Exercices de Khôlles de Mathématiques

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Mn,p(K)r
Jn,p,r
EKn f E fn= 0
fn−16= 0 f g E
f
p A1, A2, . . . , ApGln(K)p
tr p
X
i=1
Ai!≡0[p].
A Mn(K)
A
X Y Mn(K)A=XY −Y X
22
A∈Mn(R)
A I ⊂J1, nKJ⊂J1, nK
A Ai∈I,j∈JCard(I)×Card(J)

σ∈Snσ{a1,σ(1), a2,σ(2), . . . , an,σ(n)}
A0
A r ×s
A
1
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24
E|| · ||E
G E R
xRy⇐⇒ x−y∈G.
F=E/G xbx
|| · ||FF
||bx||F= inf
y∈G||x−y||E.
(F, || · ||G)
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UR2RC1
U1

fR2RC1(x, y)∈
R2
x∂f
∂x (x, y) + y∂f ∂y (x, y) = x2+y2+px4+y4.
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A∈ Mn(C)ρ(A)
A
(Uk)k≥1=Ak⇐⇒ ρ < 1.
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Rn[X]n
µ: ]0,1[→R+
∗]0,1[ P, Q ∈Rn[X]
(P|Q) = Z1
0
µ(x)P(x)Q(x)dx.
Rn[X]
(Pi)
Pii
(Pi)Rn[X]

i1n Pii]0,1[
i1n Pii
]0,1[
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E3
SO(E)SO(E)G SO(E)
∀φ∈SO(E),∀g∈G, φ ◦g◦φ−1∈G
SO(E)
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ABCD 1
1
8
[AB]M
(AB)M
(AB)M