Аналитическая геометрия и линейная алгебра. Умнов А.Е. - 250 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
|¯ººÓÈãÓ©ËäÈ¯Ò©Ò¯ÈmÈÎÓ¯ºãmm©Ò°ãÒËãÓ©²äËºÈ²ÈãË
¯©ºÓȹ¯Òä˯Òãã°¯Ò¯Ë
˺¯ËäÈ

p°ãÒ äÈ¯ÒÈ
A
ÓËm©¯ºÎËÓÈ º ËË ¯ÈÏãºÎËÓÒË È
AQR=
 Ë
Q
º¯ººÓÈã ÓÈ« äÈ¯ÒÈ È
R
m˯²Ó««
¯ËºãÓÈ« äÈ¯ÒÈ °¹ºãºÎÒËãÓ©äÒ ÒȺÓÈãÓ©äÒªãËäËÓÈäÒ
°˰mËÒËÒÓ°mËÓÓº
iº}ÈÏÈËã°mº
¯Ë¹ºãºÎÒä º ÒäËË°«
 mÈ ¯ÈÏãºÎËÓÒ«
AQR QR==
11 2 2
 jÏ
ÓËm©¯ºÎËÓÓº°Ò
A
°ãËËÓËm©¯ºÎËÓÓº°
R
1
Ò
R
2
¹º°}ºã}
Q
1
Ò
Q
2
º¯ººÓÈãÓ©ËÒºËmÒÓº ÓËm©¯ºÎËÓÓ©Ë ºÈ ¹º°ãËÓËË ¯ÈmËÓ°mº
äºÎÓº ¹Ë¯Ë¹Ò°ÈmmÒË
QQ RR
21 21
1T
=
 Ë
R
1
1
È}ÎËm˯²Ó««
¯ËºãÓÈ«äÈ¯ÒÈ
~ÈäËÒä º
RR
21
1
˰ ÒȺÓÈãÓÈ« äÈ¯ÒÈ iË®°mÒËãÓº ° ºÓº®
°º¯ºÓ©ºÓÈm˯²Ó«« ¯ËºãÓÈ«äÈ¯ÒÈ}È}¹¯ºÒÏmËËÓÒËm˯²ÓÒ²¯Ëºã
Ó©² v ¯º® °º¯ºÓ©
RR
21
1
ºãÎÓÈ © Ò ÓÒÎÓË® ¯ËºãÓº® ¹º
°}ºã}ºÓÈº¯ººÓÈãÓÈ« }È} ¹¯ºÒÏmËËÓÒË m² º¯ººÓÈãÓ©² äÈ¯Ò
QQ
21
1T
ÒËËº¯ÈÓÈ«äÈ¯ÒÈ°ºm¹ÈÈË°¯ÈÓ°¹ºÓÒ¯ºmÈÓÓº®
|ËmÒÓº º ÒȺÓÈãÓÈ« º¯ººÓÈãÓÈ« äÈ¯ÒÈ äºÎË ÒäËÓÈÒȺÓÈãÒ
ãÒªãËäËÓ©¯ÈmÓ©Ë¹ºäºãËÒÓÒËsºÒȺÓÈãÓ©ËªãËäËÓ©
R
1
Ò
R
2
¹ºãºÎÒËãÓ© ¹º °ãºmÒ ¹ºªºä º°ÈË°« mºÏäºÎÓ©ä ãÒ°ãÈ®
RR E
21
1
=
º}ÈÒ°ãËËËÒÓ°mËÓÓº°¯ÈÏãºÎËÓÒ«
˺¯ËäÈº}ÈÏÈÓÈ
|äËÒä º m °Òã ˺¯Ëä©  ¯ËËÓÒË Ó˺Ӻ¯ºÓº® °Ò°Ëä© ãÒÓˮө²
¯ÈmÓËÓÒ®
Ax b
=
äºÎË©°mËËÓº } ¯ÈÏãºÎËÓÒ ÓËm©¯ºÎËÓÓº® äÈ¯Ò©
A
 ÓÈ¹¯ºÒÏmËËÓÒË m˯²ÓË®¯ËºãÓº®
R
Òº¯ººÓÈãÓº®
4
¹º°}ºã} m
ªºä°ãÈË°Ò°ËäÈ¹¯Ëº¯ÈÏË°«}ãË}º¯ËÈËäºämÒ
Rx Q b
=
T


 |º°ÓºmÈÓÒË °˰mºmÈÓÒ« È}ºº ¯ÈÏãºÎËÓÒ« m©²ºÒ ÏÈ ¯Èä}Ò ÈÓÓºº }¯°È ~˰ä©
º¯ÈÓÒÒä°«¯È°°äº¯ËÓÒËäãÒmº¹¯º°Èº˺ËÒÓ°mËÓÓº°Ò
 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



         |¯ˆººÓÈã Ó©ËäȈ¯Ò©Ò¯È ˆmÈÎӂ ¯ºã mm©Ò°ã҈Ëã Ó©²äˈºȲÈãË­
¯©ˆºÓȹ¯Òä˯Òãã °ˆ¯Ò¯‚ˈ
         
         
 ‘˺¯ËäÈ                   p°ãÒ    äȈ¯ÒÈ             A  ÓËm©¯ºÎËÓÈ ˆº ËË ¯ÈÏãºÎËÓÒË mÒÈ
 
                             A = Q         R  Ë Q   º¯ˆººÓÈã ÓÈ« äȈ¯ÒÈ È R
                                                                                      m˯²Ó««
                            ˆ¯Ë‚ºã ÓÈ« äȈ¯ÒÈ ° ¹ºãºÎ҈Ëã Ó©äÒ ÒȺÓÈã Ó©äÒ ªãËäËӈÈäÒ
                            °‚Ë°ˆm‚ˈÒËÒÓ°ˆmËÓÓº
              
              
  iº}ÈÏȈËã°ˆmº
      
      
          ¯Ë¹ºãºÎÒä ˆº ÒäËˈ°«  mÈ ¯ÈÏãºÎËÓÒ«                                      A = Q1           R1 = Q2            R2  jÏ
          ÓËm©¯ºÎËÓÓº°ˆÒ A  °ãË‚ˈ ÓËm©¯ºÎËÓÓº°ˆ  R1 Ò R2 ¹º°}ºã }‚ Q1 
          Ò Q2  º¯ˆººÓÈã Ó©Ë Ò ºËmÒÓº ÓËm©¯ºÎËÓÓ©Ë ‘ºÈ ¹º°ãËÓËË ¯ÈmËÓ°ˆmº
                                                                 T                            −1                   −1
          äºÎÓº ¹Ë¯Ë¹Ò°Èˆ  m mÒË Q2                             Q1 = R2            R1          Ë R1            ˆÈ}ÎË m˯²Ó««
          ˆ¯Ë‚ºã ÓÈ«äȈ¯ÒÈ
          
                                                  −1
          ~ÈäˈÒä ˆº R2 R1      ˰ˆ  ÒȺÓÈã ÓÈ« äȈ¯ÒÈ iË®°ˆm҈Ëã Óº ° ºÓº®
          °ˆº¯ºÓ©ºÓÈm˯²Ó««ˆ¯Ë‚ºã ÓÈ«äȈ¯ÒÈ}È}¹¯ºÒÏmËËÓÒËm˯²ÓÒ²ˆ¯Ë‚ºã 
                                                                     −1
          Ó©² v ¯‚º® °ˆº¯ºÓ© R2 R1    ºãÎÓÈ ­©ˆ  Ò ÓÒÎÓË® ˆ¯Ë‚ºã Óº® ¹º
          °}ºã }‚ ºÓÈ º¯ˆººÓÈã ÓÈ« }È} ¹¯ºÒÏmËËÓÒË m‚² º¯ˆººÓÈã Ó©² äȈ¯Ò
                   T          −1
              Q2       Q1          ÒË˺­¯ÈˆÓÈ«äȈ¯ÒȰºm¹ÈÈˈ°ˆ¯ÈÓ°¹ºÓÒ¯ºmÈÓÓº®
          
          
          |ËmÒÓº ˆº ÒȺÓÈã ÓÈ« º¯ˆººÓÈã ÓÈ« äȈ¯ÒÈ äºÎˈ Òäˈ  ÓÈ ÒȺÓÈãÒ
          ãÒ       ªãËäËӈ©¯Èmө˹ºäº‚ã ËÒÓÒËsºÒȺÓÈã ө˪ãËäËӈ© R1 Ò
              R2  ¹ºãºÎ҈Ëã Ó© ¹º ‚°ãºmÒ  ¹ºªˆºä‚ º°ˆÈˈ°« mºÏäºÎÓ©ä ãÒ                                                     °ã‚È®
                             −1
              R2       R1         = E ºˆ}‚ÈÒ°ãË‚ˈËÒÓ°ˆmËÓÓº°ˆ ¯ÈÏãºÎËÓÒ«
          
          
     ‘˺¯ËäȺ}ÈÏÈÓÈ
          
          
          |ˆäˈÒä ˆº m °Òã‚ ˆËº¯Ëä©  ¯Ë ËÓÒË Ó˺Óº¯ºÓº® °Ò°ˆËä© ãÒÓˮө²
‚¯ÈmÓËÓÒ® A                 x = b  äºÎˈ ­©ˆ  °mËËÓº } ¯ÈÏãºÎËÓÒ  ÓËm©¯ºÎËÓÓº® äȈ¯Ò©
    A   ÓÈ ¹¯ºÒÏmËËÓÒË m˯²ÓË® ˆ¯Ë‚ºã Óº® R  Ò º¯ˆººÓÈã Óº® 4  ¹º°}ºã }‚ m
                                                                                                                          T
ªˆºä°ã‚È˰ҰˆËäȹ¯Ëº­¯Èςˈ°«}ãË}º¯Ë ÈËäºä‚mÒ‚ R                                                 x = Q             b 
              



 |­º°ÓºmÈÓÒË °‚Ë°ˆmºmÈÓÒ« ˆÈ}ºº ¯ÈÏãºÎËÓÒ« m©²º҈ ÏÈ ¯Èä}Ò ÈÓÓºº }‚¯°È ~˰  ä©
º¯ÈÓÒÒä°«¯È°°äºˆ¯ËÓÒËäãÒ mº¹¯º°ÈºËºËÒÓ°ˆmËÓÓº°ˆÒ