Аналитическая геометрия. Часть II. Аналитическая геометрия пространства. Шурыгин В.В. - 36 стр.

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Ann (S) ⊂ V
∗
n
Ann (S) = Ann (L(S))
e
u,
e
v ∈ Ann (S) =⇒
e
u(a) =
e
v(a) = 0
a ∈ S =⇒ (λ
e
u + µ
e
v)(a) = 0 a ∈ S =⇒ λ
e
u + µ
e
v ∈ Ann (S)
⊂ S ⊂ L(S)
L(S)
S Ann (L(S)) ⊂ Ann (S)
⊃
e
u ∈ Ann (S)
a ∈ L(S)
a = λ
1
b
1
+ λ
2
b
2
+ . . . + λ
k
b
k
λ
1
, . . . , λ
k
∈ R b
1
, . . . , b
k
∈ S
e
u(a) = λ
1
e
u(b
1
) + λ
2
e
u(b
2
) + . . . + λ
k
e
u(b
k
) = 0 =⇒
e
u ∈ Ann (L(S))
Ann (L(S)) ⊃ Ann (S)
L
m
⊂ V
n
{b
1
, . . . , b
m
}
L
m
Ann (L
m
) = Ann ({b
1
, . . . , b
m
}).
Ann (L
m
)
b
1
α
w
1
+ b
2
α
w
2
+ . . . + b
n
α
w
n
= 0, α = 1, . . . , m,
w
i
e
w
(b
i
α
)
α = 1, . . . , m i = 1, . . . , n {b
1
, . . . , b
m
}
L
m
m n −m
dim Ann (L
m
) = n − dim (L
m
) = n − m
V
n
V
∗
n
Ann(Ann (L
m
)) = L
m
a ∈ L
m
h
e
w, ai = 0
e
w ∈ Ann (L
m
)
=⇒ a ∈ Ann(Ann (L
m
)) =⇒ L
m
⊂ Ann(Ann (L
m
))
dim Ann(Ann (L
m
)) = n − (n − m) = m = dim L
m
   eIfiŸh¯fMLf'
   € Ann (S) ⊂ V∗ ; ‘|³‘x|tuxv{tu}|­
   < Ann (S) = Ann (L(S)) 
                   n


   ½hjnonqfŸ¾pqkh' € ue, ve ∈ Ann (S) =⇒ ue(a) = ve(a) = 0 C13 1A*++
a ∈ S =⇒ (λe   u + µev)(a) = 0 C13 1A*++ a ∈ S =⇒ λe            u + µev ∈ Ann (S)
                                                                                        
   < .1A()) ù⊂ ù 0. .0. S ⊂ L(S)  9-3.03 1)D03 F+40O +*0B
  0A 03-3 9 17 0 *+17@)4 ,+C4+s)-29) L(S) O +*0 0)2-3 9 17  0
4)7@)4 ,+C4+s)-29) S O 2+O +()9C+O Ann (L(S)) ⊂ Ann (S) 
    .1A()) ù⊃ ù ?-27 ue ∈ Ann (S)  ?+:9+17D †1)4)2 a ∈ L(S)
4))2 9C a = λ1b1 + λ2b2 + . . . + λk bk O C) λ1, . . . , λk ∈ R O b1, . . . , bk ∈ S 
 +C0 ue(a) = λ1ue(b1) + λ2ue(b2) + . . . + λk ue(bk ) = 0 =⇒ ue ∈ Ann (L(S)) 
 0.4 +*0:+4O Ann (L(S)) ⊃ Ann (S) 
   ŸfipqkLf' ÁtuÅ L ⊂ V Æ ‘|³‘x|tuxv{tu}|  {b , . . . , b } Æ ’v“t
              É
} Lm º u|•³v
                        m      n                                 1        m


                          Ann (Lm ) = Ann ({b1 , . . . , bm }).
    .++C020E C13 0E+sC)3 0132+0 Ann (Lm) s+ )@27 -B
-2)4 1)DE +C++CE 09)D
                 b1α w1 + b2α w2 + . . . + bnα wn = 0,   α = 1, . . . , m,
+2+-2)17+ .++C02 wi 1)D+D F+4 we O 402‡0 .+2++D (biα) O
α = 1, . . . , m O i = 1, . . . , n O -+-2091)0 : .++C02 9).2++9 {b1 , . . . , bm } O
+*0:A E *0:- ,+C,+-20-290 Lm  ?+-.+17. 0 †2+D 402‡ 0B
9) m O 2+ )@)3 --2)4 +*0:A2 ,+C,+-20-29+ 0:4)+-2 n − m
       « 
-4 ª— O ®€<  2-AC0 -1)C)2
   eIfiŸh¯fMLf' dim Ann (L ) = n − dim (L ) = n − m 
   1)CA )) ,)C1+s)) -2009190)2 9:04+ +C+:0(+) -++29)2B
                                        m               m


-29) 4)sC ,+C,+-20-2904 9 Vn  Vn∗ 
   eIfiŸh¯fMLf' Ann(Ann (L )) = L 
                                         m      m
     )D-292)17+O )-1 a ∈ Lm O 2+ hw,         e ai = 0 C13 1A*++ we ∈ Ann (Lm )
=⇒ a ∈ Ann(Ann (Lm )) =⇒ Lm ⊂ Ann(Ann (Lm )) + ,+ ,)CCB
                                                                   
  )4 ,)C1+s)A dim Ann(Ann (Lm)) = n − (n − m) = m = dim Lm 
?+-.+17. ,+ +-+9+D 1)44) + 1)D+D :09-4+-2 ,+C,+-20-29+O
                                               Σ

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