Аналитическая геометрия и линейная алгебра. Умнов А.Е. - 102 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
iº}ÈÏÈËã°mº
¯Ë¹ºãºÎÒä º ÓËm©¯ºÎËÓÓÈ« äÈ¯ÒÈ
A
ÒäËË mË º¯ÈÓ©Ë
A
1
1
Ò
A
2
1
ºÈÒÏ¯ÈmËÓ°m
AA E
1
1
=
Ò
AA E
2
1
=
°ãËË
AA AA E E O
1
1
2
1−−
=−=

sº ºÈ äÓºÎÈ« °ãËmÈ ºË ȰÒ ÈÓÓºº ¯ÈmËÓ°mÈ ÓÈ
A
1
1
 ¹ºãÈËä
AAA A AOO
1
1
1
1
2
1
1
1
−−
−= =
()
ÒãÒ«º
AA E
1
1
=
¹¯Ò
²ºÒä}
AAO
1
1
2
1
−−
−=

ËääÈº}ÈÏÈÓÈ

{ȰÓºä°ãÈË}ºÈ
A
=
αα
αα
11 12
21 22
äÈ¯ÒÈ
A
1
ÒäËËmÒ
A
A
=
1
22 12
21 11
1
det
αα
αα

iã«}mȯÈÓ©²äÈ¯Ò°¹¯ÈmËãÒm©
°ãËÒË¯ÈmËÓ°mÈ
det( ) det( )det( ) ;
det( ) det ; det( )
det
.
T
AB A B
AA A
A
=
==
1
1
¯Òä˯

j°¹ºãÏ«¹¯ÈmÒãºË®°mÒ«°äÈ¯ÒÈäÒ °Ò°ËäãÒÓˮө²¯ÈmÓË
ÓÒ®
α
ξ
α
ξ
β
α
ξ
α
ξ
β
11 1 12 2 1
21 1 22 2 2
+=
+=
äºÎÓºÏȹҰÈmmÒË
Ax b=
Ë

 iã«
n
 ªÒ °ººÓºËÓÒ« ¹¯ºm˯«°« Ó˹º°¯Ë°mËÓÓº ¹º º¹¯ËËãËÓÒ  °ãÈ®
¹¯ºÒÏmºãÓºº
n
¯È°°äÈ¯ÒmÈË°«m¯ÈÏËãË
 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



   iº}ÈÏȈËã°ˆmº
          
          
                                                                                                                                                         −1
         ¯Ë¹ºãºÎÒä ˆº ÓËm©¯ºÎËÓÓÈ« äȈ¯ÒÈ A  ÒäËˈ mË º­¯ÈˆÓ©Ë A                                                                               Ò
                                                                                                                                                         1
                −1                                                        −1                                    −1
            A   2
                        ‘ºÈÒϯÈmËÓ°ˆm A                      A     1
                                                                                  = E Ò A                A    2
                                                                                                                     = E °ãË‚ˈ
                    
                                                               −1                       −1
                                                     A     A   1
                                                                    − A             A   2
                                                                                             = E − E = O 
                         
                                                                                                                                             −1
         sº ˆºÈ ‚äÓºÎÈ« °ãËmÈ º­Ë ȰˆÒ ÈÓÓºº ¯ÈmËÓ°ˆmÈ ÓÈ                                                                 A         ¹ºã‚ÈËä
                                                                                                                                             1
                −1                        −1          −1             −1                                                             −1
            A   1
                         A ( A            1
                                               − A    2
                                                           )= A      1
                                                                              O = O ÒãÒ‚ˆ«ˆº A                              1
                                                                                                                                         A = E ¹¯Ò
                                     −1          −1
         ²ºÒä}                A   1
                                          − A    2
                                                         = O 
           
           
     ËääȺ}ÈÏÈÓÈ
           
           
                                                                         α11 α12                                         −1
           {ȰˆÓºä°ã‚ÈË}ºÈ A =                                            äȈ¯ÒÈ A                               ÒäËˈmÒ
                                                                         α 21 α 22
           
                                                                    −1              1         α 22         − α12
                                                               A         =                                            
                                                                              det A          − α 21         α11
           
           
           iã«}mȯȈө²äȈ¯Ò°¹¯ÈmËãÒm© °ãË‚ Ò˯ÈmËÓ°ˆmÈ
           
                                     det ( A B ) = det ( A ) det ( B ) ;
                                                                                                                                    
                                                 T                                                    −1             1
                                         det ( A ) = det A                    ;         det ( A            )=                   .
                                                                                                                det A
           
           
           
 ¯Òä˯                     j°¹ºã ς« ¹¯ÈmÒ㺠ˮ°ˆmÒ« ° äȈ¯ÒÈäÒ °Ò°ˆËä‚ ãÒÓˮө² ‚¯ÈmÓË
                      ÓÒ®
                             

                                                                                   α11ξ1 + α12 ξ2 = β1
                                                                                                         
                                                                                   α 21ξ1 + α 22 ξ2 = β2
                             
                             äºÎÓºÏȹҰȈ mmÒË                           A      x = b Ë

  iã« n  ªˆÒ °ººˆÓº ËÓÒ« ¹¯ºm˯« ˆ°« Ó˹º°¯Ë°ˆmËÓÓº ¹º º¹¯ËËãËÓÒ   °ã‚È®

¹¯ºÒÏmºã Óººn¯È°°äȈ¯ÒmÈˈ°«m¯ÈÏËãË