Аналитическая геометрия и линейная алгебра. Умнов А.Е. - 11 стр.

UptoLike

Составители: 

Рубрика: 

cÈÏ Ëã
15
{Ë}º¯©ÒãÒÓˮөËº¹Ë¯ÈÒÒ°ÓÒäÒ
lÈ¯ÒÈ¹ºãÈÈ«°«m¯ËÏãÈË¯ÈÓ°¹ºÓÒ¯ºmÈÓÒ«äÈ¯Ò©
A
ººÏÓÈ
ÈË°«
A
T
¹¯Òªºä°ä¯Ò°
ααα α
ααα α
ααα α
αα α
αα α
αα α
αα α
11 12 13 1
21 22 23 2
123
11 21 1
12 22 2
13 23 3
12
...
...
... ... ... ... ...
...
...
...
...
... ... ... ...
...
T
n
n
mm m mn
m
m
m
nn mn
=

º Ë° ã« ªãËäËÓºm ¯ÈÓ°¹ºÓÒ¯ºmÈÓÓº® äÈ¯Ò©
A
T
¹¯Ò
∀= ∀=imjn
[, ], [, ]11
m˯Ӻ¯ÈmËÓ°mº
ijji
αα
=
T

|¹Ë¯ÈÒ«¯ÈÓ°¹ºÓÒ¯ºmÈÓÒ«Óȹ¯Òä˯ÓËÒÏäËÓ«Ë°ÒääË¯ÒË°}äÈ¯Ò
Óº¹Ë¯ËmºÒ°¯º}¯ÈÏä˯È
[
m
m°ºãË¯ÈÏä˯È
m
[
ÒÓȺº¯º
iË˯äÒÓÈÓ©º¹¯ËËãÒËãÒäÈ¯ÒºÒº¹º¯«}È
iã«}mȯÈÓ©²äÈ¯Ò°Ë°mË°¹ËÒÈãÓÈ«Ò°ãºmÈ«²È¯È}˯ҰÒ}ÈÓÈ
Ï©mÈËäÈ«lnznéuqtjtzvuÒãÒvwénlnsqznsnuÒººÏÓÈÈËäÈ«}È}
det A
iËÈã
ÓºËÒÏËÓÒËº¹¯ËËãÒËãË®Ëm©¹ºãÓËÓºm¯ÈÏËãËÏË°ÎËº¯ÈÓÒÒä°«¯È°
°äº¯ËÓÒËäº¹¯ËËãÒËãË®}mȯÈÓ©²äÈ¯ÒºÒº¹º¯«}ºm
|¹¯ËËãËÓÒË

bnznéuqtjtzvu ÒãÒ vwénlnsqznsnu }mȯÈÓº® äÈ¯Ò© º
¹º¯«}È
αα
αα
11 12
21 22
ÓÈÏ©mÈË°«Ò°ãº
det
αα
αα
αα αα
11 12
21 22
11 22 12 21
=−

|¹¯ËËãËÓÒË

bnznéuqtjtzvu ÒãÒ vwénlnsqznsnu }mȯÈÓº® äÈ¯Ò© º
¹º¯«}È
ααα
ααα
ααα
11 12 13
21 22 23
31 32 33
ÓÈÏ©mÈË°«Ò°ãº

 iË˯äÒÓÈÓ }mȯÈÓº® äÈ¯Ò© È}ÎË È°º ººÏÓÈÈ ¹¯Ò ¹ºäºÒ ºÒÓȯө² m˯Ò
}ÈãÓ©² º¯ÈÓÒÒËãË® __ l© ÓË Ëä Ò°¹ºãϺmÈ ªº ººÏÓÈËÓÒ« º© ÒÏËÎÈ
}ºÓÁãÒ}È°¹¯Ë°ÈmãËÓÒËäÈ°ºãÓ©²mËãÒÒÓäºãË®ãÒÓÒÓº¯ä
c È Ï  Ë ã                                                      15
{Ë}ˆº¯©ÒãÒÓˮө˺¹Ë¯ÈÒÒ°ÓÒäÒ



                  lȈ¯Òȹºã‚È È«°«m¯Ëς㠈Ȉˈ¯ÈÓ°¹ºÓÒ¯ºmÈÓÒ«äȈ¯Ò© A º­ºÏÓÈ
                         T
Èˈ°« A                    ¹¯Òªˆºä °ä¯Ò° 
                                                                                                                      α11          α21          ...     αm1
                                          α11 α12 α13                             ... α1n
                                                                                                       T
                                                                                                                      α12          α22          ...     α m2
                                          α21 α22 α23                             ... α 2n
                                                                                                           =          α13          α23          ...     αm3 
                                           ... ... ...                            ... ...
                                                                                                                       ...          ...         ...      ...
                                          αm1 αm2 α m3                            ... α mn
                                                                                                                      α1n          α2n          ...     αmn
                                                                                                       
                                                                                                                                           T
ˆº Ë°ˆ  ã« ªãËäËӈºm ˆ¯ÈÓ°¹ºÓÒ¯ºmÈÓÓº® äȈ¯Ò©                                                                                 A          ¹¯Ò ∀i = [1, m] , ∀j = [1, n] 
m˯Ӻ¯ÈmËÓ°ˆmº α = α ji                T
                                            ij

       |¹Ë¯ÈÒ«ˆ¯ÈÓ°¹ºÓÒ¯ºmÈÓÒ«Óȹ¯Òä˯ÓËÒÏäËӫˈ°Òääˈ¯ÒË°}‚ äȈ¯Ò‚
Óº¹Ë¯Ëmº҈°ˆ¯º}‚¯ÈÏä˯È[mm°ˆºã­Ë¯ÈÏä˯Èm[ÒÓȺ­º¯ºˆ
       
       
       
iˈ˯äÒÓÈӈ© º¹¯ËËã҈ËãÒ äȈ¯ÒºÒº¹º¯«}È
                  
                  
                  iã«}mȯȈө²äȈ¯Ò°‚Ë°ˆm‚ˈ°¹ËÒÈã ÓÈ«Ò°ãºmÈ«²È¯È}ˆË¯Ò°ˆÒ}ÈÓÈ
Ï©mÈËäÈ«lnznéuqtjtzvu ÒãÒvwénlnsqznsnu Òº­ºÏÓÈÈËäÈ«}È} det A  iˈÈã 
ÓºËÒςËÓÒ˺¹¯ËËã҈ËãË®­‚ˈm©¹ºãÓËÓºm¯ÈÏËãËÏË° Î˺¯ÈÓÒÒä°«¯È°
°äºˆ¯ËÓÒË亹¯ËËã҈ËãË®}mȯȈө²äȈ¯ÒºÒº¹º¯«}ºm
         
         
 |¹¯ËËãËÓÒË bnznéuqtjtzvu ÒãÒ vwénlnsqznsnu  }mȯȈӺ® äȈ¯Ò© º
 
                                                          α11 α12                          α11 α12
                                   ¹º¯«}È                         ÓÈÏ©mÈˈ°«Ò°ãº det           = α11α 22 − α12 α 21 
                                                          α 21 α 22                        α 21 α 22
                  
                  
    |¹¯ËËãËÓÒË                   bnznéuqtjtzvu                               ÒãÒ vwénlnsqznsnu  }mȯȈӺ® äȈ¯Ò© º
    
                                            α11 α12                               α13
                                   ¹º¯«}È α 21 α 22                             α 23 ÓÈÏ©mÈˈ°«Ò°ãº
                                            α 31 α 32                             α 33
                                   
                                                                                                                                                                                                            




 iˈ˯äÒÓÈӈ }mȯȈӺ® äȈ¯Ò© ˆÈ}ÎË È°ˆº º­ºÏÓÈÈ ˆ ¹¯Ò ¹ºäºÒ ºÒÓȯө² m˯ˆÒ
}Èã Ó©² º¯ÈÓÒ҈ËãË® __ l© ÓË ­‚Ëä Ò°¹ºã ϺmȈ  ªˆºˆ mÒ º­ºÏÓÈËÓÒ« ˆº­© ÒÏ­ËÎȈ 
}ºÓÁãÒ}ˆÈ°¹¯Ë°ˆÈmãËÓÒËäÈ­°ºã ˆÓ©²mËãÒÒÓ亂ãË®ãÒÓÒÓº¯ä