos x
f:h−π
2,π 2i−→ [−1,1]
x7−→ sin x
E E
f E F
∀(x, x0)∈E2,[f(x) = f(x0) =⇒]
f E F f(E) =
f E F ∀y∈F, ∃!x∈E;

f:E−→ F g :F−→ G
g◦f f
g◦f g
g◦f x ∈E x0∈E x 6=x0.
g◦f(x)6=g◦fx0, g [f(x)] 6=gfx0 f(x)6=fx0f
g◦f z ∈G. x ∈E g ◦f(x) = z.
y=f(x)F z =g(y)g
E F K−f∈L(E, F )
f f F
f E
f={f(x)∈F|
y∈f⇐⇒ ∃
E) =
(OE,F ) =
f E F
f
f

f∈L(E, F )
E F f E F
D F D f
E f−1(D)
f−1(D) = {x∈E|f(x)∈D}
x∈f−1(D)⇐⇒
f−1(∅) =
b F f−1({b}) =
f:C−→ C
z7−→ −z2
f−1({1})
E F K−f∈L(E, F )E
F
D F f D
D F G =f−1(D)⊂E