Fiche 7 : Applications linéaires et changements de bases en

E F
E F
f E F ∀u, v ∈E∀λ∈
f(u+v) = f(u) + f(v)
f(λu) = λf(u)
f E F λ1u1+· · · +λkuk
E λ1f(u1) + · · · +λkf(uk)f
E0FF
E F L(E, F )
α1f1+· · · +αqfqf1,· · · , fqE F
E F
L(E, F )E F
FEE F
E, F, G −f∈ L(E, F )g∈ L(F, G)g◦f∈ L(E, G)
f∈ L(E, F )E F
f−1F E F E
E F E F
E
F E
n E E n
En
f∈ L(E, F )
Ker(f) = {u∈E|f(u) = 0F}f
0E∈Ker(f)Ker(f)E
Im(f) = f(E) = {v∈F| ∃u∈E v =f(u)}f
0F∈Im(f)Im(f)F
E n BE={e1,· · · , en}E Im(f)
F{f(e1),· · · , f (en)}
A B f ∈BA
∀u, v ∈A, u 6=v⇒f(u)6=f(v)
f∈ L(E, F )⇐⇒ Ker(f) = {0E}
f∈ L(E, F )k E
k F
A B f ∈BA
∀b∈B, ∃a∈A b =f(a)
Im(f)f∈ L(E, F )⇐⇒ Im(f) = F.
f∈ L(E, F )E F

E F BEE f ∈ L(E, F )
E F f BEF
f E F E F
f E F
nn
E f ∈ L(E, F )
Ker(f)Im(f)
dim(Ker(f)) + dim(Im(f)) = dim(E).
n=dim(E)
F Ker(f)E dim(Ker(f)) 6dim(E)
Im(f)f(e1),· · · , f(en)BE= (e1,· · · , en)E
dim(Im(f)) 6n
k=dim(Ker(f)) r=dim(Im(f)) k+r=n
E k +r
(u1, . . . , uk)Ker(f) (v1, . . . , vr)Im(f)
j∈ {1,· · · , r}vj∈Im(f)u0
j∈E f(u0 j) = vj
B= (u1, . . . , uk, u0
1, . . . , u0 r)E
Bλ1, . . . , λkµ1, . . . , µr
k
X
i=1
λiui+
r
X
j=1
µju0
j= 0E
λiµj
u=
k
X
i=1
λiuiu0=
r
X
j=1
µju0
ju+u0= 0Ef(u+u0) = 0F
f(u+u0) = f(u) + f(u0) = 0Fu∈Ker(f)f(u0)=0F
f(u0) =
r
X
j=1
µjf(u0
j) =
r
X
j=1
µjvj
r
X
j=1
µjvj= 0Fvj
µju0= 0E
u+u0= 0Eu= 0E(u1, . . . , uk)
λj
Bv E
f(v)∈Im(f)v1, . . . , vrβ1, . . . , βr∈
f(v) =
r
X
j=1
βjvj=
r
X
j=1
βjf(u0
j)f(v) = f
r
X
j=1
βju0
j
f v−
r
X
j=1
βju0
j
= 0F
v−
r
X
j=1
βju0
j
∈Ker(f)α1, . . . , αk∈ v−
r
X
j=1
βju0
j
=
k
X
i=1
αiui
v=
k
X
i=1
αiui+
r
X
j=1
βju0
jB
rang(f) = dim(Im(f))
E∀f∈ L(E, F )dim(Ker(f)) rang(f)
rang(f) = dim(E)−dim(ker(f))
E F f ∈ L(E, F )
E F

E F BEBF
f E F BFf(u)
u E BEu u
dim(E) = n dim(F) = p
XunBEu∈E
YvpBFv∈F
f∈FEf∈ L(E, F )⇐⇒ ∃M∈ M(p, n, )∀u∈E Yf(u)=MXu
M f (BE,BF)
M=Mat(f, BE,BF)
M=Mat(f, BE,BF)
Yf(u)=MXuu∈E i ∈ {1,· · · , p}i
f(u)BFi M Xu
uBE
u∈E
x1
x2
xn
:= Xu
y1
y2
yp
:= Yf(u)
uBEf(u)BFmi,j M
∀i∈ {1,· · · , p}yi=Li(M)
x1
x2
xn
=mi,1x1+mi,2x2+· · · +mi,n xn
M=Mat(f, BE,BF) =
m1,1m1,2· · · · · · m1,n
m2,1· · · · · · · · · m2,n
mi,1mi,2· · · · · · mi,n
mp,1· · · · · · · · · mp,n
←→ yi=mi,1x1+mi,2x2+· · · +mi,n xn
XejjBEn
j1Yf(ej)=M Xej=Cj(M)j M
Yf(ej)
||
M=Mat(f, BE,BF) =
m1,1
m2,1
mp,1
m1,2
mp,2
· · ·
· · ·
m1,j
m2,j
mp,j
· · ·
· · ·
m1,n
mp,n
M MBF(f(e1),· · · , f (en))
f(e1),· · · , f (en)BF

E=3F=2[X]
f∈FE
f((x, y, z)) = (x+y)X0+ (x−z)X1−(y+z)X2
f∈ L(E, F )f E F
∀u= (x, y, z)∈E∀v= (x0, y0, z0)∈E∀λ∈
f(u+v) = f((x+x0, y +y0, z +z0))
= (x+x0+y+y0)X0+ (x+x0−z−z0)X1−(y+y0+z+z0)X2
= (x+y)X0+ (x−z)X1−(y+z)X2
| {z }+ (x0+y0)X0+ (x0−z0)X1−(y0+z0)X2 | {z }
=f(u) + f(v)
f(λu) = f((λx, λy, λz))
= (λx +λy)X0+ (λx −λz)X1−(λy +λz)X2
=λ(x+y)X0+ (x−z)X1−(y+z)X2
| {z }
=λ f(u)
f
Ker(f) = {(x, y, z)∈E|f( (x, y, z) ) = 0F}
=(x, y, z)∈3|(x+y)X0+ (x−z)X1−(y+z)X2= 0 2[X]
3S=
x+y= 0
x−z= 0
−y−z= 0
Ker(f) = V ect( (1,−1,1) ) 3
v= (1,−1,1)
f Im(f) = {v∈F| ∃ u∈E v =f(u)}
=P∈2[X]| ∃ (x, y, z)∈3P= (x+y)X0+ (x−z)X1−(y+z)X2.
f((1,0,0)), f((0,1,0)) f((0,0,1)) 3
Im(f) = V ect(X0+X1, X0−X2,−X1−X2)
(f((1,0,0)), f((0,1,0), f((0,0,1)) )
f((1,0,0)) −f((0,1,0)) + f((0,0,1)) = 0F
Im(f)V ect(X0+X1,−X1−X2)F
(X0, X1, X2)F
P F Im(f)V ect(X0+X1,−X1−X2, P ) = V ect(X0+X1,−X1−X2)
rang(X0+X1,−X1−X2, P ) = 2 P=a0X0+a1X1+a2X2
rang 
1 0 a0
1−1a1
0−1a2
= 2.
(C3←C3−a0C1) (C3←C3+ (a1−a0)C2)
rang 
1 0 a0
1−1a1
0−1a2
=rang 
1 0 0
1−1a1−a0
0−1a2
=rang 
1 0 0
1−1 0
0−1a0−a1+a2
2
{a0−a1+a2= 0 Im(f) (X0, X1, X2)F
f E F
v= (x, y, z)∈3E Xv=
x
y
z
f(v)F Yf(v)=
x+y
x−z
−y−z
=
110
1 0 −1
0−1−1
x
y
z
f E F M =
110
1 0 −1
0−1−1

E=2[X]F=f∈FE
f(P) = t7→ (t−1) e P0(t)−2e P(t)
e
P P
f(X0)=(t7→ −2 )
f(X1)=(t7→ −t−1 )
f(X2) = ( t7→ −2t)
f∈ L(E, F )f E F
∀P, Q ∈2[X]∀λ∈
f(P+Q) = t7→ (t−1) (
^
P+Q)0(t)−2(
^
P+Q)(t)
=t7→ (t−1) e P0(t)−2e P(t) + (t−1) e Q0(t)−2e Q(t)
=t7→ (t−1) e P0(t)−2e P(t)+t7→ (t−1) e Q0(t)−2e Q(t)
=f(P) + f(Q)
f(λP ) = t7→ (t−1)( f λP )0(t)−2( f λP )(t)
=t7→ (t−1) λe P0(t)−2λe P(t)
=λt7→ (t−1) e P0(t)−2e P(t)
=λ f(P)
f
Ker(f) = {P∈E|f(P)=0F}
=a0X0+a1X1+a2X2∈2[X]| ∀t∈,(t−1) (a1+ 2a2t)−2 (a0+a1t+a2t2)=0
=a0X0+a1X1+a2X2∈2[X]| ∀t∈,−(2a0+a1)−(a1+ 2a2)t= 0
Ker(f) = a0X0+a1X1+a2X2∈2[X]2a0+a1= 0
a1+ 2a2= 0 .
2a0+a1= 0
a1+ 2a2= 0 Ker(f)
(X0, X1, X2)E
a2
Ker(f) = a2X0−2a2X1+a2X2∈2[X]a2∈
=a2(X0−2X1+X2)∈2[X]a2∈
=V ect(X0−2X1+X2)
f2[X]v=X0−2X1+X2
f
Im(f)F=
Im(f)
E Im(f)Im(f) = V ect(f(X0), f(X1), f(X2))
(f(X0), f(X1), f(X2))
f(X0)−2f(X1) + f(X2)=0F
Im(f) = V ect(f(X0), f (X1)) F
(t7→ at +b)a, b ∈
F f
E, F, G BE,BF,BG
f∈ L(E, F )g∈ L(F, G)g◦f∈ L(E, G)
Mat(g◦f, BE,BG) = Mat(g, BF,BG)Mat(f, BE,BF)