Аналитическая геометрия и линейная алгебра. Умнов А.Е. - 260 стр.

UptoLike

Составители: 

Рубрика: 


Ë} ÒÒ}ÈÁË ¯©m©° Ë®äÈËäÈÒ}Òlnj
ÙkÓÈãÒÒ˰}È«˺äË¯Ò«ÒãÒÓË®ÓÈ«ÈãË¯Èµ

äÓºmkp
ËääÈ

{°Ë°º°mËÓÓ©ËÏÓÈËÓÒ« °È亰º¹¯«ÎËÓÓººº¹Ë¯Èº¯È
R
m
E
n
mË
˰mËÓÓ©ËÒ°ãÈ
iº}ÈÏÈËã°mº
iº¹°Òä ¹¯ºÒmÓºË ¹° ²È¯È}˯ҰÒ˰}ºË ¯ÈmÓËÓÒË °È亰º¹¯«ÎËÓÓºº
º¹Ë¯Èº¯È
R
ÒäËË}ºä¹ãË}°Ó©®}º¯ËÓ
λαβ
=+
i
Ë
β
0

º ˺¯ËäË  º¹Ë¯Èº¯
R
mªºä°ãÈË ÒäËË mä˯ӺË ÒÓmȯÒÈÓÓºË
¹º¹¯º°¯ÈÓ°mº º˰°˰mË¹È¯ÈãÒÓˮӺÓËÏÈmÒ°Ò䩲ªãËäËÓºm
x
Ò
y
È}Ò²º
Rx x y
Ry y x
=−
=+
αβ
αβ
äÓºÎÈ«ªÒ¯ÈmËÓ°mÈ°}È㫯Ӻ¹Ë¯mºË°¹¯ÈmÈÓÈ
y
mº¯ºË°ãËmÈÓÈ
x
¹ºãÒä
(
,) (,) (,)
(,
)(,)(,)
Rx y x y y y
xRy xy xx
=−
=+
αβ
αβ

ÒÈ«¹ºãËÓÓºmº¯ºË¯ÈmËÓ°mºÒÏ¹Ë¯mººÒ¹¯ÒÓÒäÈ«mºmÓÒäÈÓÒË°Èäº
°º¹¯«ÎËÓÓº°
R
 ¹¯Ò²ºÒä } ÏÈ}ãËÓÒ º
β
()xy
22
0+=
 |ÓÈ}º ªº
¹¯ºÒmº¯ËÒ¹¯Ë¹ºãºÎËÓÒººäº
β
0

ËääÈº}ÈÏÈÓÈ

ËääÈ

vº°mËÓÓ©ËmË}º¯©°È亰º¹¯«ÎËÓÓººº¹Ë¯Èº¯ÈºmËÈÒË¯ÈÏ
ãÒÓ©ä°º°mË ÓÓ©äÏÓÈËÓÒ«ä¹º¹È¯Óºº¯ººÓÈãÓ©
iº}ÈÏÈËã°mº
°ã«°È亰º¹¯«ÎËÓÓººº¹Ë¯Èº¯È
R
ÒäËä˰º¯ÈmËÓ°mÈ
Rf f
111
=
λ
Ò
Rf f
222
=
λ
 ËÓËÓãËm©Ë ªãËäËÓ©
f
1
Ò
f
2
°º°mËÓÓ©ËmË}º¯©º¹Ë¯È
º¯È
A
Ò
λλ
12
 °ººmË°mÒË Òä °º°mËÓÓ©Ë ÏÓÈËÓÒ« äÓºÎÈ« ªÒ
¯ÈmËÓ°mÈ°}È㫯Ӻ¹Ë¯mºË°¹¯ÈmÈÓÈ
f
2
mº¯ºË°ãËmÈÓÈ
f
1
¹ºãÒä
(
,)( ,)
(,
)(, )
Rf f f f
fRf f f
12 112
12 122
=
=
λ
λ
ÒãÒ
(
,) (,)
(,
)(,)
Rf f f f
fRf ff
12 112
12 212
=
=
λ
λ

ÒÈ« ªÒ¯ÈmËÓ°mÈ¹ºãËÓÓºÒÒ©mÈ«º
R
°È亰º¹¯«ÎËÓÓ©®º¹Ë¯Èº¯
¹¯Ò²ºÒä}¯ÈmËÓ°m
()(,)
λλ
1212
0−=ff
º}È
(, )ff
12
0=

ËääÈº}ÈÏÈÓÈ
 Ë }  Ò Ò    } È Á Ë  ¯ ©   m © °  Ë ®   ä È ˆ Ë ä È ˆ Ò } Ò   l n ‘ j 
ÙkÓÈã҈Ò˰}È«˺äˈ¯Ò«ÒãÒÓË®ÓÈ«ÈãË­¯ÈµäÓºmkp



 ËääÈ                 {°Ë °º­°ˆmËÓÓ©Ë ÏÓÈËÓÒ« °È亰º¹¯«ÎËÓÓºº º¹Ë¯Èˆº¯È R  m E n  mË
                Ë°ˆmËÓÓ©ËÒ°ãÈ

  iº}ÈÏȈËã°ˆmº
   
   
      iº¹‚°ˆÒä ¹¯ºˆÒmÓºË ¹‚°ˆ  ²È¯È}ˆË¯Ò°ˆÒ˰}ºË ‚¯ÈmÓËÓÒË °È亰º¹¯«ÎËÓÓºº
          º¹Ë¯Èˆº¯È R ÒäËˈ}ºä¹ãË}°Ó©®}º¯ËÓ  λ = α + β i Ë β ≠ 0 
          
          º ˆËº¯ËäË  º¹Ë¯Èˆº¯ R  m ªˆºä °ã‚ÈË ÒäËˈ m‚ä˯ӺË ÒÓmȯÒÈӈӺË
          ¹º¹¯º°ˆ¯ÈÓ°ˆmº‘º˰ˆ °‚Ë°ˆm‚ˈ¹È¯ÈãÒÓˮӺÓËÏÈmÒ°Ò䩲ªãËäËӈºm xÒy
                        =α x−β y
                      Rx
          ˆÈ}Ò²ˆº                äÓºÎÈ«ªˆÒ¯ÈmËÓ°ˆmȰ}È㫯Ӻ¹Ë¯mºË°¹¯ÈmÈÓÈ
                       
                      Ry = α y + β x
          ymˆº¯ºË°ãËmÈÓÈx¹ºã‚Òä
          
                                                              , y) = α ( x, y) − β ( y, y)
                                                         ( Rx
                                                                                                
                                                          ( x , Ry ) = α ( x , y ) + β ( x , x )
          
          {©҈ȫ¹ºãËÓÓºmˆº¯ºË¯ÈmËÓ°ˆmºÒϹ˯mººÒ¹¯ÒÓÒäÈ«mºmÓÒäÈÓÒ˰Èäº
          °º¹¯«ÎËÓÓº°ˆ  R  ¹¯Ò²ºÒä } ÏÈ}ã ËÓÒ  ˆº β ( x
                                                                                                           2          2
                                                                                                               + y ) = 0  |ÓÈ}º ªˆº
          ¹¯ºˆÒmº¯Ë҈¹¯Ë¹ºãºÎËÓÒ ºˆºäˆº β ≠ 0 
      
      
      ËääȺ}ÈÏÈÓÈ
          
          
 ËääÈ                 vº­°ˆmËÓÓ©ËmË}ˆº¯©°È亰º¹¯«ÎËÓÓººº¹Ë¯Èˆº¯ÈºˆmËÈ Ò˯ÈÏ
                ãÒÓ©ä°º­°ˆmËÓÓ©äÏÓÈËÓҫ乺¹È¯Óºº¯ˆººÓÈã Ó©

     iº}ÈÏȈËã°ˆmº
      
      

              ‚°ˆ ã«°È亰º¹¯«ÎËÓÓººº¹Ë¯Èˆº¯È R ÒäË ˆä˰ˆº¯ÈmËÓ°ˆmÈ R f 1 = λ1 f 1 Ò
              R f 2 = λ2 f 2 ËÓËӂãËm©ËªãËäËӈ© f 1 Ò f 2 °º­°ˆmËÓÓ©ËmË}ˆº¯©º¹Ë¯È
              ˆº¯È A  Ò λ ≠ λ   °ººˆmˈ°ˆm‚ ÒË Òä °º­°ˆmËÓÓ©Ë ÏÓÈËÓÒ« äÓºÎÈ« ªˆÒ
                               1      2
              ¯ÈmËÓ°ˆmȰ}È㫯Ӻ¹Ë¯mºË°¹¯ÈmÈÓÈ f 2 mˆº¯ºË°ãËmÈÓÈ f 1 ¹ºã‚Òä
              
                                     ( R f 1 , f 2 ) = ( λ1 f 1 , f 2 )            ( R f 1 , f 2 ) = λ1 ( f 1 , f 2 )
                                                                        ÒãÒ 
                                                                                                                         
                                      ( f 1 , R f 2 ) = ( f 1 , λ2 f 2 )           ( f 1 , R f 2 ) = λ2 ( f 1 , f 2 )
          
         {©҈ȫªˆÒ¯ÈmËÓ°ˆmȹºãËÓÓºÒ‚҈©mÈ«ˆº R °È亰º¹¯«ÎËÓÓ©®º¹Ë¯Èˆº¯
          ¹¯Ò²ºÒä}¯ÈmËÓ°ˆm‚ ( λ1 − λ2 ) ( f 1 , f 2 ) = 0 ºˆ}‚È ( f 1 , f 2 ) = 0 
      
      
      ËääȺ}ÈÏÈÓÈ