Chng 2. Dng song tuyn tnh Dng ton phngCHNG 2. DNG SONG TUYN TNH DNG TON PHNG____________________________________________________________I. DNG TUYN TNH DNG SONG TUYN TNH1. Dng tuyn tn!" 1.1 #$n! ng!%&"Gi s K l mt trng s. Mi nh x tuyn tnh t khng gin !"#t$r V tr%n trng K!$ K &'# g(i l mt dng tuyn tnh tr%n V. )*y+ mi ,ng tuyn tnh tr%n V l mt nhx - f V K th. ## &i/u ki0n su-12 3 2 3 2 3 f x y f x f y + +12 3 2 3 f x f x !4i m(i + x y V !K .1.2 ' ()" 56t nh x 78 7-7fx x x+ a !4i 8 72 + 3 x x x 9inh !i%n t: ki;m tr &? hi0u @V l t*A tBt # ## ,ng tuyn tnh tr%n V. Cr%n t*A @V x6t hi Ah6A t$n su&2 + 3 + 8+...+ X 8+...+ 0 0f . / !, 0 n . f 4 dng song tuyn tnh duy nh?t th@! "Au kBn ny. M tr*n 2 3 2 + X 30 , n1 ! C , n K &'# g(i l m tr*n #I ,ng s$ng tuyn tnh f&i !4i#^A #[ sG 2-, -&3. Tu fl ,ng s$ng tuyn tnh tr%n V, thJ m tr*n Fi;u ,icn #I f th"$ #^A #[ sG D-, -E&'# g(i l m tr*n Fi;u ,icn #I fth"$ -. ' ()" Tu fl ,ng s$ng tuyn tnh tr%n 7 NK K &'# x# &Enh FGi 8 7 8 7 N 8 7 8 7 N2 + X + + 3 2 32 7 N 3 f x x y y y x x y y y +thJ m tr*n Fi;u ,icn fth"$ #^A #[ sG #hnht_# l 8 7 N8 7 N111 ]i s Cuyn tnh 7. 39Chng 2. Dng song tuyn tnh Dng ton phngTu f l t#h ! h4ng #I khng gin Wu#li, thJ m tr*n Fi;u ,icn #I fth"$ mt #[sG 9 #hnh l m tr*n Grm #I #[ sG &K. #$n! @A 2.2.2" Fu dng song tuyn tnh f tGHn V *> *I* ,! tGJn )Ku dLn th2o *I* *sM N . O 48n 4Pt 4 1 . - . Q 4 ,! tGJn *huyKn * sM tR N s!ng O thS O- Q 1Q . bi m tr*n 1, -th. tnh #hBt tr%n &'# g(i l hi m tr*n t[ng &dng. TKi ##hkh#+ hi m tr*n &'# g(i l t[ng &dng !4i nhu nu #heng l m tr*n Fi;u ,icn #I#fng mt ,ng s$ng tuyn tnh.'()1"56tm tr*n#I,ngs$ngtuyn tnh8 8 8 7 7 8 7 72 + 3 7 N f x y x y x y x y x y + l,ng s$ng tuyn tnh tr%n 7&i !4i #[ sG #hnh t_# #I 7 l-8 7N 8111 ]Cuynhi%n+ mtr*n-#I ,ngs$ngtuyntnhf&i !4i #[s48 7R \ + ] - u u !4i8 728+83X 28+ M3 u u 8 8 8 7 7 8 7 72 + 3 8X 2 + 3 OX 2 + 3 7X 2 + 3 8 f u u f u u f u u f u u 8 O7 8- 11 ]' () 2" Yng s$ng tuyn tnh 8 8 8 7 8 N 7 7 N 8 N N2 + 3 7 N g x y x y x y x y x y x y x y + + #K m tr*n tr$ng #[ sG #hnh t_# l 8 7 8M 8 MN M gC 1 1 1 1 ]#$n! @A 2.2.2" . "P* kU hBu 4 G!nkDfE. C!B A- Yng s$ng tuyn tnh f&'# g(i l suy )n nu rnk2f 3 h ,im ) ! khng suy)n nu rnk2f 3 i ,im V. ' ()" CJm hng #I ## ,ng s$ng tuyn tnh tr$ng ## ! ,= tr%n. #$n! ng!%& 2.2.C" Vh$ fl ,ng s$ng tuyn tnh tr%n V. + x y V + f &'# g(i l "# x+ng nu- 2 + 3 2 + 3 f x y f y x . f&'# g(i l "# x+ng 4B*h nu 2 + 3 2 + 3 f x y f y x f&'# g(i l th!y phHn nu f2x, x 3 i M ' ()" Vh$ 7V K . 56t ## nh x f ! g &'# x# &Enh nh su- 7 78 7 7 8-2 + 3f K K Kx y x y x y + a! 7 78 7 7 8-2 + 3g K K Kx y x y x y a!4i 78 72 + 3 x x x K ! 78 72 + 3 y y y K >hi &K+ f l mt ,ng s$ng tuyn tnh &i xZng ! g l mt ,ng s$ng tuyn tnhthy Ahi%n+ &jng thi l ,ng s$ng tuyn tnh &i xZng l0#h. 9inh !i%n t: ki;m tr li kt ku tr%n9inh !i%n tJm th%m ## ! ,= kh# !/ ,ng s$ng tuyn tnh &i xZng ! &i xZng l0#h.i s Cuyn tnh 7. 40Chng 2. Dng song tuyn tnh Dng ton phng#$n! @A 2.2.D"Dng song tuyn tnh tGHn KVkhng g!n .2*toG h=u hn *hAu V 4"# x+ng kh . *h6 kh ,! tGJn *T! n> "# .: * sM no "> 4 ,! tGJn "# x+ng. C!Eng hi &K+( )R8R R8R2 3 ... ...tnnxx x x O 1Ox _ , 2,39$ snh ! Ahi 23 ! 2,3 t #K- O- O 1O 2@@@3b0 thZ# 2@@@3 #h$ thBy mi kun h0 gis hi m tr*n #I #fng mt ,ng t$n Ah[ng&i !4i hi #[ sG kh# nhu. )J O l m tr*n khng suy Fin+ n%n t #K r2-3 i r213. )*y+ hng #I m tr*n ,ng t$nAh[ng . Tu r213 i n thJ g(i l ,ng khng suy Fin. #$n! @A 2.2.2" Cho N 4 * sM *T! khng g!n .2*toG V n *hAu. C5t Inh x - f V K "P* g; 4 ,5t dng ton phng kh . *h6 kh n> "P* .t d: dngW8 82 3n n00 0f x ! x x tGong "> 8 72 + +...+ 3nx x x x 4 t;! "5 *T! x th2o * sM N .0! K .2.2 Dng 5!n! tI5 56& (ng t+Fn G!H>ng"2.2.1 C> *? 5!n! tI5 56& (ng t+Fn G!H>ngV[ sG 8 7\ + +...+ ]n. . .#I khng gin !"#t$r V tr%n trng K &'# g(i l #[ sG #hnh t_##I ,ng t$n Ah[ng nu m tr*n - #I ,ng &i !4i #[ sG &K l m tr*n #h6$. 87M ... MM ... M... ... ... ...M M ...n))-) 1 1 1 1 1 ]>hi &KFi;uthZ# t( I#K,ng7 7 78 8 7 72 3 ...n nx )t ) t ) t + + +2i!3 tr$ng&K8 8 7 7...n nx t . t . t . + + +i s Cuyn tnh 7. 42Chng 2. Dng song tuyn tnh Dng ton phngti;u thZ# 2i!3 &'# g(i l dng *hnh tX* #I ,ng t$n Ah[ng . C!B A" Yng #hnh t_# #I mt ,ng t$n Ah[ng x# &Enh khng ,uy nhBt. Tu 2 3 x #K ,ng #hnh t_# thJ t #K ## kt ku su-2 3 x x# &Enh ,[ng nu m(i M

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)2 3 x x# &Enh hi &K+ 788 8 82 3 R x ! x g ++ tr$ng &K 8gl mt ,ng t$n Ah[ng khng #hZ 8x. Tu 88M ! + nhng 87M ! thJ &^t R R8 8 7R R7 8 7x x xx x x +' >hi &K+R 7 R 787 8 7 87 8 87 7! x x ! x ! x + khi &KR 78 82 3 x )x g +!4i8glmt ,ngt$nAh[ngkhng #hZ 8x. CiA t=# ku trJnh ny t & 2 3 x !/ ,ng #hnh t_#.' ()" 83 Vh$ ,ng t$n Ah[ng7 7 78 7 N 8 7 8 N2 3 7 g O v x x x x x x x x + +. bQy & ,ng t$nAh[ng tr%n !/ ,ng #hnh t_#G9N9PA ,=ng Ah[ng AhA wrng"( ) ( ) ( )( )7 7 7 7 7 7 7 78 7 N 8 7 8 N 8 8 N 7 N 7 N 7 7 N7 77 7 7 7 7 78 N 7 N 7 N 7 7 N 8 N 7 N 7 N 777 7 78 N 7 7 7 N N N2 3 7 g O v ` 7 2O 7 3 2O 7 3 a 2O 7 3 7 gO 7 8p 8p O 7 g O 7 7N 8p 7O 7 72 v 8p 3 xx x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x xx x x x x x x x + + + + + + + + + + + + +S^t 8 8 N 77 7 NN NO 7Ot x x xt x xt x + '>hi &K+ ,ng #hnh t_# #I ,ng t$n Ah[ng l 7 7 78 7 N2 3 7 x t t t t +i s Cuyn tnh 7. 43Chng 2. Dng song tuyn tnh Dng ton phngN!,n -.t- ,ng t$n Ah[ng ny khng x# &Enh ,[ng. 73 Vh$ ,ng s$ng tuyn tnh #K m tr*n Fi;u ,icn l M 8 78 M 87 8 M1 1 1 1 1 ]. >hi &K[ ]88 7 N 8 7 N 7 8 7 8 N 7 NNM 8 72 + + 3 8 M 8 7 O 77 8 Mxx x x x x x x x x x x x xx 1 1 1 1 + 1 1 1 1 ] ]S^t R R8 8 7R R7 8 7x x xx x x +' >hi &K+ ( ) ( )( )( )R R8 8 7R R7 8 7N NR R R R R R R R R R8 7 8 7 8 7 N 8 7 NR 7 R 7 R R R R R R R R8 7 8 N 7 N 8 N 7 NR 7 R 7R 7 R 7 R R R R R 7 R R R 7 R R N N8 7 8 N 7 N 8 8 N 7 7 NR 78R2 3 72 32 3 O 77 O O 7 77 7 p 7 7 pO 77x x xx x xx xx x x x x x x x x x xx x x x x x x x x xx xx x x x x x x x x x x xx + ' + + + + + + _ + + + + + ,R 7 R 7R R R 7 R R R 7 R 7 N N8 N 7 7 N N N7R 7R R R 7 R 7 N8 7 N Nx 8v72 N 3O O 7 7N 8g7 72 37 7 7x xx x x x x x xxx x x x _+ + + + + , _ + + + , 2@3S^t N8 8R R7 7 NRN NRR7N7xt xt x xt x + +' khi &K #K ,ng #hnh t_# #I ,ng t$n Ah[ng nh su-7 7 78 7 N8g2 3 7 77t t t t +N3 56t ,ng t$n Ah[ng tr$ng khng gin N &'# x# &Enh nh su-7 7 78 8 7 7 8 N N 7 N2 3 7 O O 7 x x x x x x x x x x + + + +!4i N8 7 N2 + + 3 x x x x bQy tJm ,ng #hnh t_# #I ,ng t$n Ah[ng tr%n. 9inh !i%n t: lm nh Fi t*A nh..OK P!H>ng G!1G P&5+O9Gi s Fi;u thZ# #I ,ng t$n Ah[ng 2 3 x tr$ng #[ sG 8 72 + +...+ 3n- 2 2 2 li s Cuyn tnh 7. 44Chng 2. Dng song tuyn tnh Dng ton phng+ 82 3 2 + 3n00 0u u u ! x x + !4i 2 + 300! 2 2 >hi &K+ 88 87 878 77 78 7......2 3... ... ... ......nn0n n nn! ! !! ! !1 !! ! ! 1 1 1 1 1 ]56t ## &Enh thZ# #$n #hnh #I m tr*n 188 878 88 7 878 77X X...X ,"t2 3! !! 1! ! 2y3Tu tBt # ## &Enh thZ# #$n #hnh &/u kh# M+ tZ# l-8 7MX MX...X Mn + thJ tjn ti Ah[ng AhA+ g(i l Ah[ng AhA z#$Fi &; tJm mt#[ sG R R R8 7R\ X X...X ]n( 2 2 2s$ #h$ ,ng t$n Ah[ng 2 3 x #K ,ng #hnh t_# su &hi k Y 7+ t gii h078 77777878 77N7 Mv7N pM7 + ' '+ >hi k i N+ t gii h0N8 N7 NNN8 N7N8 NNN7 7 M7NM77 L 8 + + + ' Chy NN88g + t gii h0 &'# N8v8g ! N7878g)*y #[ sG m4i lR 88R7 8 7R 8 7 NN7p vv 878g222 2 22 2 22 ' + 5K P!H>ng G!1G g91 t4$ 49Qng" #$n! @A 2.2.1" C[ dng ton phng tGHn khng g!n .2*toG (u*4d h=u hn *hAu( "Au *> ,5t * sM *hnh tX* 4 * sM tG\* *hu]n *T! khng g!n (u*4d. CI* .2*toG *T! * sM *hnh tX* "> g; 4 *I* phng *hnh *T! dng ton phng . C!Eng hi &K+ &^t 8 M M7 8 Mg 7 8OC 1 1 1 1 ] suy r 8 7 gM 8 7M M 8C 1 1 1 1 ] ! 8 M MM 8 MM M yOC 1C 1 1 1 1 ]Chy #[ sG #hnh t_# #I N Fnng #[ sG - s$ #h$ C #hnh l m tr*n &ri #[ sG t #[ sG#hnh t_# sng #[ sG -+ hy t &Q s ,=ng Ah6A &ri Fin. R 7 R g RR 7 RRx x y %y y %% % +'' () 2" bQy & ,ng t$n Ah[ng N Fin th:# su &ng t4Qn W!Xng g9&n :M5t+4 t!Y5"2.2.1 Dng t+Fn G!H>ng -15 7$n! (H>ng" #$n! ng!%& 2.2.1" Yng t$n Ah[ng tr%n 1 khng gin !"#t$r V g(i l xI* "anhdngnu 2 3 M x >&i !4i m(ix kh# !"#t$r M. Tg'# li nu 2 3 M x "Au dng. O+* 4, nu *> dng *hnh tX*. 7 78 82 3 ...n nx )t ) t + +thS )

d e .:Y ^, f, n. N!,n -.t"Gi s V l mt khng gin !"#t$r n #hi/u tr%n . >hi &K+ mt ,ng t$n Ah[ng tr%n) &'# g(i l ,ng t$n Ah[ng th:#. Z/ 7J 2.2.2"Cho 4 ,5t dng ton phng th\*. O! *> thK tS, th?y ,5t * sM N *T!V s!o *hoW 7 7 7 78 82 3 ... ...p p Gx x x x x + + + tGong "> 8 7+ +...+Gx x x4 t;! "5 *T! .2*toG x th2oN.#$n! @A 2.2.D" C; dng *hnh tX* *T! dng ton phng th\*. 7 7 7 78 8 8 8 82 3 ... ... 2 +...+ M3p p p p G G Gx * x * x * x * x * * + + + + >"Au *> *gng s# p *I* hB s# dng. s# GVp *I* hB s# b,. #$n! ng!%& 2.2.[" 9 p ## h0 s ,[ng ! s G 1 p ## h0 s . VhZngminhrnng+nhx-V V K x#&EnhFGi8 72 + 3 2 3 2 3 x y f x f y lmt,ngs$ng tuyn tnh tr%n ). x3 Vh$ ,ng t$n Ah[ng N- _ x# &Enh FGi 7 7 7 N2 + + 3 N 87 p v 7v 87 X 2 + + 3 _ x y % x xy x% y y% % x y % + + 3 w*A m tr*n #I k tr$ng #[ sG #hnh t_#. F3 w*A m tr*n ! Fi;u thZ# t( & #I k tr$ng #[ sG 8228+8+ M3+ 2M+8+83+ 28+ M+833 -#3 w*A m tr*n ! Fi;u thZ# t( & #I k tr$ng #[ sG 7228+ M+ M3+ 2 7+8+ M3+ 2y+ 7+833 - 8M3CJm hng ! x# &Enh tnh suy Fin hy khng suy Fin #I ## ,ng t$n Ah[ngN Fin th:# su-37 7 782 + + 3 7 N y _ x y % x y % +F372 + + 3 7 7 7 _ x y % xy x% y% + +#37 7 7N2 + + 3 7 p 7 v _ x y % x xy x% y y% % + + + + +)4i m(i N2 + + 3 + x y % #h$ tr4#. 883CJm ,ng #hu}n t_# #I ## ,ng t$n Ah[ng su-7 78 7 8 7 8 N 7 N7 7 78 7 N 8 7 8 N 7 N8 7 8 N 7 O N O3 O O3 O 7 N3! x x x x x x x x) x x x x x x x x x* x x x x x x x x+ + + + + ++ + +873CJm ## gi trE &; ## ,ng t$n Ah[ng th:# su &