Аналитическая геометрия и линейная алгебра. Умнов А.Е. - 20 стр.

UptoLike

ВУЗ: 

МФТИ | Москва

Составители: 

Рубрика: 

24
Ë} ÒÒ}ÈÁË ¯©m©° Ë®äÈËäÈÒ}Òlnj
ÙkÓÈãÒÒË°}È«˺äË¯Ò«ÒãÒÓË®ÓÈ«ÈãË¯Èµ
äÓºmkp
sãËmº® mË}º¯ °ÒÈË°« }ºããÒÓËȯөä ãºä ¯ºä mË}º¯ sãËmº®
mË}º¯°ÒÈË°«}ºä¹ãÈÓȯөäãº®¹È¯ËmË}º¯ºm
|¹¯ËËãËÓÒË

{©¯ÈÎËÓÒË mÒÈ
→→→
+++
nn
aaa
λλλ
...
2211
 Ë
n
λλλ
,...,,
21
ÓË}ºº¯©Ë
Ò°ãÈÓÈÏ©mÈË°«sqtnptvprvuiqtj|qnpmË}º¯ºm
→→→
n
aaa
,...,,
21

p°ãÒkxnÒ°ãÈ
n
λλλ
,...,,
21
¯ÈmÓ©ÓãºÓºm¯ËäËÓÓºº¯ÈmÓº°ÒãÓº
°ãºmÒ
0...
21
=+++
n
λλλ
 º È}È« ãÒÓË®ÓÈ« }ºäÒÓÈÒ«
ÓÈÏ©mÈË°«zéqkqjstvp
p°ãÒ }vz¹ i vltv ÒÏ Ò°Ëã
n
λλλ
,...,,
21
ºãÒÓº º Óã« º Ë°
0...
21
>+++
n
λλλ
 º ÈÓÓÈ« ãÒÓË®ÓÈ« }ºäÒÓÈÒ« ÓÈÏ©mÈË°« tn
zéqkqjstvp
vºãÈËÓÒËº°ääÒ¯ºmÈÓÒÒ
{ ˲ °ãÈ«² }ºÈ «mÓÈ« ÏȹҰ °ää© ÓË}ºº¯ºº Ò°ãÈ °ãÈÈË䩲
ÓËËãË°ºº¯ÈÏÓÈ ÒãÒ ÓËmºÏäºÎÓÈ Óº ÒÏmË°Óº }È} ÏÈmÒ°Ò ÏÓÈËÓÒË }Èκº ÒÏ
°ãÈÈË䩲 º ˺ Óºä˯È º º¹°}ÈË°« Ò°¹ºãϺmÈÓÒË °¹ËÒÈãÓº® Áº¯ä© ÏȹҰÒ
º¹Ë¯ÈÒÒ°ääÒ¯ºmÈÓÒ«
∑
=
=++++
N
nk
kFNFnFnF
)()(...)1()(

ÒÈË°«µfyuuj
Fk()
wv
k
vz
n
lv
N
µË
N
ÒÓË}°°ääÒ¯ºmÈÓÒ«
Q
äÒÓÒ
äÈãÓºËÏÓÈËÓÒËÒÓË}°È°ääÒ¯ºmÈÓÒ«
1
äÈ}°ÒäÈãÓºËÏÓÈËÓÒËÒÓË}°È°ä
äÒ¯ºmÈÓÒ«ÒÓÈ}ºÓË
Fk
()
ºÒ®mÒ°ãÈÈË人
¯Òä˯

º°ºãÈËÓÒº°ääÒ¯ºmÈÓÒÒ°¹¯ÈmËãÒm©°ãËÒË¯ÈmËÓ
°mÈ

.
1
)1(
1
)1(
1
...
32
1
21
1
4
)1(
)1(...21
6
)12)(1(
)1(...21
1
1
2
1
22
1
33333
1
22222
)
(
N
N
kkNN
j
NN
iNN
NNN
kNN
N
k
N
j
N
i
N
k
−
=
+
=
−
++
⋅
+
⋅
=
+
==+−+++
++
==+−+++
∑
∑∑
∑
−
=
==
=
24 Ë }  Ò Ò   } È Á Ë  ¯ ©   m © °  Ë ®   ä È ˆ Ë ä È ˆ Ò } Ò   l n ‘ j 
ÙkÓÈã҈ÒË°}È«˺äˈ¯Ò«ÒãÒÓË®ÓÈ«ÈãË­¯ÈµäÓºmkp



          s‚ãËmº® mË}ˆº¯ °҈Èˈ°« }ºããÒÓËȯөä ã ­ºä‚ ¯‚ºä‚ mË}ˆº¯‚ s‚ãËmº®
mË}ˆº¯°҈Èˈ°«}ºä¹ãÈÓȯөäã ­º®¹È¯ËmË}ˆº¯ºm
          
          
 
                                                              →         →                 →
 |¹¯ËËãËÓÒË
                         {©¯ÈÎËÓÒË mÒÈ λ1 a1 + λ2 a2 + ... + λn an  Ë λ1 , λ2 ,..., λn ÓË}ºˆº¯©Ë
 
                                                                                                             → →           →
                         Ò°ãÈÓÈÏ©mÈˈ°«sqtnptvprvuiqtj|qnpmË}ˆº¯ºm a1 , a2 ,..., an 
                         
                         p°ãÒkxnÒ°ãÈ λ1 , λ2 ,..., λn ¯ÈmÓ©ӂã ºÓºm¯ËäËÓÓº ˆº¯ÈmÓº°Òã Óº
                         ‚°ãºmÒ            λ1 + λ2 + ... + λn = 0  ˆº ˆÈ}È« ãÒÓË®ÓÈ« }ºä­ÒÓÈÒ«
                         ÓÈÏ©mÈˈ°«zéqkqjstvp
                         
                         p°ãÒ }vz¹ i€ vltv ÒÏ Ò°Ëã λ1 , λ2 ,..., λn  ºˆãÒÓº ºˆ ӂã« ˆº Ë°ˆ 
                          λ1 + λ2 + ... + λn > 0  ˆº ÈÓÓÈ« ãÒÓË®ÓÈ« }ºä­ÒÓÈÒ« ÓÈÏ©mÈˈ°« tn
                         zéqkqjstvp
                         
            
            
            
vºãÈËÓÒ˺°‚ääÒ¯ºmÈÓÒÒ
       
       
       { ˆË² °ã‚È«² }ºÈ «mÓÈ« ÏȹҰ  °‚ää© ÓË}ºˆº¯ºº Ò°ãÈ °ãÈÈË䩲
ÓËËãË°ºº­¯ÈÏÓÈ ÒãÒ ÓËmºÏäºÎÓÈ Óº ÒÏmË°ˆÓº }È} ÏÈmҰ҈ ÏÓÈËÓÒË }Èκº ÒÏ
°ãÈÈË䩲 ºˆ ˺ Óºä˯È ˆº º¹‚°}Èˈ°« Ò°¹ºã ϺmÈÓÒË °¹ËÒÈã Óº® Áº¯ä© ÏȹҰÒ
º¹Ë¯ÈÒÒ°‚ääÒ¯ºmÈÓÒ«
                                                                                          N
                                            F (n) + F (n + 1) + ... + F ( N ) =          ∑ F (k ) 
                                                                                         k =n
 ҈Èˈ°«µfyuuj F ( k ) wv kvz nlv Nµ Ë NÒÓË}°°‚ääÒ¯ºmÈÓÒ« Q äÒÓÒ
äÈã ÓºË ÏÓÈËÓÒË ÒÓË}°È °‚ääÒ¯ºmÈÓÒ« 1  äÈ}°ÒäÈã ÓºË ÏÓÈËÓÒË ÒÓË}°È °‚ä
äÒ¯ºmÈÓÒ«ÒÓÈ}ºÓË F ( k ) º­Ò®mÒ°ãÈÈË人
        
        
¯Òä˯        º°ºãÈ ËÓÒ º°‚ääÒ¯ºmÈÓÒÒ­‚‚ˆ°¹¯ÈmËãÒm©°ãË‚ Ò˯ÈmËÓ
        °ˆmÈ
                                                                                   N
                                                                                                     N ( N + 1)(2 N + 1)
                                12 + 2 2 + ... + ( N − 1) 2 + N 2           =     ∑k2           =
                                                                                                              6
                                                                                  k =1
                                                                                  N
                                                                                                    N 2 ( N + 1) 2              N       2
                          13 + 2 3 + ... + ( N − 1) 3 + N 3           =     ∑ i3          =
                                                                                                          4
                                                                                                                       =        ∑j
                                                                                                                               ( )          
                                                                                  i =1                                          j =1
                                                                                N −1
                                 1    1               1                                  1               N −1
                                    +
                                1⋅ 2 2 ⋅3
                                          + ... +
                                                  ( N − 1) N
                                                                        =       ∑ k (k + 1)         =
                                                                                                          N
                                                                                                              .
                                                                                k =1
            

Страницы