Applications linéaires

K=R C
E F f :E→F
f
∀x, y ∈E, ∀λ∈K, f(x+λy) = f(x) + λf(y)
L(E, F )E F
GE=F E E E
L(E)
GE F
GE=F E E
©
GC1(I, R)→C0(I, R)
f7→ f0
GR[X]→R[X]
P7→ P0
.
Gf:R3→R2
(x, y, z)7→ (2x+z, y −z)
E F Kf∈ L(E, F )
Gf(0E)=0F
G∀x∈E, f(−x) = −f(x)
G∀n∈N,∀(xi)1≤i≤n∈En,∀(λi)1≤i≤n∈Kn, f n
X
i=1
λixi!=
n
X
i=1
λif(xi)
E F K
L(E, F )K
L(E, F )FE

E, F, G K
◦∀f∈ L(E, F ),∀g∈ L(F, G), g ◦f∈ L(E, G)
◦∀f1, f2∈ L(E, F ),∀g∈ L(F, G),∀λ∈K, g ◦(f1+λf2) = g◦f1+λg ◦f2
◦∀f∈ L(E, F ),∀g1, g2∈ L(F, G),∀λ∈K,(g1+λg2)◦f=g1◦f+λg2◦f
n∈N∗
∀f, g ∈ L(E), f ◦g=g◦f=⇒(f+g)n=
n
X
k=0 n
kfk◦gn−k
E, F Kf∈ L(E, F )
f f−1
E, F K
◦f
Im f=f(E) = {f(x), x ∈E}
F
◦f
Ker f={x∈E, f(x) = 0F}
E
E, F Kf∈ L(E, F )
◦fKer(f) = {0E}
◦fIm f=F

E F K
E
B= (e1, . . . , ep)E
(f1, . . . , fp)p F
u E F
∀1≤i≤p, u(ei) = fi
E F K
E
u∈ L(E, F )
u
rg u= dim Im u
E F K
EB= (e1, . . . , ep)E
u∈ L(E, F )
GIm u= Vect(u(e1), . . . , u(ep))
Grg u= rg(u(e1), . . . , u(ep))
Grg u≤p= dim E
E F K
EB= (e1, . . . , ep)E u ∈ L(E, F )
Gu⇐⇒ u(B)⇐⇒ rg u= dim E
Gu⇐⇒ u(B)⇐⇒ rg u= dim F
Gu⇐⇒ u(B)⇐⇒ rg u= dim E= dim F
E F K
E F
u∈ L(E, F )
u⇐⇒ u⇐⇒ u

E F K
E
u∈ L(E, F )
rg(u) + dim Ker u= dim E
EKpB= (e1, . . . , ep)
x=
p
X
i=1
xieiE
xB
X=MatB(x) = 
x1
xp
MatB:E→Mp,1(K)
x7→ MatB(x)
EKpB= (e1, . . . , ep)
FKnC= (f1, . . . , fn)
u∈ L(E, F )
uB C
A=MatB,C(u) =
u(e1). . . u(ep)
f1
fn
MatB,C:L(E, F )→Mn,p(K)
u7→ MatB,C(u)

EKpB= (e1, . . . , ep)
FKnC= (f1, . . . , fn)
u∈ L(E, F )A= (ai,j )1≤i≤n
1≤j≤p
=MatB,C(u)∈Mn,p(K)
x∈E X =
x1
xp
=MatB(x)∈Mp,1(K)
y=u(x)∈F Y =
y1
yn
=MatC(y)∈Mn,1(K)
Y=AX
y=u(x)
Y=A X
∈Mn,1(K)∈Mn,p(K)∈Mp,1(K)
A∈Mn,p(K)
KnMn,1(K)
Kp→Kn
X7→ AX
AKpKn
GKer A={X∈Kp, AX = 0}
GIm A={AX, X ∈Kp}
Grg A= dim Im A
EKpB= (e1, . . . , ep)
FKnC= (f1, . . . , fn)
u∈ L(E, F )A= (ai,j )1≤i≤n
1≤j≤p
=MatB,C(u)∈Mn,p(K)
Gx∈Ker u⇐⇒ X∈Ker A X =MatB(x)
Gy∈Im u⇐⇒ Y∈Im A Y =MatC(y)
Grg u= rg A