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a∈Rfafa(x) = eax (fa)a∈R
n∈N?a1< a2< . . . < an
λ1, λ2, . . . , λnλ1fa1+λ2fa2+. . . +λnfan= 0
∀x∈R, λn=−
n−1
X
k=1
λke(ak−an)x.
ak−an<0k < n x +∞λn= 0
E u, v ∈ L(E)
Im(u+v)⊂Im(u) + Im(v)
Ker(u) + Ker(v) = EIm(u+v) = Im(u) + Im(v)
x∈E u(x)∈Im(u)v(x)∈Im(v)u(x) + v(x)∈Im(u) + Im(v)
u v =−uIm(u) + Im(v) = Im(u)6={0}
u+v= 0 Im(u+v) = {0}
zIm(u) + Im(v)x y ∈E z =u(x) + v(y)
x=xu+xvy=yu+yvxu, yu∈Ker(u)xv, yv∈Ker(v)u(x) = u(xv)
v(y) = v(yu)z=u(w) + v(w)w=xv+yu
E F V E
LV(E, F ) = {u∈ L(E, F )V⊂Ker(u)}
L(E, F )
LV(E, F )
L(E, F )3u7→ u|V∈ L(V, F )V
W V E
φ:LV(E, F )3u7→ u|W∈ L(W, F )W
φ(u)=0 u W u ∈ LV(E, F )u V u V +W=E
v∈ L(W, F )u v W V
V+W=ELV(E, F )φ(u) = vLV(E, F )L(W, F )
dim(LV(E, F )) = dim(L(W, F ))
= dim(W) dim(F)
= (dim(E)−dim(V)) dim(F).
BV+W W
V E u LV(E, F )B
V
z }| {
0··· 0
W
z }| {
?··· ?
0··· 0?··· ?
.
?dim(W) dim(F)LV(E, F ) (dim(E)−
dim(V)) dim(F)
E F f ∈ L(E, F )f g ∈ L(F, E)
f◦g= IdF

g∈ L(F, E)f◦g= IdFx∈F f(g(x)) = x
x∈Im(f)
SKer(f)E f S
Im(f) = F g :F→S x ∈F f(g(x)) = x
n0< x0< x1< . . . < xn<1n+ 1 (n+ 1)
(ω0, ω1, . . . , ωn)
∀P∈Rn[X],Z1
0
P(x) dx=
n
X
i=0
P(xi)ωi.
φ:P7→ Z1
0
P(x) dx
Rn[X]i∈ {0,1, . . . , n}φiP7→ P(xi) (φ0, φ1, . . . , φn)
λ0, λ1, . . . , λnλ0φ0+λ1φ1+. . . +λnφn= 0
1xi0xjj6=i λi= 0 (φ0, φ1, . . . , φn)
n+ 1 Rn[X]Rn[X]
ω0, ω1, . . . , ωnφ(φ0, φ1, . . . , φn)
E u ∈ L(E)
φu:L(E)→ L(E)
v7→ u◦v.
φu
u◦v= 0 Im(v)⊂Ker(u)
Ker(φu) = L(E, Ker(u)) ⊂ L(E, E).
rg(φu) = dim(L(E)) −dim Ker(φu)
= (dim E)2−dim L(E, Ker(u))
= (dim E)2−(dim E)(dim Ker(u))
= (dim E)(dim E−dim Ker(u))
= (dim E)(rgu).
Im(u◦v)⊂Im(u) Im(φu)⊂ L(E, Im(u))
L(E, Im(u)) φu
Im(φu) = L(E, Im(u)).
A, B ∈ Mn(R)AB =BA F A B(F)⊂F
X∈F λ X AX =λX
A(BX) = BAX
=B(AX)
=B(λX)
=λ(BX),
BX ∈F

E n u ∈ L(E)un= 0
u
u u = 0
λ u x un(x) = λnx x
λn= 0 λ= 0
uKer(u)u
Ker(u)n u = 0
A=
1··· ··· 1
0 0
0 0
1 0 0 1
∈ Mn(R).
A
A
λ∈RX= (x1, x2, . . . , xn)∈RnAX =λX
(1 −λ)x1+x2+x3+. . . +xn= 0
(λ−1)x2=x1
(λ−1)x3=x1
(λ−1)xn=x1,
λ6= 1
(1 −λ) + n−1
λ−1x1= 0
xi=x1
λ−1∀i∈ {2, . . . , n}
.
X6= 0 x16= 0 (1 −λ) + n−1
λ−1= 0 (λ−1)2=n−1
λ= 1±√n−1
λ1= 1 + √n−1, Eλ1= Vect 
1
(n−1)−1/2
(n−1)−1/2
λ2= 1 −√n−1, Eλ2= Vect 
1
−(n−1)−1/2
−(n−1)−1/2
.
λ= 1 AX =X
(x2+x3+. . . +xn= 0
x1= 0 ,
1n−2
λ3= 1, Eλ3= Vect 
0
1
0
−1
,
0
0
1
−1
,...,
0
0
1
−1
.
Eλ1Eλ21Eλ3n−2
n A

uRn[X]u(P)=(X2−1)P00
u
uRn[X]
u u
P n
u(P)P P 00 n−2u(P) = (X2−1)P00
n
u(1) = u(X)=0 u(Xk) = k(k−1)Xk−k(k−1)Xk−2k≥2
(1, X, X2, . . . , Xn)Rn[X]u
0 0 −2 0 0
0 0 0 −6
0 2 0 0
0 6 −n(n−1)
0
0n(n−1)
.
uRn[X] (0,0,2, . . . , n(n−1))
χu(x) = x2
n
Y
k=2
(x−k(k−2)).
u χu{0,2,6, . . . , n(n−1)}n−1
0 2 1 X
E0= Ker(u)E02 2
E00χu
u
A, B ∈ Mn(R)
AB BA
λ AB EλFλAB BA
λ B(Eλ)⊂FλA(Fλ)⊂EλEλFλ
AB BA
M N det(MN) = det(M) det(N)
Mdet(M−1) = det(M)−1
χAB(x) = det(xIn−AB)
= det(xIn−ABAA−1)
= det(A(xIn−BA)A−1)
= det(A)χBA(x) det(A−1)
=χBA(x).
λ BA X ∈FλBAX =λX AX ∈Eλ

AB(AX) = λ(AX)
AB(AX) = A(BAX)
=A(λX)
=λAX.
A(Fλ)⊂EλA
dim(A(Fλ)) = dim(Fλ) dim(Fλ)≤dim(Eλ)A
B A(Fλ)⊂Eλdim(Eλ)≤dim(Fλ)
dim(Eλ) = dim(Fλ)
S AB BA
AB ⇔X
λ∈S
dim(Eλ) = n
⇔X
λ∈S
dim(Fλ) = n
⇔BA .
AB =B−1(BA)B AB BA B
AB BA