Algèbre linéaire 1. Fiche n 4 Applications linéaires et

◦4
f g R3R3
f(x, y, z) = (2x+ 3y−4z, 2y+z, 2z)g(x, y, z) = (3x, 2x+ 3y, x +y+ 3z).
f g f ◦g g ◦f f +gR3
fngnn∈N
P1(X)=3X+ 2 P2(X)=2X+ 3 f:R1[X]→R1[X]
f(P) = P0
(P1, P2)R1[X]f
Rn[X]n
R{1, X, X2, ..., Xn}
Rn[X]→Rn[X]
P(X)7→ P0(X)
P(X)7→ 1
2(P(X) + P(−X))
P(X)7→ P(X)−P(0)
P(X)7→ 7P(X)
P(X)7→ P(X+ 1) −P(X)
(e1, e2, e3)R3
f1=−e1+e3, f2= 2e1−e2+ 2e3, f3=e1−e2+e3.
(f1, f2, f3)R3
(f1, f2, f3)
(f1, f2, f3)
(x, y, z)
(f1, f2, f3)
u:R2→R2u(x, y)=(x−2y, 3y)
u(e1, e2)R2
f1=e1f2=e1+e2(f1, f2)R2
u
A=
2 1 0
−2 1 2
−1 1 3
u A
(e1, e2, e3)
f1=e1−e2+e3f2=e1+e3f3=e2+e3(f1, f2, f3)
R3
u(f1, f2, f3)

e= (e1, e2, e3)R3fR3
A=Me,e(f) = 
223
−2−1−2
112
.
f(bi)b1= (1,−2,1) b2= (0,2,−1) b3= (0,0,1)
b= (b1, b2, b3)R3
T=Mb,b(f)f
N2N=
0 1 3
0 0 2
0 0 0
Nkk≥0
n∈NTnn N n
Ann
uR3(e1, e2, e3)
M=
15 −11 5
20 −15 8
8−7 6
e0= (e0
1, e0 2, e0 3)R3
e0
1= 2e1+ 3e2+e3, e0 2= 3e1+ 4e2+e3e0 3=e1+ 2e2+ 2e3.
e0
e0
M0u e0
Mnn∈N
u u−1