Аналитическая геометрия и линейная алгебра. Умнов А.Е. - 196 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
},...,,{
21
n
ggg
˰ÈÏÒ°m
n
Λ
È
},...,,{
21
m
fff
ÈÏÒ°m
m
Λ
ºÈäºÎÓº°ËãÈ°ãË
ËËººËÓÒËº¹¯ËËãËÓÒ«
|¹¯ËËãËÓÒË


lÈ¯ÒÈ
A
gf
¯ÈÏä˯È
nm
×
 °ºã© }ºº¯º® ˰ }ºº¯ÒÓÈÓ©Ë
¯ÈÏãºÎËÓÒ« ªãËäËÓºm
n
gAgAgA
ˆ
,...,
ˆ
,
ˆ
21
¹º ÈÏÒ°
},...,,{
21
m
fff

ÓÈÏ©mÈË°« ujzéq|np sqtnptvmv vzviéjntq¹
A
m ÈÏҰȲ
},...,,{
21
n
ggg
Ò
},...,,{
21
m
fff

|äËÒä ºm}ºÓËÓºä˯Ӻä °ãÈË °¯Ë}ÒmÓº° ºº¯ÈÎËÓÒ« ºÏÓÈÈË
m©¹ºãÓËÓÒË °ãºmÒ«
m
Λ=Θ
 È ÒÓË}ÒmÓº°  °ãºmÒ«
{}
oA
=
ˆ
ker
 |°È °ãËË
º ¯ÈÓ äÈ¯Ò© ãÒÓˮӺº º¹Ë¯Èº¯È «mã«˺°« °¯Ë}ÒmÓ©ä ºº¯ÈÎËÓÒËä
¯ÈmËÓ Ò°ã ËË °¯º} È ¯ÈÓ äÈ¯Ò© ÒÓË}ÒmÓºº ºº¯ÈÎËÓÒ« ¯ÈmËÓ Ò°ã ËË
°ºãºm sÈ}ºÓË ºº¯ÈÎËÓÒË «mã«Ë˰« ºÓºm¯ËäËÓÓº Ò ÒÓË}ÒmÓ©ä Ò
°¯Ë}ÒmÓ©äËmÏÈÒäÓººÓºÏÓÈÓ©äÒãÒÒË}ÒË®°äº¹¯ËËãËÓÒË
jÏº¹¯ËËãËÓÒ«°ãËËºäÈ¯ÒÈãÒÓˮӺººº¯ÈÎËÓÒ«ÏÈmÒ°Ò}È}
ºm©º¯ÈÈÏÒ°È
},...,,{
21
n
ggg
È}Òºm©º¯ÈÈÏÒ°È
},...,,{
21
m
fff
¯ÈmÒãºÒÏäËÓË
ÓÒ«ªº®äÈ¯Ò©¹¯ÒÏÈäËÓËÈÏÒ°ºmÈË
˺¯ËäÈ

lÈ¯ÒÈ ãÒÓˮӺº ºº¯ÈÎËÓÒ«
A
mÈÏҰȲ
},...,,{
21
n
ggg
Ò
},...,,{
21
m
fff
A
gf
′′
°m«ÏÈÓÈ°äÈ¯ÒË®ªºººº¯ÈÎËÓÒ«mÈÏҰȲ
},...,,{
21
n
ggg
Ò
},...,,{
21
m
fff
A
gf
°ººÓºËÓÒËä
,
ˆˆ
1
GAFA
gffg
=

Ë
F
äÈ¯Ò
È ¹Ë¯Ë²º
È º ÈÏÒ°È
},...,,{
21
m
fff
}ÈÏÒ°
},...,,{
21
m
fff
È
G
äÈ¯ÒÈ¹Ë¯Ë²ºÈºÈÏÒ°È
},...,,{
21
n
ggg
}È
ÏÒ°
},...,,{
21
n
ggg
iº}ÈÏÈËã°mº
kÓÈãºÒÓºº}ÈÏÈËã°m˺¯Ëä©

{ºËä°ãÈËÒ°°ã˺mÈÓÒË°mº®°mº¹Ë¯Èº¯È}ºº¯ºººãȰÏÓÈËÓÒ®
ÓË°ºË¯ÎÒ°«mºãȰÒ˺º¹¯ËËãËÓÒ«äºÎËº}ÈÏÈ°«º°ÈºÓº°ãºÎÓº®ÏÈÈ
Ë®p°ãÒÎËºãȰÏÓÈËÓÒ® ÒäËË }ºÓËÓ ¯ÈÏä˯Ӻ°ÓË¹¯Ëm©È ¯ÈÏ
ä˯Ӻ°ºãȰÒº¹¯ËËãËÓÒ«º¹ºãÏ«°˺¯Ë亮ºÒϺ亯ÁÒÏäËäºÎÓº
¹º¹©È°« °m˰Ò Ò°°ã˺mÈÓÒË ºº¯ÈÎËÓÒ« } Ò°°ã˺mÈÓÒ ¹¯Ëº¯ÈϺmÈÓÒ«
°ÈÓºmÒm ÒϺ亯ÁÒÏä äËÎ ºãȰÏÓÈËÓÒ® ºº¯ÈÎËÓÒ« Ò ÓË}ºº¯©ä
¹º¹¯º°¯ÈÓ°mºäºãȰÒ˺º¹¯ËËãËÓÒ«
 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



{g1, g 2 ,..., g n } ˰ˆ ­ÈÏÒ°m Λn È { f1, f 2 ,..., f m } ­ÈÏÒ°m Λm ‘ºÈäºÎÓº°ËãȈ °ãË
‚ Ë˺­º­ËÓÒ˺¹¯ËËãËÓÒ«
      
      
                                            A
 
 |¹¯ËËãËÓÒË            lȈ¯ÒÈ                      ¯ÈÏä˯È m× n  °ˆºã­© }ºˆº¯º® ˰ˆ  }ºº¯ÒÓȈөË
                                                 gf
 
                         ¯ÈÏãºÎËÓÒ« ªãËäËӈºm                       Aˆ g1, Aˆ g 2 ,..., Aˆ g n  ¹º ­ÈÏÒ°‚ { f1, f 2 ,..., f m } 
                         ÓÈÏ©mÈˈ°« ujzéq|np sqtnptvmv vzviéj ntq¹ A  m ­ÈÏҰȲ
                         {g1, g 2 ,..., g n } Ò { f1, f 2 ,..., f m } 
            
            
            |ˆäˈÒä ˆº m }ºÓËÓºä˯Ӻä °ã‚ÈË ° ¯žË}ˆÒmÓº°ˆ  ºˆº­¯ÈÎËÓÒ« ºÏÓÈÈˈ
m©¹ºãÓËÓÒË ‚°ãºmÒ« Θ = Λm  È ÒӞË}ˆÒmÓº°ˆ   ‚°ãºmÒ« ker Aˆ = { o}  |ˆ° È °ãË‚ˈ
ˆº ¯ÈÓ äȈ¯Ò© ãÒÓˮӺº º¹Ë¯Èˆº¯È «mã« Ëº°« ° ¯žË}ˆÒmÓ©ä ºˆº­¯ÈÎËÓÒËä
¯ÈmËÓ Ò°ã‚ ËË °ˆ¯º} È ¯ÈÓ äȈ¯Ò© ÒӞË}ˆÒmÓºº ºˆº­¯ÈÎËÓÒ« ¯ÈmËÓ Ò°ã‚ ËË
°ˆºã­ºm sÈ}ºÓË ºˆº­¯ÈÎËÓÒË «mã« Ë˰« ºÓºm¯ËäËÓÓº Ò ÒӞË}ˆÒmÓ©ä Ò
° ¯žË}ˆÒmÓ©ä­‚ˈmÏÈÒäÓººÓºÏÓÈÓ©äÒãÒ­ÒË}ÒË® °äº¹¯ËËãËÓÒË 
        
        
        jϺ¹¯ËËãËÓÒ«°ãË‚ˈˆºäȈ¯ÒÈãÒÓˮӺººˆº­¯ÈÎËÓÒ«ÏÈmҰ҈}È}
ºˆm©­º¯È­ÈÏÒ°È {g1, g 2 ,..., g n } ˆÈ}Òºˆm©­º¯È­ÈÏÒ°È { f1, f 2 ,..., f m } ¯ÈmÒãºÒÏäËÓË
ÓÒ«ªˆº®äȈ¯Ò©¹¯ÒÏÈäËÓË­ÈÏÒ°ºmÈˈ
        
        
 ‘˺¯ËäÈ              lȈ¯ÒÈ ãÒÓˮӺº ºˆº­¯ÈÎËÓÒ« A  m ­ÈÏҰȲ {g1′ , g ′2 ,..., g n′ }  Ò
 
                        { f1′, f 2′ ,..., f m′ }  A            °m«ÏÈÓȰäȈ¯ÒË®ªˆºººˆº­¯ÈÎËÓÒ«m­ÈÏҰȲ
                                                        g ′f ′

                        {g1, g 2 ,..., g n } Ò { f1, f 2 ,..., f m }  A         gf
                                                                                         °ººˆÓºËÓÒËä
                        
                                                                                         −1
                                                                    Aˆ            = F         Aˆ        G   ,
                                                                         g ′f ′                    gf
                        
                        Ë       F   äȈ¯ÒÈ ¹Ë¯Ë²ºÈ ºˆ ­ÈÏÒ°È { f1, f 2 ,..., f m }  } ­ÈÏÒ°‚
                        { f1′, f 2′ ,..., f m′ } È G äȈ¯Òȹ˯˲ºÈºˆ­ÈÏÒ°È {g1, g 2 ,..., g n } }­È
                        ÏÒ°‚ {g1′ , g ′2 ,..., g n′ } 
         
  iº}ÈÏȈËã°ˆmºkÓÈãºÒÓºº}ÈÏȈËã °ˆm‚ˆËº¯Ëä©
         
         
         {º­Ëä°ã‚ÈËÒ°°ã˺mÈÓÒ˰mº®°ˆmº¹Ë¯Èˆº¯È‚}ºˆº¯ººº­ãȰˆ ÏÓÈËÓÒ®
Ó˰º˯Î҈°«mº­ãȰˆÒ˺º¹¯ËËãËÓÒ«äºÎˈº}ÈÏȈ °«º°ˆÈˆºÓº°ãºÎÓº®ÏÈÈ
Ë® p°ãÒ ÎË º­ãȰˆ  ÏÓÈËÓÒ® ÒäËˈ }ºÓËӂ  ¯ÈÏä˯Ӻ°ˆ  ÓË ¹¯Ëm© È ‚  ¯ÈÏ
ä˯Ӻ°ˆ  º­ãȰˆÒ º¹¯ËËãËÓÒ« ˆº ¹ºã ς«°  ˆËº¯Ë亮  º­ ÒϺ亯ÁÒÏäË  äºÎÓº
¹º¹©ˆÈˆ °« °m˰ˆÒ Ò°°ã˺mÈÓÒË ºˆº­¯ÈÎËÓÒ« } Ò°°ã˺mÈÓÒ  ¹¯Ëº­¯ÈϺmÈÓÒ«
‚°ˆÈÓºmÒm ÒϺ亯ÁÒÏä äË΂ º­ãȰˆ  ÏÓÈËÓÒ® ºˆº­¯ÈÎËÓÒ« Ò ÓË}ºˆº¯©ä
¹º¹¯º°ˆ¯ÈÓ°ˆmºäº­ãȰˆÒ˺º¹¯ËËãËÓÒ«