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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
N
f∈ O(E)
Edim E= 2k+ 1 f
Pf=−X2k+1 +... + det f
lim
x→−∞ Pf(x)=+∞lim n→+∞Pf(x) = −∞
x7→ Pf(x)
f
Spec(f)6=∅

f−1f
fC1
f f
f−1∈Spec f
Spec f=−1,1x y x ∈E1y∈E−1
(x|y)=(f(x)|f(y)) = (x| − y) = −(x|y)
2(x|y) = 0 (x|y) = 0
E1E−1
fB= (e1, . . . , en)
MatBf=
ε10
ε2
0εn
εi=±1
f f ◦f=IdE
MatB(f◦f) = (MatBf)2=
ε2
10
ε2
2
0ε2
n
=In
f E1E−1E1E−1
f
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=1
53 4
4−3
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) = AT
a b E
a b