Аналитическая геометрия и линейная алгебра. Умнов А.Е. - 345 стр.

UptoLike

Составители: 

Рубрика: 

¯ÒãºÎËÓÒË


wãËäËÓ©ËÓϺ¯ÓººÒ°Ò°ãËÓÒ«
|¹Ë¯ÈÒ«º¹°}ÈÓÒ«ÒÓË}°È
|¹¯ËËãËÓÒË
¯
°m
E
n
ÏÈÈÓ ËÓϺ¯ Ò¹È
),( pq
q
p
jjj
iii
...
...
2
21
1
α
 Ë
1
q
 ËÓϺ¯ Ò¹È
)1,1(
+
pq
q
p
jj
iiii
...
...
2
21
0
β
ÓÈÏ©mÈË°«énoyszjzvuvwnéj|qqvwyxrjtq¹rvt
zéjkjéqjtztvmv qtlnrxj
1
j
y zntovéj
q
p
jjj
iii
...
...
2
21
1
α
Ë°ãÒ m }Èκä ÈÏÒ°Ë
ÒäËËäË°º¯ÈmËÓ°mº
q
p
q
p
jjj
iii
ji
jj
iiii
...
...
...
...
2
21
10
2
21
1
0
α
γ
β
=

~ÈäËÒäºÒ°¹ºãϺmÈÓÒËºË}ã«}ÈÏÈÓÒ«¹º¯«}È°ã˺mÈÓÒ«ÒÓË}°ºmm
ªº®º¹Ë¯ÈÒÒº}ÈÏ©mÈË°«Ó˺²ºÒä©äº©°ËãÈËËºÓºÏÓÈÓº®jÓÈËÓ˹º
Ó«Óº}È°ãËËº¹°ÒÒÓË}°
|¹Ë¯ÈÒ«¹ºÓ«Ò«ÒÓË}°È
|¹¯ËËãËÓÒË
¯
imÈΩ}ºÓ¯ÈmȯÒÈÓÓ©®ËÓϺ¯}ºä¹ºÓËÓ©}ºº¯ººmãºäÈ
ÏÒ°Ë Ëm}ãÒºmÈ ¹¯º°¯ÈÓ°mÈ
E
n
°ºm¹ÈÈ ° äÈ¯ÒË® º¯ÈÓº®
äÈ¯ÒË¯ÈäÈÓÈÏ©mÈË°«rvtzéjkjéqjtztuunzéq·nxrquzntovévu
ËÒä°«mÓÈÈãËºäÈ¯ÒÈº¯ÈÓÈ«äÈ¯ÒË¯ÈäÈÏÈÈËm}ÈκäÈ
ÏÒ°Ë ËÓϺ¯ Ò¹È
)0,2(
 jäËËä
ΓΓ
=
gg
SS
T
 j°²º« ÒÏ ªºº °ººÓºËÓÒ«
¹ºãÈËä°ãËËË¹¯ÈmÒãº¹¯Ëº¯ÈϺmÈÓÒ«º¯ÈÓº®äÈ¯Ò©¯ÈäÈ¹¯ÒÏÈäËÓËÈ
ÏÒ°È
ΓΓ Γ Γ
−−
== =
gg g g
SSS SS S
1
1
1
1
11
() () ()
TT
T

¹º°}ºã}ÒÏ
EE SS S S
== =
−−
T
TT
T
()()
11
°ãËËºã«ÓËm©¯ºÎËÓ
Óº® äÈ¯Ò©
S
°¹¯ÈmËãÒmº ¯ÈmËÓ°mº
()()
T
T
SS
=
1
1
 k ªº Ò ºÏÓÈÈË º
º¯ÈÓÈ«äÈ¯ÒÈ¯ÈäÈº¹¯ËËã«Ëmºm°Ë²ÈÏҰȲmÈΩ}ºÓ¯ÈmȯÒÈÓÓ©®ËÓϺ¯
γ
ij

ºÈÓÈãºÒÒ°º¹Ë¯ÈÒË®º¹°}ÈÓÒ«ÒÓË}°ÈÈÒä
¯ÒãºÎËÓÒË
wãËäËӈ©ˆËÓϺ¯ÓººÒ°Ò°ãËÓÒ«



|¹Ë¯ÈÒ«º¹‚°}ÈÓÒ«ÒÓË}°È
             
             
             
                                                                                                           α i1i22...i p q  Ë q ≥ 1  ‘ËÓϺ¯ ˆÒ¹È
                                                                                                                j1 j ... j
    |¹¯ËËãËÓÒË            ‚°ˆ  m E n  ÏÈÈÓ ˆËÓϺ¯ ˆÒ¹È ( q, p )
    ¯
                                                  • j 2 ... j q
                            (q − 1, p + 1) β i                      ÓÈÏ©mÈˈ°«énoyszjzvuvwnéj|qqvwyxrjtq¹rvt
                                                   0 i1i 2 ...i p



                            zéjkjéqjtztvmv qtlnrxj j1  y zntovéj α i 1i 2...i
                                                                                                                 j j ... jq
                                                                                                                                Ë°ãÒ m }ÈÎºä ­ÈÏÒ°Ë
                                                                                                                 12       p
                                                                           • j ... j
                            ÒäËˈäË°ˆº¯ÈmËÓ°ˆmº β i i 2i ...iq = γ i 0 j1 α i 1i 2...i q 
                                                                                                           j j ... j
                                                      0 1 2      p              1 2      p
        
        
        ~ÈäˈÒ䈺Ұ¹ºã ϺmÈÓÒˈºË}ã«‚}ÈÏÈÓÒ«¹º¯«}È°ã˺mÈÓÒ«ÒÓË}°ºmm
ªˆº®º¹Ë¯ÈÒÒº}ÈÏ©mÈˈ°«Ó˺­²ºÒä©äˆº­©°ËãȈ Ë˺ӺÏÓÈÓº®jÓÈËÓ˹º
Ó«ˆÓº}‚È°ãË‚ˈº¹‚°ˆÒˆ ÒÓË}°
        
        
        
        
|¹Ë¯ÈÒ«¹ºÓ«ˆÒ«ÒÓË}°È
             
             
             
    |¹¯ËËãËÓÒË            imÈΩ}ºÓˆ¯ÈmȯÒÈӈө®ˆËÓϺ¯}ºä¹ºÓËӈ©}ºˆº¯ººmã ­ºä­È
    ¯
                            ÏÒ°Ë Ëm}ãÒºmÈ ¹¯º°ˆ¯ÈÓ°ˆmÈ E n  °ºm¹ÈÈ ˆ ° äȈ¯ÒË® º­¯ÈˆÓº®
                            äȈ¯ÒË€¯ÈäÈÓÈÏ©mÈˈ°«rvtzéjkjéqjtzt€uunzéq·nxrquzntovévu
             
             
             ­ËÒä°«mÓÈÈãˈºäȈ¯ÒȺ­¯ÈˆÓÈ«äȈ¯ÒË€¯ÈäÈÏÈÈˈm}Èκä­È
                                                                                   T
ÏÒ°Ë ˆËÓϺ¯ ˆÒ¹È ( 2,0)  jäËËä Γ                               g′
                                                                          = S          Γ       g
                                                                                                   S  j°²º« ÒÏ ªˆºº °ººˆÓº ËÓÒ«
¹ºã‚ÈËä°ãË‚ Ë˹¯ÈmÒ㺹¯Ëº­¯ÈϺmÈÓÒ«º­¯ÈˆÓº®äȈ¯Ò©€¯Èäȹ¯ÒÏÈäËÓË­È
ÏÒ°È
       
       
                       −1                                                  −1                                                 −1                 −1 T
                                                   S ) −1 = S
                                     T                                                             T −1
                  Γ    g′
                             =( S        Γ    g
                                                                                 Γ     g
                                                                                           ( S         )       = S                 Γ   g
                                                                                                                                           ( S    ) 
             
             
                                         T                          −1 T               −1 T                T
¹º°}ºã }‚ÒÏ E = E                          =( S           S        ) =( S                )       S           °ãË‚ˈˆºã«ÓËm©¯ºÎËÓ
                                                                                           −1                         T
Óº® äȈ¯Ò© S  °¹¯ÈmËãÒmº ¯ÈmËÓ°ˆmº ( S  ) T = ( S ) −1  k ªˆº Ò ºÏÓÈÈˈ ˆº
º­¯ÈˆÓÈ«äȈ¯ÒÈ€¯ÈäȺ¹¯ËËã«Ëˆmºm°Ë²­ÈÏҰȲmÈΩ}ºÓˆ¯ÈmȯÒÈӈө®ˆËÓϺ¯
γ ij 
             
             
             ºÈÓÈãºÒÒ°º¹Ë¯ÈÒË®º¹‚°}ÈÓÒ«ÒÓË}°ÈÈÒä