Аналитическая геометрия. Шурыгин В.В. - 15 стр.

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a
1
, a
2
a
1
||a
2
a
1
, a
2
, a
3
a
1
, a
2
, a
3
, a
4
a
1
a
2
a
3
O a
i
=
−−→
OA
i
i = 1, 2, 3, 4 A
4
OA
1
OA
2
OA
3
B
1
B
2
B
3
−−→
OB
i
−−→
OA
i
λ
i
i = 1, 2, 3
−−→
OB
i
= λ
i
−−→
OA
i
a
4
= λ
1
a
1
+ λ
2
a
2
+ λ
3
a
3
V {a
1
, a
2
, . . . , a
n
}
{b
1
, b
2
, . . . , b
m
}
a
α
= λ
1
α
b
1
+
λ
2
α
b
2
+ . . . + λ
m
α
b
m
α = 1, 2, . . . , n
a
1
= λ
1
1
b
1
+ λ
2
1
b
2
+ . . . + λ
m
1
b
m
a
2
= λ
1
2
b
1
+ λ
2
2
b
2
+ . . . + λ
m
2
b
m
. . .
a
n
= λ
1
n
b
1
+ λ
2
n
b
2
+ . . . + λ
m
n
b
m
.
n > m {a
1
, a
2
, . . . , a
n
}
m = 1 a
α
= λ
1
α
b
1
α = 1, 2, . . . , n a
1
= 0
a
α
a
1
6= 0 λ
1
1
a
2
− λ
1
2
a
1
= 0
m 6 k
m = k + 1
m = k + 1 n > m
b
1
   òƒ ö” ”Ž‰žŠ a , a ‹ŒŽ¦Š ¢”ŒŒ’‘ žŠ©ª Œ žŠ‹Ÿ‰Š žŠ©ª¥ ‰Š©ª
ŠŒ ‰Š‹‹ŒŽ‘Û a ||a §         1       2

   _ƒ ¨Œ ”Ž‰žŠ a , a , a ‹ŒŽ¦Š ¢”ŒŒ’‘ žŠ©ª Œ žŠ‹Ÿ‰Š žŠ©ª¥ ‰Š©ª
                             1       2


ŠŒ ¡‹‹Ž‹Ÿ‘ ŠªŠ¦ ¡‹Š‰ŠžŒ§
                             1       2       3


   „610* b*18-3 ,6[b6J8/9 ‰Š’¡‹‘’Œ= €27*b-M b*18-3 /)086*8/9
.63677*7?,; b/91-M .7-/1-/80=
   ムôÀ¬‘Ž ¤Žž‘Ž ”Ž‰žŠ a , a , a , a ‹ŒŽ¦Š ¢”ŒŒ’‘§ H30 L-16[6K
8*7?/8b* ~8-:- /b-M/8b6 ;-|,- /)0868?\ )8- b*18-3 a \ a \ a · 70,*M,-
                                                         1       2   3   4


,*[6b0/0; ‚b /72)6*\ 1-:L6 -,0 70,*M,- [6b0/0;\ 28b*3|L*,0* -)*b0LK                1       2   3


,-ƒ= ˆ87-|0; b/* )*83* b*18-36 -8 -L,-M 8-)10 O = H2/8? a = −OA          −→ \
                   ­                        )
i = 1, 2, 3, 4 = .3-*18032*; [68*; 8- 12 A ,6 16|L2J 0[ .39;N OA \
                                                                                                         i             i

                                               H
OA 0 OA .63677*7?,- Lb2; L32:0; = 2/8? .3-*1 09;0 9b79J8/9\ /-K
                                                                             4                                         1


-8b*8/8b*,,-\ 8-)10 B \ B 0 B = „61 161 b*18-3 −OB
     2             3
                                                             −→ 0 −−→ 1-770,*K
63,\ 8- ,6ML28/9 8610* )0/76 λ \ i = 1, 2, 3 \ )8- −OB
                                                                  OA
                                                                = λ OA = „-:L6
                                     1           2           3                           i           i
                                                            −→      −−→
                                                                 i                       i       i           i
a =λ a +λ a +λ a =
 4
   ˜Txw™ušxXWx ÷϶XutXÍ´ ™xµµÍ u ™WXxÌXuÌ êÍtW¶Wµu¶zW txyzuTutøZ ›œžŸ
             1 1       2 2       3 3


” ”Ž‰žŠŠ’ ¡Šžž”Ž V ¢ª‘ ª” ¬Š ”Ž‰žŠŠ” {a , a , . . . , a }
Œ {b , b , . . . , b } ¥ Œ ¡œžŸ ¡Œ ފ’ ‰»ª‘¦ ”Ž‰žŠ ¡Ž”Š©Š ¬Š ‹Œº                       1       2             n


Ž¦Š ”‘»Žž£ ¤ŽŽ¢ ”Ž‰žŠ‘ ”žŠŠ©Š ¬Š¥ žŠ ŽžŸ a = λ b +
         1    2         m

                                                   ª
                                                                                                                 1

λ b + . . . + λ b ¥ α = 1, 2, . . . , n ¥ Œ‹Œ¥ ” ¡Š Š¬Š¦ ¢¡ŒŒ¥
                                                                                                     α           α 1
 2                     m
 α 2                   α m
                             
                             
                              a1 = λ11 b1 + λ21 b2 + . . . + λm
                                                               1 bm
                             
                                                                                                                   ‚_ƒ
                             
                              a = λ1 b + λ2 b + . . . + λm b
                                2     2 1      2 2             2 m
                             
                              ...
                             
                             
                              an = λ1 b1 + λ2 b2 + . . . + λm bm .
                                      n        n                n

¨Š©ª ¡Œ n > m ”Ž‰žŠ‘ {a , a , . . . , a } ‹ŒŽ¦Š ¢”ŒŒ’‘=
  éuyÍêÍzx™ë¶ztuZ H30;*,0; ;*8-L ;68*;680)*/1-M 0,L21 00=
                                                     1       2           n


   H30 m = 1 0;**; a = λ b \ α = 1, 2, . . . , n = ¼/70 a = 0 \ 8- b*18-3
                                        ¼
                                                     1

a 70,*M,- [6b0/0; .- /b-M/8b2 ]ƒ = /70 a 6= 0 \ 8- λ a − λ a = 0 =
                                     α               α 1                               1

   H3*L.-7-|0; 8*.*3?\ )8- 28b*3|L*,0* /.36b*L70b- .30 b/*N m 6 k \
                                                                                     1           1
 α                                                                               1   1 2         2 1


0 L-16|*;\ )8- -,- /.36b*L70b- 0 .30 m = k + 1 = H2/8? 0;**8 ;*/8-
,6+-3 70,*M,N 1-;+0,6 0M ‚_ƒ\ :L* m = k + 1 0 n > m = ¼/70 b/* 1-K
~OO0 0*,8 .30 b*18-3* b b 70,*M,N 1-;+0,6 09N ‚_ƒ 36b, ,27J†
                                                     1
                                     ܾ

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