Аналитическая геометрия и линейная алгебра. Умнов А.Е. - 171 стр.

UptoLike

Составители: 

Рубрика: 

cÈÏ Ëã
175
ÒÓˮӺË¹¯º°¯ÈÓ°mº
äËÓ©
m
xxx
,...,,
21
 ¹ºãÒä
],1[;
1
ljxy
m
i
ijij
==
=
β
 ¯Ò¯ÈmÓ«Ëä ÓãËmºä
ªãËäËÓ ¹¯ºÒÏmºãÓãÒÓË®Ó}ºäÒÓÈÒ m©¯ÈÓÓºº ÓÈº¯È
yy y
l
12
,,...,

oxxy
i
m
i
l
j
jji
m
i
iji
l
j
j
l
j
jj
===
∑∑
=====
11111
)(
µββµµ

º°}ºã}ªãËäËÓ©
[[ [
P12
,,...,
ãÒÓˮӺ ÓËÏÈmÒ°Òä©º }ºªÁÁÒÒËÓ©
µ
i
ºãÎÓ©ºmãËmº¯«°ãËË®ºÓº¯ºÓº®°Ò°ËäËãÒÓˮө²¯ÈmÓËÓÒ®
=
==
l
j
jji
mi
1
],1[,0
µβ

sºªÈ °Ò°ËäÈ ÒäËË ¹º ˺¯ËäË
0rg
> mll
ji
β
ãÒÓˮӺÓËÏÈ
mÒ°Ò䩲È°ã˺mÈËãÓº tntysnk} ¯ËËÓÒ® ¹º°}ºã}
m
ji
β
rg

¯ÒÓÒäÈ« mº mÓÒäÈÓÒË º
l
Ò
m
ÓÈ¯ÈãÓ©Ë Ò°ãÈ ¹ºãÈËä
1rg
mll
ji
β
 º ˰ °˰mË ÓË¯ÒmÒÈãÓÈ« ãÒÓË®ÓÈ« }ºäÒÓÈÒ«
ªãËäËÓºm
l
yyy
,...,,
21
¯ÈmÓÈ«
o

˺¯ËäÈº}ÈÏÈÓÈ
ҹ˯¹ãº°}º°
|¹¯ËËãËÓÒË

lÓºÎ˰mº
Γ
º¯ÈϺmÈÓÓºËÒÏªãËäËÓºmÈ
0
xx
+
Ë
Λ
0
x
˰
ÓË}ºº¯©® ÁÒ}°Ò¯ºmÈÓÓ©® ªãËäËÓ ãÒÓˮӺº ¹¯º°¯ÈÓ°mÈ
Λ
 È
x
ãº®ªãËäËÓÓË}ºº¯ºº¹º¹¯º°¯ÈÓ°mÈ
Λ
ÓÈÏ©mÈË°«mqwnéw
svxrvxzíÒãÒsqtnptuutvmvviéjoqnumãÒÓˮӺä¹¯º°¯ÈÓ°mË
Λ .
~ÈäËÈÓÒ«
°
 {ºËä°ãÈËҹ˯¹ãº°}º°ÓË«mã«Ë°«¹º¹¯º°¯ÈÓ°mºä
°
 p°ãÒGLP
k
ººmº¯«º
k
ä˯Ӻ®ҹ˯¹ãº°}º°Ò
ÈÈ

Ívrjojz ·zv nxsq ësnuntz
x
q
y
wéqtjlsnjz tnrvzvévp mqwnéwsv
xrvxzq
Γ
zvëzvpnmqwnéwsvxrvxzqiylnzwéqtjlsnjzqësnuntz
zx y=+
αα
()1
mln
α
síivn·qxsv
c È Ï  Ë ã                                                      175
ÒÓˮӺ˹¯º°ˆ¯ÈÓ°ˆmº



                                                                                                        m
                      äËӈ© x1, x 2 ,..., x m  ¹ºã‚Òä y j =                                       ∑ β ji xi ;              j = [1, l ]  ¯Ò¯ÈmÓ«Ëä ӂãËmºä‚
                                                                                                      i =1
                      ªãËäËӈ‚           ¹¯ºÒÏmºã ӂ                                 ãÒÓˮӂ                      }ºä­ÒÓÈÒ                          m©­¯ÈÓÓºº                      ÓÈ­º¯È
                       y1 , y 2 ,..., y l 
                                                                l                      l          m                      m       l
                                                              ∑ µ j y j = ∑ µ j ∑ β ji xi = ∑ ( ∑ β ji µ j ) xi = o 
                                                               j =1                  j =1        i =1                  i =1 j =1
                      
                      º°}ºã }‚ ªãËäËӈ© [1 , [2 ,..., [P  ãÒÓˮӺ ÓËÏÈmÒ°Òä© ˆº }ºªÁÁÒÒËӈ© µi
                      ºãÎÓ©‚ºmãˈmº¯«ˆ °ãË‚ Ë®ºÓº¯ºÓº®°Ò°ˆËäËãÒÓˮө²‚¯ÈmÓËÓÒ®
                      
                                                                                           l
                                                                                      ∑ β ji µ j = 0 ,                  i = [1, m] 
                                                                                      j =1
                                                                                                               
                      sº ªˆÈ °Ò°ˆËäÈ ÒäËˈ ¹º ˆËº¯ËäË   l − rg β ji ≥ l − m > 0 ãÒÓˮӺ ÓËÏÈ
                      mÒ°Ò䩲 È °ã˺mȈËã Óº tntysnk€} ¯Ë ËÓÒ® ¹º°}ºã }‚ rg β ji ≤ m 
                      ¯ÒÓÒäÈ« mº mÓÒäÈÓÒË ˆº l Ò m ÓȈ‚¯Èã Ó©Ë Ò°ãÈ ¹ºã‚ÈËä
                      l − rg β ji ≥ l − m ≥ 1  ˆº ˰ˆ  °‚Ë°ˆm‚ˈ Óˈ¯ÒmÒÈã ÓÈ« ãÒÓË®ÓÈ« }ºä­ÒÓÈÒ«
              ªãËäËӈºm y1, y 2 ,..., yl ¯ÈmÓÈ«o
              
              
       ‘˺¯ËäȺ}ÈÏÈÓÈ
            
            
            
€Ò¹Ë¯¹ãº°}º°ˆ
                  
                  
    |¹¯ËËãËÓÒË                    lÓºÎ˰ˆmº Γ º­¯ÈϺmÈÓÓºËÒϪãËäËӈºmmÒÈ x + x 0 Ë x 0 ∈ Λ ˰ˆ 
    
                                    ÓË}ºˆº¯©® ÁÒ}°Ò¯ºmÈÓÓ©® ªãËäËӈ ãÒÓˮӺº ¹¯º°ˆ¯ÈÓ°ˆmÈ Λ  È x
                                    ã ­º®ªãËäËӈÓË}ºˆº¯ºº¹º¹¯º°ˆ¯ÈÓ°ˆmÈ Ω ⊂ Λ ÓÈÏ©mÈˈ°«mqwnéw
                                    svxrvxzí ÒãÒsqtnpt€uutvmvviéjoqnu mãÒÓˮӺ乯º°ˆ¯ÈÓ°ˆmË Λ .
                  
                  
                  
 ~ÈäËÈÓÒ«                        ° {º­Ëä°ã‚ÈËҹ˯¹ãº°}º°ˆ ÓË«mã«Ëˆ°«¹º¹¯º°ˆ¯ÈÓ°ˆmºä
                                    
                                    ° p°ãÒGLP Ω kˆººmº¯«ˆºkä˯Ӻ®ҹ˯¹ãº°}º°ˆÒ
                  
                  
 ~ÈÈÈ                          Ívrjojz ·zv nxsq ësnuntz€ x q y wéqtjlsn jz tnrvzvévp mqwnéwsv
 
                                  xrvxzq Γ zvëzvp nmqwnéwsvxrvxzqiylnzwéqtjlsn jzqësnuntz
                                  z = αx + (1 − α ) y mlnαsíivn·qxsv