Série d'exercices d'algèbre linéaire MP2

2
u v
λ6= 0 u◦v v ◦u
u v R[X]
u(P) = P0v(P) = ZX
0
P(t)dt
ker(u◦v) ker(v◦u)
λ= 0 E
A= (aij )∈Mn(R)kAk
1≤i≤n
n
P
j=1
|ai,j |
(A)⊂[−kAk,kAk]
M∈Mn(C) (Mp)p⊂GLn(R) lim
p→+∞Mp=M
A, B ∈ Mn(C)
χAB =χBA A∈GLn(C))
a, b ∈RM n
M=
a b · · · b b
b a · · · b b
b b · · · a b
b b · · · b a
∈ Mn(R), b 6= 0
M−(a−b)In1
M
M

EKn u E H = ker(u)
f∈ L(E)
H f u ◦f u
A=MB(f)L=MB,BK(u)
H f tLtA
R3
f: (x, y, z)7−→ (2x+y+z, x + 2y+z, x +y+ 2z)
EKn f, g ∈ L(E)f◦g=g◦f
f n
λ∈(f)Eλ(f) = 1
f g
E f g
D=α,..., αn)∈ Mn(K)λi6=λj,∀i6=j
DM =MD M
A=
3 0 1
3 1 −2
1 0 3
A
M3(K), X2=A
n≥3
A=
0 (0) 1
1
1
1 (0) 0
∈ Mn(R)
A A2
fRnA
ker f⊕f=Rn
A

0 (0)
0
(0) B
B∈GLn(R)
B B2
B A A
f g f ◦g=g◦f
f g
A B ∈ Mn(K)AB =BA
P∈GLn(K)P−1AP P −1BP
A+B AB
fKE
Γf{g∈L(E); f◦g=g◦f}
f sp(f) = {λ1, ..., λr}Ei=Eλi(f)
g∈Γf⇔ ∀1≤i≤r, Eig
ϕ: Γf−→ L(Eλ1(f)) ×... × L(Eλr(f))
g7−→ (g|E1, ... , g|Er)
Γf
fKE n
x0∈E{x0, f(x0), ..., fn−1(x0)}E
g∈ L(E)f◦g=g◦f
P∈Kn−1[X]g(x0) = P(f)(x0)
f◦g=g◦f g ∈Kn−1[f].
A∈ Mn(R)B=0In
A0
B A
A B

A∈ Mn(C)A= 1
A A 6= 0
Xn+anXn−1+... +a1X+a0∈Cn[X]
CP=
0 0 · · · 0−a0
1 0 · · · 0−a1
0
1 0 −an−2
0· · · 0 1 −an−1
χCP=P
λ∈(Cp)Eλ
CPP
J=
0 1 · · · 0 0
0 0 · · · 0 0
0
0 0 1
1· · · 0 0 0
χJ(J)M=
n−1
P
i=0
aiJi
u E DuL(E)
f∈ L(E); Du(f) = u◦f
P∈C[X]P(Du) = DP(u).
P(u)=0 P(Du)=0
u Du
a∈E f ∈ L(E)Ia={P∈K[X], P (f)(a) = 0}
IaK[X]
ΠaIa= ΠaK[X]
d= Πa
{a, f(a), ..., fd−1(a)} {P(f)(a), P ∈K[X]}

A, B ∈ Mn(C)χA(B)=0
A B
M=A0
0BA, B ∈ Mn(C)
∀P∈K[X]P(M) = P(A) 0
0P(B)
A B
M=A B
0AA, B ∈ Mn(C)AB =BA
k∈NMk
∀P∈R[X], P (M) = P(A)P0(A)B
0P(A)
M A B = 0
A∈ Mn(K)P∈K[X]
P(A)P∧QA= 1
A∈Mn(C)A2A
λ∈sp(A)∀k∈N2λk∈sp(A)
A
P∈K[X]P(0) = 0 A∈Mn(K)
A P (A)
EKn∈N∗f∈ L(E)χf=
p
Q
i=1
(X−λi)mi
i∈ {1, ..., p}Fi= ker(f−λiIdE)miNi=f−λiIdE
E=⊕p
i=1Fi
FiNiNi|Fi
∀i∈ {1, ..., p}Fi=mi