Аналитическая геометрия. Часть III. Многомерные пространства. Гиперповерхности второго порядка. Шурыгин В.В. - 84 стр.

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(ϕ
α
β
)
ϕ {e
α
}
[ϕ] [ψ]
ϕ = λψ λ ∈ R ϕ = λψ [ϕ] = [ψ]
[ϕ] = [ψ] ψ : e
α
7→ e
0
α
ϕ : e
α
7→ λ
α
e
0
α
λ
α
α = 1, . . . , n + 1
ϕ : e
1
+e
2
+. . .+e
n+1
7→ λ
1
e
0
1
+λ
2
e
0
2
+. . .+
λ
n+1
e
0
n+1
= µ(e
1
+e
2
+. . .+e
n+1
)
λ
α
= µ α = 1, . . . , n+1
(ϕ
α
β
)
[e
α
]
P
n
[ϕ
α
β
]
P
n
GP (P
n
) GL(V
n
)/H
GL(V
n
)
V
n
H λ id :
x 7→ λx
GP (P
n
)
GL(n, R )/H H
(ϕ
α
β
) = (λ δ
α
β
)
RP
n
ϕ : V
n+1
→ V
n+1
x x
0
x
α
{e
α
} {e
0
α
}
e
0
α
= ϕ(e
α
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ϕ : x = x
α
e
α
7−→ x
0
= x
α
e
0
α
,
eϕ : P
n
→ P
n
[x] [x
0
]
[x
α
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α
] [e
0
α
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eϕ : P
n
→ P
n
E
α
7→ E
0
α
α = 1, . . . , n + 1 E 7→ E
0
9D* (ϕαβ) F :5730½5 60,*”,-9- .3*-+35E-B5,08 ϕ B +5E0/* {eα} <
    ’3-*170B,A* .3*-+35E-B5,08 [ϕ] 0 [ψ] /-B.5D5‡7 7-9D5 0 7-6>1- 7-9D5
1-9D5 ϕ = λψ  λ ∈ R < *”/7B07*6>,-  */60 ϕ = λψ  7-  -)*B0D,-  [ϕ] = [ψ] <
’2/7> 7*.*3> [ϕ] = [ψ] 0 ψ : eα 7→ e0α  7-9D5 ϕ : eα 7→ λαe0α  9D* λα
E5B0/07 •.30-30‹ -7 α = 1, . . . , n + 1 < -  751 “*  151 0 B D-15E57*6>/7B*
.3*D6-“*,08 ,5 / < I» .-/1-6>12 ϕ : e1 +e2 +. . .+en+1 7→ λ1e01 +λ2e02 +. . .+
                                          )
λn+1 e0n+1 = µ(e1 +e2 +. . .+en+1 )  .-62 5*: λα = µ .30 B/*C α = 1, . . . , n+1 <
    7/‡D5 B )5/7,-/70  /6*D2*7 )7- :5730½5 (ϕα) .3-*170B,-9- .3*-+35E-
B5,08 •º·‹ -.3*D*6*,5 / 7-),-/7>‡ D- 2:,-“*,08       β
                                                       ,5 )0/6-  5 .3-*170B,A”
3*.*3 [eα] 2/75,5B60B5*7 BE50:,- -D,-E,5),-* /--7B*7/7B0* :*“D2 .3-*1
70B,A:0 .3*-+35E-B5,08:0 .3-/735,/7B5 Pn 0 ./*BD-:5730½5:0 [ϕαβ] <
    @,-“*/7B- .3-*170B,AC .3*-+35E-B5,0” .3-/735,/7B5 P -+35E2*7 932.
.2 GP (Pn)  0E-:-3„,2‡ „517-3932..* •/:< ºà  Í·¶‹ GL(Vn)/H 932..A
                                                                  n

                                          ‹
GL(Vn ) 60,*”,AC •,*BA3-“D*,,AC .3*-+35E-B5,0” B*17-3,-9- .3-/7
35,/7B5 Vn .- ,-3:56>,-” .-D932..* H  /-/7-8ˆ*” 0E 9-:-7*70” λ id :
x 7→ λx < ’-/1-6>12 B 1--3D0,575C 60,*”,A* .3*-+35E-B5,08 E5D5‡7/8
,*BA3-“D*,,A:0 :5730½5:0  932..5 GP (Pn) 0E-:-3„,5 „517-3932..*
                              )
GL(n, R)/H .-6,-” :5730 ,-” 932..A .- ,-3:56>,-” .-D932..* H  /-
/7-8ˆ*” 0E /15683,AC :5730½ (ϕαβ) = (λ δβα) <
    ’3*-+35E-B5,0* .3-*170B,AC 1--3D0,57 •º²‹ .3*D/75B68*7 /-+-” .3-*1
70B,-* .3*-+35E-B5,0* /75,D537,-9- .3-*170B,-9- .3-/735,/7B5 RPn <
    ’-/1-6>12 60,*”,-* .3*-+35E-B5,0* ϕ : Vn+1 → Vn+1 2/75,5B60B5
*7 BE50:,--D,-E,5),-* /--7B*7/7B0* :*“D2 B*17-35:0 x 0 x0  0:*‡ˆ0
:0 -D0,51-BA* 1--3D0,57A xα .- -7,-‰*,0‡ 1 +5E0/5: {eα} 0 {e0α}  9D*
e0α = ϕ(eα )  /--7B*7/7B*,,-~

                         ϕ : x = xα eα 7−→ x0 = xα e0α ,
7- .3-*170B,-* .3*-+35E-B5,0* ϕe : Pn → Pn 2/75,5B60B5*7 BE50:,--D
,-E,5),-* /--7B*7/7B0* :*“D2 7-)15:0 [x] 0 [x0]  0:*‡ˆ0:0 -D0,51-BA*
1--3D0,57A [xα] .- -7,-‰*,0‡ 1 3*.*35: [eα] 0 [e0α] /--7B*7/7B*,,- < 7
/‡D5 /6*D2*7 •/:< .3*D6-“*,0* ,5 / < I» )7- .3-*170B,-* .3*-+35E-B5
,0* ϕe : Pn → Pn -D,-E,5),- -.3*D*68*7/8 /--7B*7/7B08:0 Eα 7→ Eα0 
                                        )
α = 1, . . . , n + 1  E 7→ E 0 :*“D2 7- 15:0  E5D5‡ˆ0:0 .3-*170B,A* 3*
                                        Ï´

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