Applications linéaires

E F
Ef(u+v) = f(u)+f(v)λ f(λu) = λf(u)
(1) f(u+λv) = f(u) + λf(v)
E F f
λ= 0 E
F
x7→ ax
E
E F L(E, F )
E F E L(E)
E
E
E
E
EC∞
D:f7→ f0E
I:f7→ R1
0f(x)dx E

S E f E F
S f f(S)
F
E F
fIm f f
Im(E) = F
f S0
F E f−1(S0)f
f−1¡{0F}¢
fKer fKer f
f(v) = 0
fu v f(u) = f(v)⇒u=vf
f(x) = 0F⇐⇒ x=0E
Ker f{0E}fKer f{0E}Im f F
0F
fF= (vi)E
G= (f(vi)) F
FEG
Im f f
B= (vi)1≤i≤nE
B0= (v0
i)1≤i≤nf f
f(
n
X
i=1
aivi) =
n
X
i=1
aiv0
i
c`F
KpE c`F((ai)1≤i≤p7→
p
P
i=1
aiviF= (vi)1≤i≤p
E
c`F((ai) + λ(bi)) =
p
P
i=1
(ai+λbi)vi=
p
P
i=1
aivi+λ
p
P
i=1
bivi=c`F((ai)) + λc`F((bi))
F F
FE
KpEdim(E) = p
(x1,· · · , xp)Kp(x1,· · · , xp)F
EFIm c`F
Vect F F Ker c`F
F(ai)∈Kp
p
P
i=1
aivi=0E

frg(f) Im f
Frg(f)≤dim(F)
f
(f(vi)) viE E
rg(f)≤dim(E)frg(f) = dim(F)f
⇐⇒ rg(f) = dim(E)
f g L(E, F )
h h(v) = f(v)+g(v)
f+g k(v) = af(v)a
af
L(E, F )
S
EL(S, F )L(E, F )
f∈ L(E, F )g∈ L(F, G)
g◦fL(E, G)
(g+g0)◦f=g◦f+g0◦f
g◦(f+f0) = g◦f+g◦f0g
(λg)◦f=g◦(λf) = λ(g◦f)
(h◦g)◦f=h◦(g◦f)◦
E
IdE:v7→ v
g◦f6=f◦g g ◦f=˜ 06⇒ (f=˜ 0) (g=˜ 0)
g◦f= IdEf◦g6= IdE
E=RX f :P7→ Q Q(X) = XP (X)
g:P7→ Q Q(X) = P(X)−P(0)
X
◦f◦f−1=f−1◦f= IdE
(g◦f)◦(f−1◦g−1) = IdEIdE−IdE
Im f=A
vf(v)A
A

Ker f=B
v∈B f(v) = 0Fv/∈B f(v)6=0F
BE
f(B) Im fKer f
(vi)1≤i≤mKer f
E(vi)1≤i≤n
Ker f f(vi)m+1≤i≤n
Im f
n
P
i=m+1
αif(vi)
n
P
i=m+1
αiviKer f
vi
Im f n −m
dim(Ker f) + rg(f) = dim(E)
E F
f
Ker f={0}f
rg(f) = dim(E)f
f f
f f ◦f=f S1S2
EvE
v=v1+v2v1v2S1S2
f f(v) = v1
E S1
S2p
Im p⊕Ker p=E p Im pKer p
B= (ui)1≤i≤mB0=vi)1≤i≤nE F
f(ui) =
n
P
j=1
ajivjf(ui)B0
M= (aji)fB
B0Mui
M n ×m V
x=
m
P
i=1
ciuif(x)B0
W=MX
MB B0f M

L(E)Mn
A B f g
A+λB f +λg BA
g◦f
EB=B0
MnL(E) Φ Φ(M)
MΦ(A+λB) = Φ(A) + λΦ(B)
Φ Φ−1(f)f
BΦ
Φ(AB) = Φ(A)◦Φ(B) Φ
dim(L(E, F )) = dim(E) dim(F) dim(L(E)) = dim(E)2
VB
B0P V P
B0B B B0
P−1B B0
EB0F E
B B1B2E
B0
1B0 2F B f
B2B0
2A f B1B0 1
B=Q−1AP
PB1B2QB0
1B0 2
B=P−1AP.
A B
rg(A) = rg(B)
E
n f Im f=R
dim Ker f=n−1 Ker f
EB= (vi)1≤i≤nE ai=f(vi)
x=
n
P
i=1
xivif(x) =
n
P
i=1
aixix∈Ker f⇐⇒
n
P
i=1
aixi=
0 Ker f
E
f