05--2 - |‰uuz }{Xx ‹‹ ŒŠ ‰ † ‰ –z ˆs˜|...

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Unformatted text preview: |‰uuz }{Xx ‹‹ ŒŠ ‰ † ‰ –z ˆs˜| š ‰ † ™z ˆo˜† XˆŒ‚!© Ÿžžª‹ ŸžœŠ Xˆ‚`› ™¨ˆ¨8˜| x † ‡z ‰ † ‰ –z ˆs˜| œ° œ !Xž !© Ÿžžª‹ XˆŒ‚!© š ‰ † ™z ˆo˜† ŸžœŠ Xˆ‚`› … u u –z }!s˜| ™ x † ‡z ¨ˆ¨8˜| ¶ ‹¨Œœ ¶µŠ œ° œ !Xž !© … u u –z }!s˜| ™ x † ‡z ¨ˆ¨8˜| ¶ ¨Œœ ‹¶µŠ µ¼µ »‹° ³}C¨!}¶ š ‰ † ™z ˆo˜† ŸžœŠ Xˆ‚`› x†uz ˆ}{—– †¨8Xx † šz | | |z oXx †¨8Xx † šz | | |z oXx x†uz ˆ}{—– | | |z oXx § | ™ |z ¨oXx §¨oXx | ™ |z x†uz ˆ}{—– ‡ † …z ˆoXx • “‘ Ž   ”’b‚1 •Ž´ ² ¨`S³R§ •¹ ¯ ¸ ƒ ´ ²  ºŒº¹ „!S³R· •Ž´ ² ¨`S³R§ « « ¯ ‘ ® Ž «¬ « ~  Œˆ¦mˆ­m)‚€~ • “‘ Ž   ”’b‚1 «Œˆ¦mˆ­m)‚€~ « ¯ ‘ ® Ž «¬ « ~  • “‘‘¥ƒ ~  £  ¦”¦€`t¤¢z€~ •“¦‘”‘¦€`t¤¢z€~ ¥ƒ ~ £  • “‘ Ž   ”’b‚1 ƒƒ  „¢‚€~ x|uz ˆ}{Xx x|uz ˆ}{Xx u }{Xx |uz x|uz ˆ}{Xx u }{Xx |uz x|uz ˆ}{Xx x|uz ˆ}{Xx u }{Xx |uz xˆ}{Xx |uz x|uz ˆ}{Xx u }{Xx |uz | ƒ x ½ u ± u ± xu }¡ xu }¡ u¨¡ u uu ¨¡ …¡ u x yw x yw vx Xu vXu u v… ˆ}u vˆ‰ v ˆ† v ‡ vˆ™ v š v Œ– v x v Xu ” ‡ p r ˆ r’ n s s r p p l s’ n  “ ˆ l tE8’ )s•¢ vP"Eq8•n o•ƒmE¢8xu h g – j —h f d ˜ — k™bige™†– f ” “’  ‰ ˆ •8ƒ‘"t‡ … „ ‚ €y w u s r p †xƒvxvtEqi e ca Y W U bdb`XVT G¥¥Q I G  D%A S"ERPH¢FE!CB@ 75 3 98642 0 ( '% #      1)!!&$"! © § ¥£ ¡ ¨¦¤¢ x  • “¹ ¬ “ ¨º{  ”‘ Ž @ ƒ® Ž`´ “¦ ¥ “@ ¥ ƒ F ¥ “ • • “¹  Ž¬ ƒ ¥ ¥ Ž ƒ •¹¬ •¹  ¥ ¦T!UV­¹ ­{Bº„¦S¨!­ºV­¨Ž G„ƒ 9 Ž « Aº¦B{’G‘  5 “ @ ‘¹  Ž ‘  H @ “ 5 ¥ “@ • “¬ ƒ ±U¦Y­„!´ 9 ƒ }¦¥ “ ‘ •ƒ ¦P„!•“  50“ 9 ƒ!´¦e“”‘ ‘„ƒm“¦¥ƒ„Q­¬¨Ž•@“ }“¦¥1ŽE‘!­¨Ž P•“ 9 ¬ •¹ H “  • ƒ F • ƒ F «¬ ¥ Ž •¹¬ ƒ ¥ ”‘ P„!ƒ  !"­”¨ƒ ººR¦¨Ž F 9 Fƒ¥ ƒ ´ Vƒ  V”t!¦d‘  & ‚!—{oz }”€­¬  d!¤º¹ ´  ¯ •Ž¥ “¥ ¬ @ “ • ‘ A H AT y = b b AT AT @‘ “ !• 5  ¨“ 9 !¦c—Ž ƒ ¦F!{b”¨Ž ´ 9 !’ ¬ ƒ ´ ‘ 5Ž‘ ƒ ´ ƒ¥ ¹´ H `¹ ¥  5 ƒ ´  @ “ ‘  “ ƒ ´ a¦¦BŽ Y!¦X6’X”¥ !{ AT ‘ƒ ¹‘ F •Ž ´  ´ ¹  ƒ „ ¦‘  Ž 9 ³¸ „!F ¦†¨`’PI ¦¥ ƒ • “¹  W ƒ ƒ ´  v F ƒ F ƒ ƒ • ‘ F ¥ “  ƒ ¥ ­{BŽ  6R!&P‚V!6„!¤!U¨6¦“ 5 D v 9 D b ƒ 0¦P¦"º¦  ¨¦‘ “ ¤¨`´ 5 “ ‘ ¥ “@ • “¹ ¬ “ • ‘Ž AT y = b H H  ‘ Ž ƒ¬   v G¨­’BŽ — ƒ ¹ @ “ ¯ ¨{¥ • 5  “ 9 !{T‘  ­¹ 5 •Ž ¬ ƒ ´ • ‘ º¹ ƒ 9 Ž  º­¬  q!{EB“ ­¦!ƒ 5 !S!& D „¦ 9 ¨e¦„R!qƒ “ G¨­¬ BŽ ‘¬ • ƒ ´  @ • “¹ ‘ • ¹ F ƒ ´ ¥ “ ƒ ® “¥ƒQz • “ • • ‘ Žƒ  A 1 n    ‘ •¹  • „„6!­¨Ž P“ 9 H A @“ 1ƒ 9 Ž  ºº¬  • !& ‚¸ D ‘¬ ƒ´ v v ‘ 5  F • G“ EBŽ ¨Ž D m @ 6Pƒ 6{B{A‘ ”{A‘ 1ºx´ 9 !’ ¹  • 5 ƒ  Ž  ¥ ƒ I • “ ¥  Ž ‘¹ ¹ ´ n−1 A m<  ‘ „„6­¹ n−1 ‘ 5  A“ EBŽ @ ¯•Ž ƒ´ v “ !¨{¥ !& Ž D C ƒ¬ ¸¹ ‘ ‘ Bº!­¦¦¨“  ¨Ž º¦BŽ A¦­¹ ƒ  {"´  8¨Ž ‘ • “¹  5 ¥ “@ • ¥ 9 5 ‘ ´ ¹  ƒ  ƒ 5 “ ² {7’t¦6º¬  0S § c 42! 0(¥ &b# £ &Y %! !Œ¦da "¦ ’ ˆ¸ 31 )' c#Yc « ‘ º¹ ”“  ƒ‘    # $ UW#c! ´¹ A m n © § ¥ £ ¡ h ¨¦¤¢¦ f |  @ ¹ ­!¨“ —Ž 7Vº¦  ¨¦‘ t—`´ «¬ • F • @¹ • “¹ ¬ “ Ž ‘ Ž ƒ 9 „t •ƒ  H • “¹ ¹ F • ƒ ´ ‘ ƒ ‘ ¨º{7!“ 9 !{ „”“  5 ¹ ”„„Q ­¨¤“ }¦k{UBC†!{ ¨Ž ‘ ƒ “ ¥ ƒ ¬¬ Ž @  “ ¥ 5 “   “ ¸ ƒ ´ F • F• ¨Ž @ ‘ ¬Ž ƒ´ @ “ „ƒ  —® !{ “ ‘ ‘ ƒ¬ F ¥ Ž I ƒ F ƒ ¢ ‘¹ Ž ‘ ƒ • ‘  “ “  ”„­!U¨V¦¥ V`¡º{ {³¸ ¨Ž 9 ’}¦¥ ¨o  ¦ !¦ {¤º¹ “  „”¦“ 9 !¨º{ Ž  Vƒ ¨ƒ ºº!& “ ƒ ´ “  I • F • ‘ ƒ ¥ ¥ ‘ • “¹ W ¥ Ž •¹¬ ƒ ´ b3 − 2b2 + b1 = 0 Ax = b b2 .  cS ƒI ƒ ‘ ­¹ “   H [A b] A 1 2 3 `¹ ¥  7”{BŽ 5 6{P„ƒ 0I  Ž ƒ { º¹  6¦½}’`¸ D Fƒ• 5 ´ I • 9 Fƒ¥ “  v @ B“ ƒ 9 Ž  V• 5  “ 9 !¦T¦Tº”¨`¸ ’º¹ ‘ ¬ ƒ ´  ¥ “@ ‘¹ ‘ Ž Ž ‘ H 2 3 4 H 1 1 1  b3 − 2b2 + b1 = 0 b1 1 2 3 4 b1 0 −1 −2 −3 b2 − 2b1 0 0 0 0 b3 − 2b2 + b1 1 2 3 v Ž D 1 c ) a ¥ 2! ) 0 −1  Ax = “ • “¹ ¬ “ ‘ ƒ  ƒ 5 ƒ ´  F • { ¨º{  ”1{6­¬  0“ 9 !¦"º¹  v ‚F D ¤ ¥ 1  vX¦¥¨Žº¬  —Ž  9 ¦&­¹ I• ƒ¥ ‘ ƒ ”‘  Ž 9 C’­!{T­¹ D ¦¨„¦¥ ƒ ¸ ‘¹ ´  ‘ • “ ‘ Ž ƒ `¹ ¥  a¦¦BŽ 5 †ˆ¸ – !&­R”„!& v 9 D |« “ • ‘¹ ƒ ¥ ƒ ´ ´ 9 !’ ¦@ ¹´ ¥“ A F Y6® 6 †ŒF| D ““I Ž ƒ ¹ ¨ v| «¸ AB = I B  Ax = b ƒ ®¹ ¨ ¦ • “¹ ¬ “ ¨­1©§º¦  ”‘ Ž `´ ƒ®Ž ‘ƒ“ !F b 1 b = b2 b3 b1 , b2 , b3 ‘ º¹ @ “ ƒ 9 Ž  V• 5  “ 9 ƒ {T¦@ º”¨`¸ 9 £¹ 9 ƒ  i§ Ž D ‘ ¬ ´  ¥ “ ‘¹ ‘ Ž ¢ ‘ v A  • “ ‘ • “¹ ¹ F • ¨&!º¦ !“ 9 ‘¥“ ƒ ¦¨{ 9 ® ´ 9 !’F`¥‚¸ D ¹´  ¥“¤ v A= 2 3 4 5 3456 1234 C 7”{BŽ Y– ŒF’­!{T­¹ `¹ ¥  5 « ¸ | ‘¹ ´  ‘ A ƒ‘    ”“  g © § ¥ £ ¨¦¤¡ ¡ ‘ • “¹ ¬ “ ‘ ¬ Ž ¥ ƒ • ƒ I ƒ ´ ­¹ ­{  ¦³¨¦„!d!¦ ƒ 9 „t •ƒ  F• ¨Ž – x = xp + xn = xp + cs3 + ds4 . A £ @“ tƒ 9 Ž  º­¬  ¤!¦ S­¦¨`¸ Rº¹ U¦@ § c ) a ¥ 2! V§ ! ¦a ' # V§ ‘¬ • ƒ ´ @ “ ‘¹ ‘ Ž Ž I • 5 ¥ “ 1) W 1 −2 s3 = 1 0 F• ¨Ž H • “¹ ¬ “ ¥ Ž  ¹  ¥ º¦  ”‘ ¨º¬ V9 {A—Ž  t­¹ Ž‘ 1 2 3 4 1 2 −3 s4 = 0 1 −→ 0 −1 −2 −3 −2 −→ 0 1 0 0 0 0 0 00 2 3 0 0 ƒ¥ ¦¨Ž 2 0 0 −3 1 1234 2 0 1 0 −1 −2 −3 •ƒ´ !& • “¹ 9 F ƒ ¥  “ ´ 5 ¥ “@ ¥ ¨º{  6¦½}”¥ ƒ { A¦Aƒ  ƒ ¥`F D ¤v  I “• ‘Ž !1¨`´ ƒ 0¦&¨¦@ º¦  ”‘ 5 “ ‘ ¥ “ • “¹ ¬ “ ƒ¸ ³xF ¬  S¨º{  ”F!{ ­­¨Ž   9 § “  • “¹ ¬ “ ‘ ƒ ´ «¬¬  Ž`{ }¦FSt`¸ D G¨Ž  º6  £ ´ Ž‘ ƒ  v ¥ •¹ ¢ H H H •­¹{BŽ  6ƒ ¤¥„ƒ¨ƒ ¨Ž 1H}¦¥ “ W « ® F• @“ | “ @ x “ “ $}¦¥ @“ u “ ’½}¦¥ ƒ ®¬ “ ¨­”‘ H @ v ¥ R¡ `¸ D A—Ž  º†’}!¦&¨Ž ¯ ¨{sz X”•­¬  @ !&­¹ •¹ •  “ ´ ‘ ‘ •Ž¥ “¥ ¬ “• ‘ ƒ ”‘  Ž 9 ³¸ º&­!& v 9 D ƒ ‘¹ ‘¹ ´ Ax = b I Ab3 = Ax = b F ¬  “ 9 ¨†!{ ƒ  •ƒ´ I Ab2 =  xp = 2 3 4 5 0 3 4 5 6 −1 b x = Bb Ab1 = AB = I A H ƒ ´ « Fƒ • • !¦¤ˆ¸ 6!!¨Ž  ‘ !º­¬ ƒ {c‘  ¹ ƒ •¹ ´   ¥“ F• ¨¦@ ¨Ž H ¥ “ ¥ ƒ ¹ ‘ ¦@ D •!F ƒ 9 Ž  †• 5  “ 9 º!{   ¸ v ‘ ¬ ¥¹ ƒ ´ B ƒ {6!¦e¨}!´ 5Ž‘ ƒ ´ ƒ®Ž H š v (1, 2) ¥“ ƒ ¦ 9 ¨® H ¥“ ƒ ƒ´ « Fƒ•• ¨{ 9 ® !{ ˆ¸ 6!!¨Ž  ‘ !º­¬ !¦E‘  ¹ ƒ •¹ ƒ ´   ¥ “ ƒ ® ƒ ´  « ¸ F ƒ • • ‘ ƒ •¹ ƒ ´ ¨{ 9 6!{"ˆq6!!¨Ž  6!º­¬ ! D ƒ 9 Ž  º­¬  • ‘¬ (1, 1) (1, −1) B −1 = (P A)−1 = A−1 P −1 11 11 A= B= 11 22 F• ¨Ž Cƒ 5 Bº¬  Ž ` H  ƒ ¬ @ ¹ • ¨¦­‘¨dŽTº‘PP„ƒ A 5ƒ ‘ ƒ ´ V¦BŽ G6!& v 9 D v “ ‘ F • ƒ¬ ¸¹  ¥ ƒ ® •¹ ‘ ”†¨Ž ­!º¦Aˆº&­¹ `¹ ¥  a¦¦BŽ 5 ¨º{ Ž   A„ƒ  • “¹ 5¥ H H “ ¯•Ž ƒ´ ¦‘ !¨{¥ !¦ ƒ!¦dC P„ƒ 5  A¨Ž „!¦!t§ D —º!­{Aˆº¹ ‘  ¦dº¹ ¨Ž º„¨1¨x!—{¥ ­¬  E¨`´ ´  • I¥ ¥ƒ ´“ •  ƒ¬ ¸¹  ¥ ƒ ® • ´  ‘ F • ¬¬ ƒ  ‘ Ž ¯ • Ž ¬ @ ‘ Ž • “  9 •©ƒ !6¨!´ ’}¦¥ !¦ º{  U€~ ³¯ ¨{)­¬  c¨`´ ƒ  “ • ‘ Ž ‘  “ ƒ ´ I •¹ 5 ¥ ƒ  • Ž ¥ ¬ @ ‘ Ž Ž ´ •Ž B`¦c‘ ¨ƒ 5 º!­{G„ˆ­†º„³¸ ƒ¬ ¸¹  ¥ ƒ ® •¹ I •¹ ƒ v `¸ D P B A A 100 000 0 0 0 , 0 1 0 000 000 000 010 , 0 0 0 , 1 0 0 001 000 001 000 , 0 0 0 , 0 0 1 100 010 . ‘¹ ‘¹ ‘ Ž ¸ ¬ Ž  Ž •  ‘ 5 ƒ ´ v ­&º”¨`¢¨¦¥  B!’A“ c!& Ž D 1 c ) a ¥ 2! )  H ¥   ƒ ‘¬ Ž@  ¹  ´ “  ‘ ¹ ¥  5 x F • ¨¦­¨dT‘  ½X“ ¦‘ {Tƒ 9 ”{BŽ bq`¢ # 1 U Y¥ ) c § a § a ¥  ¨¦  ‘ ¨¦  « ´  • “‘Žƒ Ž ƒ ¹ ©ˆ’q”¨”¥ F® I H # 1 U ¥ § a § a ¥ ©§¥¤¨ƒ 9 Ž  • 5  “ 9 ƒ B{t!¦b`´ ¬ 5Ž‘ ƒ ´ ƒ®Ž H «ƒ´ •ƒ´ !¦ !{ ƒ 9 Ž  º­¬  1ƒ ¦‘ ƒ { ¨Ž ƒ  `¦1ƒ {6!¦e¨`&„ƒ 9 ¦¦BŽ 5 R £v 9 D ‘¬ • 5 Ž ´ F • Ž ´‘ 5Ž‘ ƒ ´ ƒ®Ž ´ ‘ ¹¥ x@¢ ¦ ƒ¬ ¸¹  ¥ ƒ ® • “ ‘¬ º!­{Aˆº¹ ”º¨Ž • “¹  º¦BŽ   Uƒ  5¥ ‘­¹ Ž´ “• B`¦ }!Œ¯  ¨†“ F }t¥ “« “  `¹ ¥ a¦{ Ž 5  !• R I { ƒŽƒ“ Ž ˆ¸ ’X”6{ ¹ {  Uƒ ¤ "¡•a¦¦BŽ Y­!º¦A„¨ˆº¹ ¦¨Ž  A‘ Ž º¹ « ‘  “ ¥ ‘ ƒ 5 ¥  “  `¹ ¥  5 ƒ¬ ¸¹  ¥ ƒ ® • ƒ ¥ W ‘ ƒ‘    v ”“  ‚¸ D P B B A ‘ ¹¥ „ƒ 9 ”{BŽ 5 9 ¦¦6ƒ 5 5 ¤‘ | ˆ¸ ³¨”`º¨¢¤ƒ 9 Ž  S{ 9 ’!¦ ¦Xº”¨`¸ Ž ­¹ ¹¥ « « | ¬ Ž ƒ ¥ ¬¬ Ž @ “ ‘ ¥ “ ƒ ® ƒ ´ ¥ “@ ‘¹ ‘ Ž F• ¤ © § ¥ £ ¡ h ¨¦¤¢¦ v ˆŽ D h ™ H ) c ) a § # 3 bY !{ ”{ 9 ¨FP!­¨Ž P“ 9 º”¨`&º¹ {Eƒ 9 ­¹  c a ƒ ´ ‘ ¥ “ ƒ ® ™ ‘ •¹  • ‘¹ ‘ Ž ¸ ‘ ´  • 0 S2 = 0 0 0 S5 = 0 1 0 0 0 0 1 ¨ 0 0 1 0 0 0 0 0 0 0 00 S © !§ a 1 S1 = 0 0 0 S4 = 1 0 0 0 1 0 0 00 0 S3 = 0 0 0 S6 = 0 0 1 0 0 0 0 0 1 0 1 0 00 C ƒ 5 Ž ƒ ‘¹ ´  ¥ “ ­¬  B!` ’º!¦T¦@ v 9 ¦¦6ƒ 5 5 ”•º¨Ž D º¦—`1º!­¦”“  !¨&º6¦t ¹¥ « ‘ ¬¬ ‘¹ ‘ Ž ¸ ƒ¬ ¸¹ ‘ ‘ ƒ • “ ‘¹ ƒ ¥ ƒ  S •¹ v `¹ ¥  ­¨!7”{BŽ 5 D { 9 ¨V”„¨6ƒ  ”¥  A‘  T!­{B`­!t0“ 9 º!&’x ¥ “ ƒ ® « ¥ ƒ ® ƒ 9 F “  5 ‘ • “¹  Ž •¹ ¸ 5 ¥¹ ƒ ´     • ƒ F • ƒ F • ¬ ¥ Ž ƒ •¹¬ ƒ ¸  P„!„ƒ  !­¹ « ¦¨„!º­1CPG‘  5 ¦¦ 9 ¨&º¦—`6!&’u ‘ ¥ “ ƒ ® ‘¹ ‘ Ž ¸ ƒ ´   C‘ • 5ƒ¥ Wƒ “  ƒ¥ Ž ƒ¥ƒ ´ •ƒ ´   ‘ ¹¥ 6{P„ƒ V”º¹  6¦¥ Sod”¨R”„!¦ !&ƒ 9 ”{BŽ 5 9 ¦¦6ƒ 5 5 ¤e| ˆ¸ q¦—Hƒ & © # ' W V§ ¥ 1 § # ¥ c 4# a ! ¥ § 1 3 £ a  )¥  ¹¥ «‘ « | ƒ¥Ž « ´ U ) ¨ § U ¨' # § a § W ¥ #   H C VQ ¦—`´  Vƒ ³t{T‘ V„ƒ • Pº“  « ¨R&B`{ VQ º—¦E Ž ` ƒ ¨Ž V­{G„ƒ  W ­!{ 6¨¦—¨¥!$¤ Fƒ ¹‘Ž 5 ƒ ¸ “  F ƒ  •¹ ƒ ¯ Ž  Ž ´ F ƒ ¹¬ Ž ƒ ¥ 5 • • “ • “¹  ‘ ‘¹ ´ F ƒ ¯ ‘ Ž • ƒ ´ ¥ © U ) ¨ § a § W ¥ W bda ¨3 # ! ¦) U ¤Ca¦{ Ž 5 9 ¦{ ƒ 5 5 ”F­º¦A‘ £  `¹ ¥ ¹¥ « ‘ Ž ¬¬¹  Yc ©§ H H ‘  ‘ƒ¥ ƒ ´ ‘¥ƒ ¸ ­¹ 7¬  ¦t!¦ ¦„³t5  €¨”ˆV« ¬  ¦ ¬  5 “ „ƒ 9 ”{BŽ 5 9 ¦{ ƒ 5 5 ”V¨o Ž  R  ƒ 9 Ž  !¸  ‘ • ¬Žƒ¥ « ¸ ¹ ¥ ‘ ¹¥ ¹¥ «‘ “  F F “« @ ‘ •¹ ‘ ¹ ¥  ºFƒ 9 ”{BŽ 5 9 ¦{ ƒ 5 5 ”‘ !&  ¹¥ « ƒ´ M Ž 5¥ “ kA¦@ S S S M ƒ 9 Ž  ‘ { 9 ¨xŽ ¥“ ƒ® 5¥“ ‘ ¹¥ U¦@ „ƒ 9 ¦¦BŽ 5 | ˆx’¨„¦¥ !& «¸ | ¬Žƒ ƒ´ vƒI Ž  „¨1!{"ƒX“ • D ¸ƒ  ƒ ´ •“  ¡U ¢!W 3# ...
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This note was uploaded on 06/21/2011 for the course CIVIL 1011 taught by Professor Juan during the Spring '11 term at HKU.

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