Аналитическая геометрия. Часть III. Многомерные пространства. Гиперповерхности второго порядка. Шурыгин В.В. - 12 стр.

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bα : V
n
→ V
0
m
bα(
−→
AB) =
−−−−−−→
α(A)α(B) A, B ∈ A
n
.
α
O
0
= α(O) O ∈ A
n
bα
O
M
O
0
M
0
α
M ∈ A
n
M
0
= α(M)
−−−→
O
0
M
0
= bα(
−−→
OM)
α(π
k
) π
k
= {M
0
, L
k
} ⊂ A
n
α : A
n
→ A
0
m
π
0
p
= {α(M
0
), bα(L
k
)} ⊂
A
0
m
p = dim bα(L
k
) 6 k
A
n
α {α(M
0
), im(bα)} ⊂ A
0
m
M
0
M
v
α
α(M
0
)
α(M)
bα(v)
im(bα) = V
0
m
α A
n
A
0
m
im(bα) = 0 α A
n
A
0
m
α
−1
(π
0
k
) π
0
k
= {M
0
0
, L
0
k
} ⊂ A
0
m
α : A
n
→ A
0
m
α
−1
(L
0
k
)
gig|fjymbam
                                       0
                             α
                             b : Vn → Vm
ijvgmw sig
              α
                −→    −−−−−−→
              b(AB) = α(A)α(B)
                                   not o{|cr   A, B ∈ An .


   E ¨7-9- -.3*D*6*,08 /6*D2*7)7- 5„„0,,-* -7-+35“*,0* α -D,-E,5),-
-.3*D*68*7/8 -+35E-: O0 = α(O) 6‡+-” -D,-” 7-)10 O ∈ An 0 60,*”,A:
                                 -7-+35“*,0*: αb <
                                      *”/7B07*6>,-  .2/7> M ∈ An F .3-0E
                        0
      M               M

             α                   B-6>,58 7-)15 0 .2/7> M 0 = α(M ) < -9D5
                                              OM ) <
                                 −−0−→0       −−→
   O                             O  M   = αb (
                                      51 /6*D/7B0* .-62)5*:
                          0
                       O
   ŽXYZ[—¡Y\]Y^ Ô|fjl α(π ) eoghvghia π = {M , L } ⊂ A efa j``abp
bgd gig|fjymbaa α : An → A0m tkotmiht eoghvghi‚{ πp0 = {α(M0), αb(Lk )} ⊂
                            k                  k     0  k     n


A0m fjldmfbghia p = dim α  b(Lk ) 6 k €
   ¹ sjhibghiaw g|fjlgd j``abbgxg       efghifjbhikj An efa j``abbgd gig|p
fjymbaa α tkotmiht eoghvghi‚ {α(M ), im(bα)} ⊂ A0 €
                                     0             m




                               α                           α(M )
              v   M                            α
                                               b(v)
         M0
                                                  α(M0 )


   /60 im(bα) = V0  7- α /‡3Ö*170B,- •-7-+35“5*7 .3-/735,/7B- A ,5
B/* .3-/735,/7B- A0m ‹<
                    m                                              n

   /60 im(bα) = 0  7- α -7-+35“5*7 .3-/735,/7B- A B -D,2 7-)12 .3-/7
35,/7B5 A0m <
                                                    n


   ŽXYZ[—¡Y\]Y^ žfgg|fjl α−1(π0 ) eoghvghia π0 = {M 0 , L0 } ⊂ A0 efa
j``abbgd gig|fjymbaa α : An k→ A0m tkotmiht        oa|g e}hicd dbgymp
                                                k       0  k     m


hikgd oa|g eoghvghi‚{ h bjefjkot{qad egnefghifjbhikgd α−1(L0k ) €

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