Classe de TSI2 - Exercices de mathématiques

E
A Mn(R)A−1=AT
ATA=InAAT=In
A A
Rn
E E
O(n) = On(R)Mn(R)
GLn(R)
In
O(n)O(n)O(n)
O(n)O(n)

E f ∈ L(E)f
f
∀x∈E, kf(x)k=kxk
E f ∈ L(E)
f
f∀x, y ∈E, (f(x)|f(y)) = (x|y)
f
f
f
f
O(E)E
GL(E)
IdE
O(E)O(E)O(E)
O(E)O(E)
f∈ O(E) det f= 1 det f=−1
SO(E) = {f∈ O(E),det f= 1}
O(E)
f∈ O(E) det f= 1 det f=−1 dim E= 2

f∈GL(E)
f∈ O(E)⇐⇒ ∀x, y ∈E, (f(x)|y)=(x|f−1(y))
A∈ O(n) det A= 1 det A=−1
SO(n) = SOn(R) = {A∈ O(n),det A= 1}
O(n)
EB= (e1, e2, . . . , en)
B0detBB0= det(P ass(B, B0)) = ±1
detBB0= 1 B0B
EB= (
~
i,~ j,~ k)
(
~
i,~ j,~ k)
f∈ O(E) Spec f⊂−1,1
x6= 0 f(x) = λx λ ∈Rkxk=kf(x)k=kλxk=|λ|×kxk
fMn(C)
1f
R2A
•Spec f=1A=1 0
0 1 =I2
•Spec f=−1A=−1 0
0−1=−I2
•Spec f=−1,1A=1 0
0−1A=0 1 1 0 
•Spec f=∅A=0−1
1 0 
f∈ O(E)
ESpec f6=∅
f−1∈Spec f

Spec f=−1,1E1E−1
f f idE
−idE
f∈O(E)F f F ⊥f
EB= (e1, e2)
f A =a c
b d A∈ O(2)
a2+b2= 1, c2+d2= 1, ac +bd = 0
det A=±1c d a b
det A=1: ac +bd = 0
−bc +ad = 1 A=a−b b a 
det A=−1 : ac +bd = 0
−bc +ad =−1A=a b b−a
EE= (e1, e2)
f∈ O(E)⇐⇒ 
B(f) = a−b
b a a2+b2= 1
B(f) = a b
b−aa2+b2= 1
EB= (e1, e2)
f∈ L(E)
B(f) = a b
b−aa2+b2= 1
Spec f=−1,1f
B0
B0(f) = 1 0
0−1
f E1

fB(f) = A=√2
21 1
1−1
E
n−1n
EB= (e1, e2)
f∈ L(E)A=B(f) = a−b
b a a2+b2= 1
B0f
?B0(f) = a−b
b a =A
?B0(f) = a b
−b a =B(f) = tA
a b
E
a b
E r E
r
a−b
b a 
a r b r
θcos θ=asin θ=b
r θ0+ 2πZr
θcos θ=asin θ=b θ ]−π, π]
rθθ
cos θ−sin θ
sin θcos θ
2SO(E)
rθ◦rθ0=rθ+θ0=rθ0◦rθ