Applications linéaires 1 Aspect vectoriel

E F E F
u:E→F
• ∀(x, y)∈E2u(x+y) = u(x) + u(y)
• ∀λ∈R∀x∈E u(λx) = λu(x)
u:E→F∀(λ, µ)∈R2∀(x, y)∈E2
u(λx +µy) = λu(x) + µu(y)
u k
X
i=1
λiei!=
k
X
i=1
λiu(ei)
•u:R2→R3u(x, y) = (2x−3y, 4x+y, −x+ 2y)
•u:R2→R3u(x, y) = (2x−3,4+y, −x+2y)
u(2x, 2y)6= 2u(x, y)
•u:M3(R)→ M3(R)u(M) = AM
A∈ M3(R)
•u:M3(R)→ M3(R)u(M) = M2
(M+N)26=M2+N2
•E f :E→R3f(un) = (u0, u8, u35)
•E C∞R R u:E→E u(f) = f0
•E[0; 1] u:E→R
u(f) = Z1
0
f(t)dt
u:E→F E F
L(E, F )E
F
u:E→E
L(E)E
E GL(E)

u:E→F Ker(u) = {x∈E|
u(x) = 0}
u:E→F Ker(u)
E
(x, y)∈Ker(u)2u(x) = u(y) = 0 u(x+y) = u(x)+u(y) = 0
x+y∈Ker(u)x∈Ker(u)u(λx) = λu(x) = 0 λx ∈Ker(u)
u x ∈E u(x) = u(x+0) = u(x)+u(0)
u(0) = u(x)−u(x) = 0
uR3(x, y, z)7→ (x−y+z, 3x−
2y+ 5z, −x−3z) (x, y, z)
x−y+z= 0
3x−2y+ 5z= 0
−x−3z= 0
2L1−L2=L3
(−3z, −2z, z)z∈Ru
1R3(−3,−2,1)
Ker(u) = {0}
u
(x, y)∈E2u(x) = u(y)u(x−y) = u(x)−u(y) = 0 x−y∈Ker(u)
x−y= 0 x=y u
0u0
E Ker(u)
u:E→F Im(u) = {y∈F| ∃x∈E, u(x) =
y}
u:E→F u
Im(u) = F
F
u(0) = 0 y∈Im(u)
λ∈Ry=u(x)u(λx) = λu(x) = λy λy ∈Im(u)y y0
u y =u(x)y0=u(x0)u(x+x0) = y+y0
u:E→F E F (e1, . . . , en)E
Im(u) = V ect(u(e1), . . . , u(en))
u(e1), . . . , u(en)Im(u)
V ect(u(e1), . . . , u(en)) ⊂Im(u)y∈Im(u)y=u(x) (e1, . . . , en)
E x =
i=n
X
i=1
λieiy=u(x) = u i=n
X
i=1
λiei!=
i=n
X
i=1
λiu(ei)
y∈V ect(e1, . . . , en)

(u(e1), . . . , u(en)) Im(u)
R2R3u(x, y) = (2x−y, x + 2y, −2x+y)
(a, b, c)Im(u)
2x−y=a
x+ 2y=b
−2x+y=c
a=−c
Im(u) = {(a, b, −a)|a, b ∈R2}=V ect((1,0,−1); (0,1,0))
R2
(1,0) (0,1) u(1,0) = (2,1,−2) u(0,1) = (−1,2,1)
Im(u) = V ect((2,1,−2); (−1,2,1))
(e1, e2, . . . , en)E n (f1, f2, . . . , fp)
F p u :E→F
x E u(x) = u i=n
X
i=1
xiei!xix
u(x) =
i=n
X
i=1
xiu(ei) =
i=n
X
i=1
xi
j=p
X
j=1
ai,j ej=
j=p
X
j=1 i=n
X
i=1
ai,j xi!ejai,j
i ej(f1, . . . , fp)
u:E→F
B= (e1, e2, . . . , en)EB0= (f1, . . . , fp)F
uB B0M∈ Mp,n(R)i
u(ei)B0u(ej) =
n
X
i=1
λifiMi,j =λi
u:R3→R2u(x, y, z) = (4x−3x+z, −2x+y−5z)u
M=4−3 1
−2 1 −5
u E BE
MatB(u)uB B
X=
x1
xp

Bx∈E u(X) = 
y1
yn
B0u(X) = MX M u B B0
x=
p
X
i=1
xieiM u(ei) =
n
X
j=1
Mjifj
u(X) =
p
X
i=1
xi
n
X
j=1
Mjifj=
n
X
j=1 p
X
i=1
xiMji!fjj
MX
p
X
i=1
xiMji
u∈ L(E, F )λ∈RM u B B0
λu λM
(u, v)∈ L(E, F )2M N u +v M +N
u∈ L(E, F )v∈ L(F, G)M N
v◦u NM
u(X) = MX v(X) = NX λu(X) = λMX u(X) + v(X) = MX +
NX = (M+N)X v ◦u(X) = v(MX) = NMX
u∈ L(R3,R2) (x, y, z)7→ (x−y, 2x+z)v∈ L(R2,R3)
(x, y)7→ (x+y, 3x−y, −x+ 2y)
M=1−1 0
201N=
1 1
3−1
−1 2 
NM =
3−1 1
1−3−1
3 1 2 
v◦u(x, y, z) = (3x−y+z, x −3y−z, 3x+y+ 2z)
M u−1
M−1
u u−1∀X∈E u−1(u(X)) = X
u−1Y1Y2E X1
X2u X1+X2Y1+Y2u−1(Y1+Y2) = X1+Y2
N u−1
u−1◦u=u◦u−1=id
NM =MN =I
I N =M−1
u:E→F λ u
x∈E u(x) = λx
u λ
uR2u(x, y) = (2x+y, −3x−2y)
u M =2 1
−3−21
u(1,−1) −1
u(1,−3) u(1,−3) = (−1,3) = −(1,−3)

((1,−1); (1,−3)) R2
u
M0=1 0
0−1
u
u∈ L(E)BE
u
E u
B= (e1, . . . , en)B0= (f1, . . . , fn)
EB B0P= (pi,j )pi,j
i fjB
u∈ L(E)A
A0B B0A0=P−1AP
((1,−1); (1,−3)) P=1 1
−1−3P−1=3
2
1
2
−1
2−1 2P−1MP =
1 0
0−1
P MP −1