Аналитическая геометрия и линейная алгебра. Умнов А.Е. - 225 стр.

UptoLike

Составители: 

Рубрика: 

cÈÏËã


sËãÒÓˮөËÏÈmÒ°Ò亰ÒmãÒÓˮӺä¹¯º°¯ÈÓ°mË
° ¯Ò
n=
mãºäÈÏÒ°Ë
)(x
=
2
111
ξ
ϕ
p°ãÒ
ϕ
11
0
=
ºä©ÎËÒäËËä}È
ÓºÓÒ˰}Ò®  ˰ãÒ ÎË
ϕ
11
0
 º m©¹ºãÓ«« ÏÈäËÓ ¹Ë¯ËäËÓÓ©²
1111
ξ
ϕ
ξ
=
¹¯Ò²ºÒä}}ÈÓºÓÒ˰}ºämÒ
° ¯Ë¹ºãºÎÒä º m˯ÎËÓÒË ˺¯Ëä© m˯Ӻ ã« }mȯÈÒÓ©² ÁÓ}
ÒºÓÈãºm ÏÈmÒ°«Ò² º
n
 ¹Ë¯ËäËÓÓº® Ò ¯È°°äº¯Òä °ãÈ®
n
¹Ë¯ËäËÓÓ©²
rËä °ÒÈ º
ϕ
11
0
 wºº äºÎÓº ºÒ°« ÒÏäËÓËÓÒËä Óä˯ÈÒÒ
¹Ë¯ËäËÓÓ©² m °ãÈË È ²º« © ºÓº ÒÏ Ò°Ëã
],2[, ni
ii
=
ϕ
ÓË ¯ÈmÓº
Óãp°ãÒÎËm°Ë
],1[, ni
ii
=
ϕ
¯ÈmÓ©ÓãºÓºm¯ËäËÓÓººËÏº¯ÈÓÒ
ËÓÒ« ºÓº°Ò äºÎÓº °ÒÈ º
ϕ
12
0
 ºÈ m©¹ºãÓ««
ÓËm©¯ºÎËÓÓÏÈäËÓ¹Ë¯ËäËÓÓ©²
nn
ξ
ξ
ξ
ξ
ξ
ξ
ξ
ξ
ξ
ξ
=
=
=
+
= ,...,,,
33212211

¹ºãÈËä ÏȹҰ }mȯÈÒÓºº ÁÓ}ÒºÓÈãÈ ° ÓËÓãËm©ä ÒȺÓÈãÓ©ä
ªãËäËÓºä
)(
x
),...,,,(22
321
2
221
2
112
n
F
ξ
ξ
ξ
ξ
ξ
ϕ
ξ
ϕ
+

Ë
),...,,,(
321
n
F
ξ
ξ
ξ
ξ
ÓË°ºË¯ÎÒ}mȯÈºmº
1
Ò
2

°
{ÏȹҰÒ}mȯÈÒÓººÁÓ}ÒºÓÈãÈ°¯¹¹Ò¯Ëä°ãÈÈËä©Ë°ºË¯ÎÈÒË
¹Ë¯ËäËÓÓ
ξ
1

)(
x
∑∑∑∑
=====
++=
n
i
n
k
kiiki
n
i
i
n
i
n
k
kiik
222
1
11
1
2
111
11
)2(
ξ
ξ
ϕ
ξ
ξ
ϕ
ϕ
ξ
ϕ
ξ
ξ
ϕ
Òm©ËãÒä¹ºãÓ©®}mȯÈmº°¹ºãϺmÈmÒ°°ººÓºËÓÒ«äÒ
.2)(2
)()()()(
222
1
2
1
2
2
2
1
2
1
2
2
1
2
11111
∑∑∑∑
∑∑
=====
======
++=++=
=+===
n
k
n
k
n
i
ikk
n
k
n
k
kk
n
k
k
n
k
k
n
k
n
i
i
n
k
k
n
i
ik
ααααααααα
ααααααα
ºãÒä
)(
x
∑∑∑∑
=====
+++
n
i
n
k
ki
ik
ik
n
i
n
k
ki
ik
i
n
i
i
22
11
11
22
2
11
11
2
1
11
1
2
111
)()2(
ξ
ξ
ϕ
ϕϕ
ϕ
ξ
ξ
ϕ
ϕϕ
ξ
ξ
ϕ
ϕ
ξ
ϕ
Òº}ºÓÈËãÓº
)(
x
.)()(
22
11
11
2
2
11
1
1
11
∑∑
===
++
n
i
n
k
ki
ik
iki
n
i
i
ξ
ξ
ϕ
ϕϕ
ϕ
ξ
ϕ
ϕ
ξ
ϕ
{ ¹º°ãËÓË® Áº¯äãË ¹Ë¯mºË °ãÈÈËäºË ˰ ¹ºãÓ©® }mȯÈ È mº¯ºË 
}mȯÈÒÓ©® ÁÓ}ÒºÓÈã ÓË ÏÈmÒ°«Ò®º
ξ
1
Ò¹¯Òmº«Ò®°« °ºãȰӺ
¹¯Ë¹ºãºÎËÓÒÒÓ}ÒÒ}}ÈÓºÓÒ˰}ºäÓË}ºº¯º®ÓËm©¯ºÎËÓ
cÈÏËã 
sËãÒÓˮөËÏÈmÒ°Ò亰ˆÒmãÒÓˮӺ乯º°ˆ¯ÈÓ°ˆmË



           ° ¯Ò n=mã ­ºä­ÈÏÒ°Ë  (x ) = ϕ 11ξ 12 p°ãÒ ϕ11 = 0 ˆºä©‚ÎËÒäËËä}È
                 ÓºÓÒ˰}Ò® mÒ ˰ãÒ ÎË ϕ 11 ≠ 0  ˆº m©¹ºãÓ«« ÏÈäËӂ ¹Ë¯ËäËÓÓ©²
                      ξ1′ =     ϕ11 ξ1 ¹¯Ò²ºÒä}}ÈÓºÓÒ˰}ºä‚mÒ‚
           
           ° ¯Ë¹ºãºÎÒä ˆº ‚ˆm˯ÎËÓÒË ˆËº¯Ëä© m˯Ӻ ã« }mȯȈÒÓ©² Á‚Ó}
                 ÒºÓÈãºm ÏÈmÒ°«Ò² ºˆ n ¹Ë¯ËäËÓÓº® Ò ¯È°°äºˆ¯Òä °ã‚È® n
                 ¹Ë¯ËäËÓÓ©²
           
                 r‚Ëä °҈Ȉ  ˆº ϕ 11 ≠ 0  wˆºº äºÎÓº º­Òˆ °« ÒÏäËÓËÓÒËä ӂä˯ÈÒÒ
                      ¹Ë¯ËäËÓÓ©² m °ã‚ÈË }ºÈ ²ºˆ« ­© ºÓº ÒÏ Ò°Ëã ϕ ii , i = [2, n]  ÓË ¯ÈmÓº
                      ӂã p°ãÒÎËm°Ë ϕ ii , i = [1, n] ¯ÈmÓ©ӂã ºÓºm¯ËäËÓÓºˆº­ËϺ¯ÈÓÒ
                      ËÓÒ« º­Óº°ˆÒ äºÎÓº °҈Ȉ  ˆº ϕ 12 ≠ 0  ‘ºÈ m©¹ºãÓ««
                      ÓËm©¯ºÎËÓӂ ÏÈäËӂ¹Ë¯ËäËÓÓ©² ξ 1 = ξ 1′ + ξ 2′ ,ξ 2 = ξ 1′ − ξ 2′ ,ξ 3 = ξ 3′ ,...,ξ n = ξ n′ 
                      ¹ºã‚ÈËä ÏȹҰ  }mȯȈÒÓºº Á‚Ó}ÒºÓÈãÈ ° ÓËӂãËm©ä ÒȺÓÈã Ó©ä
                      ªãËäËӈºä
                                                   (x )           2ϕ12ξ1′ 2 − 2ϕ 21ξ 2′ 2 + F (ξ1′ , ξ 2′ , ξ 3′ ,..., ξ n′ ) 
                      
                      Ë F (ξ1′ , ξ 2′ , ξ 3′ ,..., ξ n′ ) Ó˰º˯Î҈}mȯȈºmºˆ ξ 1′ Ò ξ 2′ 
  
             °{ÏȹҰÒ}mȯȈÒÓººÁ‚Ó}ÒºÓÈãȰ¯‚¹¹Ò¯‚Ëä°ãÈÈËä©Ë°º˯ÎȝÒË
                  ¹Ë¯ËäËÓӂ  ξ 1 
                                                 n        n                                  n
                                                                                                  ϕ1i             n n
                                    (x )       ∑ ∑ϕ ik ξ iξ k = ϕ11(ξ12 + 2 ∑                        ξ1ξ i ) + ∑ ∑ϕ ik ξ iξ k 
                                                i =1 k =1                                   i = 2 ϕ11           i =2k =2
             
                      Òm©ËãÒ乺ãÓ©®}mȯȈmº°¹ºã ϺmÈm Ò° °ººˆÓº ËÓÒ«äÒ
                  
                               n   n                  n              n               n                     n
                              ∑ ∑α kα i = ( ∑α k )( ∑α i ) = ( ∑α k ) 2 = (α1 +                           ∑α k ) 2 =
                              k =1i =1               k =1           i =1         k =1                     k =2
                                                                   n               n                         n               n      n
                                               = α12 + 2 ∑α1α k + ( ∑α k ) 2 =α12 + 2 ∑α1α k +                              ∑∑α kα i .
                                                                  k =2            k =2                     k =2            k = 2i = 2
                  
                      ºã‚Òä
                      
                                                              n
                                                                ϕ1i           n n
                                                                                       ϕ ϕ           n n
                                                                                                              ϕ ϕ
                           (x )         ϕ11 (ξ12 + 2 ∑             ξ1ξ i + ∑ ∑ 1k 2 1i ξ iξ k ) + ∑ ∑ (ϕ ik − 1k 1i )ξ iξ k 
                                                          i = 2 ϕ11         i = 2 k = 2 ϕ11        i =2k =2    ϕ11
                  
                                                                                 n
                                                                                     ϕ1i 2 n n             ϕ ϕ
                      Òº}ºÓȈËã Óº  (x )                      ϕ11(ξ1 + ∑            ξ i ) + ∑ ∑ (ϕ ik − 1k 1i )ξ iξ k . 
                                                                               i = 2 ϕ11        i =2k =2    ϕ11
             
                      { ¹º°ãËÓË® Áº¯ä‚ãË ¹Ë¯mºË °ãÈÈËäºË ˰ˆ  ¹ºãÓ©® }mȯȈ È mˆº¯ºË 
                      }mȯȈÒÓ©® Á‚Ó}ÒºÓÈã ÓË ÏÈmÒ°«Ò® ºˆ ξ1  Ò ¹¯Òmº«Ò®°« °ºãȰӺ
                      ¹¯Ë¹ºãºÎËÓÒ ÒÓ‚}ÒÒ}}ÈÓºÓÒ˰}ºä‚mÒ‚ÓË}ºˆº¯º®ÓËm©¯ºÎËÓ