AdvancedLinearAlgebra2022-06-21

p “ê: l\€ (þþ) ] huy@sustech.edu.cn 2022 c 6 21 F °Ï 8¹ có vii 'uPÒÚâŠ 1"Ù 0.1 0.2 0.3 ½ xiii ‚5“ê•=„ ÆS‚“•Ÿo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ‚5“êÆŸo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ‚5“êNoÆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 4 10 1˜Ù Ý Ú‚5•§| 1.1 )‚5•§| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 •§| Ð C† . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 XêÝ ÚÝ ž { . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 ^1 F/(½)85 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 SK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Ý Žâ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Ý Ä $Ž . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Œ_Ý ÚÐ Ý . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Ý d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 ©¬Ý . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 SK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 13 13 18 26 31 34 34 42 51 53 61 1 Ù •þ˜m9Ùf˜m 71 2.1 1½ •þ˜m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.1.1 •þ˜m K n 9Ùf˜m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.1.2 ‚5|Ü†‚5ƒ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 2.1.3 •þ| •Úf˜m‘ê . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2.1.4 SK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 2.2 ˜ A^ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 2.2.1 Ý • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 2.2.2 2Ø‚5•§| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 2.2.3 • 8†‹IAÛ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 2.2.4 SK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 2.3 Ä–•þ˜m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 2.3.1 Ä–B, Ä– ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 2.3.2 Ä–? ~fõõÃõ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 iii 8¹ iv 2.3.3 * û˜m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 2.3.4 †Ú††È . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 2.3.5 SK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 1nÙ ‚5N 3.1 3.2 3.3 ‚5N Ä 4.3 4.4 123 Vg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 “ê½Â†AÛ†* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 3.1.2 •õ~f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 3.1.3 Ø†”!‚5˜m 3.1.4 * ‚5N 3.1.5 * 3.1.6 SK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Ä Ü ‚5N Ó . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 ½n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Ý L« . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 3.2.1 ‹IÚ‹IC† . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 3.2.2 ‚5N 3.2.3 •–"zÝ½n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 3.2.4 SK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 ÄC††Ý Ý . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 ƒq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 3.3.1 ‚5N 3.3.2 A 3.3.3 ØCf˜m!Oé 3.3.4 SK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 $ Ý Š!A ÄC†úª . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 •þÚÝ ª9ÙA^ 1 é z . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 zÚþn z . . . . . . . . . . . . . . . . . . . . . . . . . 163 173 ª . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 4.1.1 4.2 L« 3.1.1 1oÙ 1 4.1 9ÙÝ 1 ª†‹IAÛ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 1 ª†˜mAÛ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 4.1.2 n 4.1.3 ˜mAÛ¥ 4.1.4 SK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 p 1 ål¯K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 ª . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 4.2.1 1 4.2.2 N‘Ý 4.2.3 OŽÞ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 4.2.4 1 4.2.5 SK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 1 ª ª¼ê9Ù5Ÿ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 ª † Laplace Ðm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 •35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 nØA^ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 4.3.1 Cramer {K†)‚5•§| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 4.3.2 Ý 4.3.3 A 4.3.4 Cayley–Hamilton ½n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 4.3.5 SK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 •†1 ª . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 õ‘ª!,†A * ü !˜††1 Š . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 ª . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 8¹ v N¹ A ˜ •£Ö¿ 221 A.1 êX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 A.2 8Ü†N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 P Q A.3 ë\ÎÒ Úë¦ÎÒ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 A.4 õ‘ª9ÙÏª©) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 A.5 SK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 N¹ B ÿ †•ÁÁKÀ6 249 B.1 ÿ I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 B.2 ÿ II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 B.3 ÿ III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 B.4 ÿ IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 B.5 2019 c¢GÆÏÏ¥•ÁÁK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 B.6 2019 c¢GÆÏÏ"•ÁÁK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 B.7 2021 c¢GÆÏÏ¥•ÁÁK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 B.8 2021 c¢GÆÏÏ"•ÁÁK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 ë•©z 263 vi 8¹ có ‚5“ê´ŒÆnó‰•-‡ êÆÄ:‘ƒ˜. 3·I, éêÆÄ:‡¦' p ;’(XÄ :êÆ!nØÔn!OŽÅ‰Æ ), m ‚5“ê‘§‡ùÇ˜ nØ5ÚÄ–5•r S N. Ïdù ;’ÆS ‚5“ê‘§kžÿ¬±, ˜‡‘§¶Ñy, =p “ê. p “ê• Ð, ‚5“ê•Ð, ÙÌNSN å•ØŽéŒ, Ù‘§-‡5•ÑØó ’. Ø=3·IXd, 3 I ŒÆ•Ä þÑ´Xd. ØL, k‡y–Š 5¿. 3·I, ‚5“ê (½p “ê) ‘§ 8¥3ŒÆ$c?m , Ê Ï‚5“ê~~•m ˜‡ÆÏ, p “ê½•p?O ‚5“êX ‘§ÆžK˜„ü‡Æ Ï. 3ù ƒé' ;Ü žmS, ·Ip ‚5“ê½p “ê (AO´ ö) ‘§ NXÚ I p Ó ã‘§ƒ', %~~•¹•2! • SN. d , éuIS á' ý- ˜ SN (~X1 ª ¢ÔŽE|, ±9p “ê¥¦^õ‘ª‚þk•)¤ nØ)º Jordan IO/), I Nõ áK¿Ø:u3ÐÆ ã\±rN. ~X, Sheldon Axler ¤ á [3] A ÏŸæ £;1 ª Ý, %E,U^|ïå‚5“ê ÌZSNNX. TÖ•vkLõÐ mõ‘ªnØ, Ïd•!Øþ¦^õ‘ª‚þ k•)¤ (IS ákžrùÜ©nØo(• λ-Ý nØ). Axler $–u 1995 c3{IêÆ r (American Mathematical Monthly) uLL˜ ŸK•5 Down with determinants !6 ©Ù, T©„J¼{IêÆ ¬ (Mathematical Association of America) •)`. Š…u Lester R. Ford ø (•¡ Paul R. Halmos–Lester R. Ford ø). (, I ŒÆ ‚5“ê‘§Ú á†ISƒ'k˜½ •?. …, 8ž“ p Ã T¡•-., 5¿ÓIS ;. ¤±·‚3 Æ¥•AT/•I ² , ÆSI ˜ k?n g. ´, duI p ˜ ªÚN›†·I œ¹ØÓ, ŒÆ)\Æž •£NXÚ( •Ø Ó, Ïd·I p “ê‘§ ÆAT†I k¤«O, ù•ń¤ , . QŽ ±ISzÀ•! ¢yÓIS ; ¿ , q‡ˆ¤ISIO Æ8IÚ ‡¦, ¦äN ÆSNÚ•{·A IÆ) ¢Sœ¹, ùÒ‡• 3 Æ˜‚¢‚ “‚XÛ3 öƒm‰Ün ²ï . äN ‚5“ê‘§, XÛÀJ`kùÇ SN, XÛòÆ)n)' (J VgÚnØ‰Ð¿© Á= ÚòÐ, ~ Æ)ÆS JÝ, XÛdf\ /ÅÚÔöÆ)ˆ v p “êÆÄ õ, ÃXda Ù¢kéõ¯KŠ üV•þ. ¢Sþ, Nõ‚5“ê á cóÜ©Ñkéù ¯K \&?. , , ÖcóØ‹Ž2éù ¯KÐm•Ÿ Øã. ùÙ¥ Ì‡ Ï´, )ö <¿Ø´l ¯p “ê Æ 1[pÃ, ¿vkpÂï_ ú1ÚUå, Ïdl‡<Y²þØ„ UÚ•Ïl ¯p “ê Æ Ó1‚ \&? Çù€‘ ² . )ö•ŽL²gC®²5¿ ù ¯K, 3Ö¥äN Ù!SN¥¬ž•5¿NygCéù ¯K g•, NyÑgCéù äN¯K ? n•ª. Óž, Ö? L§¥•2•ë• yk ¥©Ú © á, Ù¥Ì‡•) Ø=•u [2], [3], [6], [13], [14], [15], [18], [22], [23], [24], [25], [28], [30], [33], [35]. ƒ&[Ö Ö Ó1;[U é Ö?1` ” ¬µ. duY²k•, Ö¥ ¦!Øþƒ?3¤J•, ¹žÖö1µ• . 3 e5 ãáp·‚Ž•ÐÆ‚5“ê Öö‚ ÀÜ. ·‚ú ùÜ©Öö3ÆSL§ ¥éŒU¬‘ ˆ«¯K!(JD–%nþ (¾. Ïd, ·‚Ž ˜ ‚5“ê Ì‡SNÚA :, ¦ÐÆöõ 3%nþk‡O Ú·A L§, •8 ÆS~”˜ Kú. vii viii có ‚5“ê ,áu“êÆ ˜Ü©. ù˜Ü©¢SþQ´˜‡LÞ ã‡ÆS SN, q´± • \SN -‡Ä:. XJ‡XÚ/Ýºy““êÆ, ÖöI‡lÙG ¥Æ“êÆ (Ä á uÐ “êÆ), ÅÚ \ •\î‚z!XÚzÚÄ–z p “êÆSN. lÐ p 1˜‡LÞ ã, Ò´‚5“ê ÆSSN. Óž, ‚5“ê SN¢Sþ•ä “êÆ Ä A , …•´• \ÆS“êÆ 7 Ä:. ùÒńØ ¡þ‚5“ê -‡¿Â. 3 ªm©ÆS‚5“êƒc, ·‚ú Š 2&?˜e“êÆ Ÿ´Ÿo, Ï•éõÖö ŒUf`““ê”ù‡c®²éÈ , ™7 \ )L“êù€Æ‰ 59 ó!cÏ JXÛ. Ä k, ¥ © ““ ê”˜ c € È g = © algebra ù ‡ ü c. = © ¥ algebra ˜ c 5 g u C . Ë Š al-jabr , Ù ¿ •“» ” ¡ ä - à”. ü c al-jabr K g u ˜ Ö Ö ¶: Ilm al-jabr wa’l-mukābala . ù Ö Šö´˜ Åd (˜`C.Ë) êÆ[!U©Æ[. ¦ ¶i C –s. f’ (al–Khwārizmi), )¹c“Œ •ú 780 c– 850 c (¥I ¥ /žÏ). ¦ Ñ)/s.f (Khwārizm) ½N¬Ï•7Tk) ¶Í5 H=<D6 k¶í. s.f’ Ö¥?Ø SN38Uw5Eáu¥ÆêÆ. ÙÖ¶ Ilm al-jabr wa’l-mukābala ¿ gŒN•“„ †éž Eâ”. üc al-jabr zÑ“„ ” ¿g, §´•ò•§¥ K‘£ Ò ,˜>C¤ ‘; mukābala ˜cJø“éž”½“z{” ¹Â, ´••§üàŒ±ž ƒÓ \{‘ ½ö¦{Ïf. 8Uw5, ùü«öŠDt²~, Ï•öŠ é–´“ê”½ö“^5L«™•ê Î Ò”. <‚m©ïÄù«öŠ´ÄŒ±é“ê”± êÆé–5–1, ¢SþÒI“X““êÆ”m ©Åìr•¤ÙÚpà ¸. . C . Ë © al-jabr { ² õ « © i (X Ü • ß ©! . ¶ © ) = È ƒ , • ª 3 = © X Ú È • algebra . â•y, ù‡=©üc•Ð•´'u?n›ú½øÏ ‰Ãâ ˜‡šÆc®, Ù êÆ¹Â 16 -VâÄg ¦^. 3·I, =©üc algebra •Ð´d²"˜Ð 5uD ¬ÑÈ •“C 9n.”. ù« % È{w,J±-<÷¿. 1859 c, ˜êÆ[!€È[oõ=3†= IÇÆ[ Alexander Wylie ÜŠ€È Auguste de Morgan 5Elements of Algebra6˜ÖžME5/ ¦^ ““ê”ù‡¿ÈcŠ. ˜ c , u„•†=I€È[ John Fryer (¦k‡¥©¶ F=ä) ÜÈ=I< W. Wallance 5Algebra6ž`: ““êƒ{, ÃØÛê Œ±?ÛPÒ“ƒ”. Œ±`, ““ê”˜c˜‡Ä ¹Â´fw ŸÈ : “êö, “Oê•. 3 “ ê Æ Ð ? ã, · ‚ Ï ~ ‘ “ O ê À Ü, Ù ¢ • ´ ˜ Î Ò ½ i 1, X a, b, c ½ x, y, z. ·‚^§‚5“Oê?1Úê˜ ~„$Ž, AO´\~¦ØoK•Ä $Ž. û“ Oê i13?1ˆ«$Žƒ , Ï~I‡÷v·‚Ï– ˜ ª. ù ·‚¢ŸþÒr¢S¯ K3êÆþÄ–¤ )•§ ¯K. ¤±, “êÆ• © ïÄSN, Ò´¦)¹k™•ê •§. ISQ²k˜ãžm (•N–8E3±Y), ¥ÆêÆ Ì‡SN ©¤““ê”Ú“AÛ”üŒ¬. Ù ¥“ê˜¬Ù¢Ò´Ì‡ïÄ¹k™•ê •§ (½ö•õ žÿ±¼ê /ª). ØL, ŒÆ ã, ·‚Ï~r“êÆ¥ïÄ •§2‰˜‡•›, ‡¦3•§¥~êÚ™•ê¤²{ $ŽÑ √ 3 g˜ •õ•k\!~!¦!Øùo«$Ž. XJ¦^ Ù¦$Ž, ~X¦Ãnêg˜ (X x √ x 3 )! éê!¦n ¼ê , K ŸþI‡^ ¦4• L§. ù 3ŒÆ ãÏ~ y8 ©Ûa (~XêÆ©Û!¢©Û!E©Û ) êÆ‘§NX¥. ¤±, Ööl?\ŒÆå, Œ±r“ êÆ¥ïÄ •§, V)••k\~¦Øùo«$Žë†/¤ •§. (5¿: x2 êg˜Œ± @•´¦{ ‡ES“, x−1 ¢Sþ´‰Ø{, Ïdù •áu“ê•§#N $Ž.) ¶gÂ, ‚5“êïÄ Ì‡´äk“‚5”A êÆé–. Ò•§ ó, ·‚Ì‡ïÄ‚5 •§, ù´•™•ê3•§¥Ñy /ªÑ•k˜g˜ /ª. ÃX x3 + 4x2 − 3x + 1 = 0 ù«¹k ™•ê•pg˜ “ê•§ò¬´8 •p g “êÆ‘§ïÄ ‘K. ¤¢‚5•§, Ù• {ü /ªÒ´ ax = b, Ù¥ x ´˜‡™•ê, a Ú b Ñ´‰½ ~ê. ¦) ax = b ù«•{ü/ ª •§, ™ùéA ¤kÖö5`, Ñw,k:ÏYœû. ¤±·‚˜½‡Þ‡•kSº ~f. ix ~X (1)  3x + y = 5 x − 2y = 4 y3·‚kü‡™•ê, …kü‡•§éáå5 ¤˜‡•§|. ,, )ù‡•§|E,Ø´Ÿ oJ¯, ¤±·‚ØI‡wŠÖöTXÛ¦). 2E,˜ , ·‚•Ä˜‡kn‡•§Ún‡™•ê ‚5•§|    3x + 2y + z = 39 (2) 2x + 3y + z = 34    x + 2y + 3z = 26 ù˜g, ·‚ŽžÖö<˜|)˜Ü’üŽ˜e, wwsI‡sõ•žmâU)Ñù‡•§|. b X·‚Ž–, s¦)•§| (2) Ñž'¦) (1) ‡²w••, @os´Äú gCfâ¦^ •{ Ø´•`, ´Ä•Ä I‡˜«XÚ !(&Œ1 ¿…BuöŠ •{5• ·‚¦)˜„ ‚ 5•§|Q ? fâ ~fŒUk ÖöE,ú ØLŠ, ¤±·‚Zy† Ñ‚5•§| •˜„/ª, =   a11 x1 + · · · + a1n xn = b1     a x + · · · + a x = b 21 1 2n n 2 (3)   ··· ··· ··· ···     am1 x1 + · · · + amn xn = bm ùp·‚ Ñ m ‡‚5•§, ^‘keI i1 x1 , . . . , xn L« n ‡™•ê (#N m 6= n), ^5L «~ê i1 aij Ú bi •±‘eI /ª¥y, Ù¥ aij ù‡PÒ$–‘kü‡•I, 1˜‡•I i Š‰Œ´l 1 mù ê, 1 ‡•I j Š‰Œ´ 1 nù ê. 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T©C½ û 5G Ï&?è 4zèEâ•YÄ:. 3 2010 cuL ,˜Ÿ Ø©¥, Arikan Ç?˜Ú)º 4zè êÆ n. ã 0.3 Ò´T© Ä•Ü© ã. ƒ&lùÜ Ä• ã?ÛÖöÑŒ±²waú Ù¥%¹X‚5“ê gŽÚSN. ã 0.3: Arikan Ç'u4zè ˜ŸØ©Ä• ã –d, ƒ&Öö®²Œ±aú , Ø Ä:nØ‰Æ (XêÆ!ÔnÆ) ƒ , 3Ù¦NNõõÆ ‰ ¥, $^‚5“ê •£5©ÛÚ)û¯KÑ´š~~„ . ù Æ‰CXg,‰Æ!ó§ Eâ‰Æ!²LÆD– ¬‰Æ Ãõ€a, Ù ‰Œƒ2•, ^•Û –5/N•Ø•L. Š• (, Ò4·‚±e¡˜Ä ˜5ðˆÖö?\‚5“ê Û õç -.j. ‚“Ð••=„, e¡+°þ¡U. AÛ“ê ¢^, – d˜‚V. 0.2 ‚5“êÆŸo !L ÆS‚5“ê ÄÏÚnd, ·‚„Ž é‚5“ê31˜‡ ãò‡ÆS •£SN •ÖöÎn˜‡{ü óä. {ü5`, ‚5“êïÄ ´äk“‚5”A •§!¼ê±9ddû)ÚuÐ ½Ä–½ä N ˆ«nØÚ¯K. ØJN¬ , äk“‚5”A êÆ¯KáuƒéN´ êÆ¯K. ùÙ¥q ±¦)‚5•§|••Ä ¯K. Ï•••Ä , ¤±Ù¢•´•N´‘ ã¯K. (0.2.1) ¤¢‚5•§ (linear equation), Ù¢Ò´•3•§¥¤k ™•ê (•¡C !Cþ) Ñ• ±˜g•˜ /ªÑy. ~X, •k˜‡™•ê (± x L«) ‚5•§˜„/ª• Ù¥ a, b •~ê. (0.2.1.1) ax = b äkü‡™•ê (± x Ú y L«) ‚5•§K/X (0.2.1.2) a1 x + a2 y = b Ù¥ a1 , a2 , b þ•~ê. 0.2 ‚5“êÆŸo 5 ‚5•§Œ±@•Ò´˜g•§. ·‚r“˜g”ù‡/Nc†¤“‚5”, Œ±4<•N´éŽ ùa““ê”¯KÚ“AÛ” éX, Ï•“‚5”ù‡ck²w AÛ†*. (5-n)“êÚAÛ éX, Ù¢´ÆÐ‚5“ê ‡üƒ˜.) 3²¡‹IAÛ¥, XJ‡£ã˜^†‚þ?¿: ‹I, I‡^ •§Ò´/X (0.2.1.2) ª ‚5•§. Ï~œ¹e( a1 , a2 ü‡~êØ • 0), (0.2.1.2) ª¥ •§û½ ´˜^†‚, ´ 1 ‘ ‚5AÛé–. (0.2.1.1) ª£ã Ù¢´•{ü AÛé–: a 6= 0 ž, (0.2.1.1) ª(½Ñ ê¶þ ˜‡:, Œ±@•´ 0 ‘ AÛé–. XJń‡™•ê, P• x, y, z, @oa' (0.2.1.1) Ú (0.2.1.2) •§˜„/ª• (0.2.1.3) a1 x + a2 y + a3 z = b •‡ a1 , a2 , a3 ùn‡~êØ ¯¢þ, éu?¿ (0.2.1.4) Ù¥ a1 , a2 , a3 Ú b þ•~ê. • 0, (0.2.1.3) ª¤û½ AÛé–ń‘˜m¥ ê n ≥ 1, ·‚Œ±•Ä‘k n ‡™•ê a1 x1 + a2 x2 + · · · + an xn = b ˜‡²¡. ‚5•§. Ù˜„/ª• Ù¥ a1 , . . . , an Ú b þ•~ê. (ùp·‚^‘keI i1 x1 , . . . , xn 5L«™•ê. ù PÒw,•••B˜ .) AÛþ`, a1 , . . . , an Ø • 0 ž, (0.2.1.4) ª£ã AÛé–´ n ‘˜m¥ ˜‡‡²¡. ( ,3, Ž–‡ L 3 ‘ ˜mI‡Öök˜½ MÉ. Ù¢, Ž–Ø •v'X3. AÛŽ–C (Jž, Öö• I‡²x“êþ ¹ÂÒÐ .) (0.2.2) XJÓž•Äõ‡•§, ¯KÒC¤˜‡•§| ¯K. XJ•§|¥z‡•§Ñ´‚5 , ·‚Ò`T•§|´‚5•§| (system of linear equations). žÏ ¥I “ Žâ¶K“/êÓ<¯K”Ù¢Ò´‡)‚5•§| êÆÍŠ5šfŽ²6¥, K8SN : ¯K. TKP1u·IH 8kW!êÓ<, þkn›ÊÞ, ekÊ›ov. ¯: W!êÂÛ? ‰ =†¤êÆŠó, ù‡¯KÒC¤ (0.2.2.1) : W ›n, ê˜› Š . Xe‚5•§|  x + y = 35 2x + 4y = 94 3ÔnÆ¥, ·‚• Ô‚´ÔN$Ä •~„;,ƒ˜. b ·‚ÏL¢ ÿ½Ñ, Ô‚;,þ n‡:, ~X, ‹I• (1, 1), (2, 2), (3, 0) n‡:, @o‡(½T Ô‚ ;,•§ y = ax2 + bx + c, •I‡ò¤• n‡:‹I“\•§, 'uXê a, b, c n‡‚5•§ ¤ ‚5•§| (0.2.2.2)    a + b + c = 1 4a + 2b + c = 2    9a + 3b + c = 0 •‡)ÑT•§|¥™•ê a, b, c Š (ù‡óŠ·‚3‰k, ¤•Ä Ô‚•§ y = ax2 + bx + c äNLˆªXÛ. ‡ 3zÆÆ‰¥, ²zÆ•§ªŒ±`´˜‘Ä ² zÆ•§ª• õ. ±‡Ô ÖögC }Á), ÒŒ±(½ 1ÜŠ^ù˜zÆ‡A•~, I x1 CO2 + x2 H2 O = x3 O2 + x4 C6 H12 O6 1"Ù ‚5“ê•=„ 6 Ù¥ CO2 , H2 O, O2 , C6 H12 O6 ©O• z%!Y! íÚÄ:0 ê. w,, ‡¦Ñ x1 , . . . , x4 äNêŠ, Ò´‡¦)‚5•§|    x1 − 6x4 = 0 (0.2.2.3) zÆª, x1 , . . . , x 4 • 2x1 + x2 − 2x3 − 6x4 = 0    2x2 − 12x4 = 0 ±þ==´·‚‘ ‚5•§| A‡~f. XJs´9O}ï¯K Öö, ƒ&s3Ù¦é õ|Ü (Ø=•uêÆÆ‰) ¥ÑŒ±ª„/‘ 'u‚5•§| ¯K. (ÖöŒ±£w §0.1 !.) ^êÆŠóV)‚5•§| ˜„/ª, Œ± ¤   a11 x1 + · · · + a1n xn = b1     a x + · · · + a x = b 21 1 2n n 2 (0.2.2.4)   ··· ··· ··· ···     am1 x1 + · · · + amn xn = bm ùp, ± i1 a L« ~ê| aij kü‡êi‰eI. c˜‡êi i Š‰Œ´ 1 5I£¤éA •§3=˜1; ˜‡êi j Š‰Œ´ 1 n, §^5I£¤éA Ò´1A‡. m, §^ ™•êS (0.2.3) l (0.2.1.1) ¥•{ü ˜ ‚5•§, (0.2.1.4) ¥¹õ‡C ‚5•§, 2 ¹keZ ‡‚5•§ •§| (0.2.2.4), ¯Kwþ ĆE, . @o, ¦)•§ (0.2.1.1) •{, XÛâU í2¤v±?n˜„‚5•§| •{Q ? ù‡¯KÙ¢Ò´·‚ÆS‚5“ê•k‡)û ¯K. ¯K äN‰YÖö¬3 © Ù! ¥g, ,/é . ·‚3ùp==`²˜:: ˜„/ª ‚5•§| (0.2.2.4) – 3/ªþŒ± C¤Ú•{ü •§ (0.2.1.1) ¬Ü /ª. k`˜e·‚•ŸoŽr (0.2.2.4) C¤Ú (0.2.1.1) ƒÓ /ª. * ˜e (0.2.1.1) ª¥ •§ª. ù‡/ªƒ¤±{ü, ´Ï•Ù¥™•ê x Ú§ Xê~ê a´ ©l , …§‚ƒm•²{˜g$Ž (= a ¦± x). ¤±, 'uù‡•§Ûžk)±9X Û¦), ‰Y´š~˜ß : 1. XJ~ê a k¦{_, =, Œ±^Ù§êØ± a ( ,, ù du` a 6= 0), KÃØm>~ê b ŠXÛ, •§ (0.2.1.1) ˜½k), …•k•˜ ), = x = a−1 b. (ùp a−1 = 1/a Ò´ a 'u¦{$Ž _ (inverse), =, ^ 1 Ø± a ¤ ê.) 2. XJ~ê a vk¦{_ (= a = 0), K•§ (0.2.1.1) ´Äk)„•6u b Š: e b = 0, K ax = b ÒC¤ 0x = 0. ùž •§ ,k), …(3·‚Ï~n) ê ‰ŒS)kÃ¡õ‡ ); e b 6= 0, K 0x = b w,Ã). y3•Ä (0.2.1.1) 1˜ í2/ª, = (0.2.1.2) ª!(0.2.1.3) ª, ½ö•˜„/, (0.2.1.4) ª. ùž•§¥kõ‡Cþ, z‡Cþ xi (Ù¥ i ∈ [[1, n]] = {1, 2, . . . , n}) Ñkˆg ~êXê ai . 3 (0.2.1.4) ª†>Ñy ¦{Ú\{ü«$Ž, §‚ò NCþ x1 , . . . , xn Ú NXê a1 , . . . , an ·Ü å5. @o, ´Ä•3˜«•ªòCþ NÚXê N©lm, , •?1˜g$ŽÒ (0.2.1.4) ª †>Q ? ‰YÙ¢AT´ÖöÙG . ·‚I‡Óž?n˜|êiž, ~~Œ±r§‚ü ¤˜1½ ö˜ f, , r§‚ ¤˜‡•þ (vector) é–. (ÖöŒ±6žU‹IAÛ½öÔnÆ * :5n)“ •þ”ù‡c.) XJr n ‡Cþü¤˜‡äk n ‡‹I©þ •þ X = (x1 , . . . , xn ), r 0.2 ‚5“êÆŸo 7 n ‡Xê•ü¤˜‡k n ‡‹I©þ •þ A = (a1 , . . . , an ), @o (0.2.1.4) ª†>Ñy Lˆª Œ±n)• A Ú X ü‡•þ IþÈ½êþÈ (scalar product), •¡SÈ (inner product) ½:È (dot product). ^êÆúªL«=• A · X = (a1 , . . . , an ) · (x1 , . . . , xn ) = a1 x1 + · · · + an xn . (0.2.3.1) þ¡ù‡Lˆª¥•þ A Ú X / ´é , †AÚX ˜ØK•Lˆª ¹ÂÚ•ª Š. A Š•~ê|¤ •þ, X Š•Cþ|¤ •þ, ·‚„´¬ú r§‚ ‡«Oé–• Ð˜ . ¤±·‚~~r A ¤1 /ª, ò X ¤ /ª∗ . (u´, Œ±¡ A ´‡1•þ (row vector), X ´‡ •þ (column vector).) ¤±·‚8 ¬~~P  (0.2.3.2) ,  x1    x2   X=  ..   .  xn A = (a1 , . . . , an ) , ·‚^ (0.2.3.1) ª5½ úª5½Â A Ú X ˜‡¦È:   x1    x2   AX = (a1 , . . . , an )   ..  := a1 x1 + · · · + an xn .  .  xn (0.2.3.3) ùp, 3 A Ú X ƒm ¦È$ŽŽÎ·‚•ZyŽÑ ! Ú\ (0.2.3.3) ª½Â ¦{ƒ , ·‚uy (0.2.1.4) ªÒC¤ f: (0.2.3.4) Ú (0.2.1.1) ª /q AX = b . ¤ù ƒ kvk]mú 5öšP• ªf˜ef {z Nõ ? ù´Ø´é Û ? ,, ˜½•¬kÖö`: “Ÿo ? ùÒ´\¤¢ {z ? ùØÒ´Ü†Vgí ? 5 Lˆª ·Ü \{Ú¦{. \y3L¡þ´r§‚C¤ ˜g¦{, Ÿþ\ØÒ´r 5 õg$Ž |Üå5, -#å‡¶ií ? \ù ¶Âþ´•k˜g$Ž, ù«$Ž ½Â Ò%¹ 5 õg$Ž |Ü. ù ‘{z’ý ¬k¿Âí ?” Äk, 4·‚•JÑù«¦¯ Öö‚:‡7 ! Ï•ù«Ÿ¦Ú1 g‘ ° ´‰?ÛÆ¯ Ñš~I‡ ŒB° . ÆS?ÛÆ¯ÑØUü‚ Ä/ É¤¢ýnÝºö /Ñ, ÆSö7L ‡ÏLgCÕá g•, JÑgC ¯K, $–´1 ½öŸ¦, , 3EÛ¯K L§¥âU-ý n•E•², ÆSöé¯KÚ•£ n)•âUˆ • ¸.. éuÖöc¡ Ÿ¦, êÆ[‚¬N £‰? •NØÓ <¬kØÓ )º, k˜:´(½ : êÆ[‚Ñ´„`Œ“, ¦‚ýØ¬«@gC“Ü†Vg” (¯¢þ, 3é–êÆ ¯Kþ, êÆ[ ‚•ýØ¬ù ‰). ¢Sþ, êÆ[‚¬`¦‚´3MEVg! v†, Ò´“ME”ù‡c. ƒ¤±ê Æ[‚cuÎØ^J/¦^ùož- •Œ c®, ´Ï•¦‚ý Œ±y², ¦‚zgu²Ú¦ ^˜‡# Vg, $–•´˜‡# PÒ, ÑØ´%É5Œ ˜ž,å, ´k¿© rºÚ¯¢U y², ¦‚#u² Vg½ÎÒ¬‘5 • C€!ãŒ B|Ú2• A^. (0.2.3.3) ª¥½ Â ¦{, Ò´ù ME ˜‡~f. –u§ -‡¿Â, ·‚3ùpñ‡'f, 3‰k% Öö3 ∗ Ù¢, éu¦{$Ž Ï„žÅ™ . ó, ƒ¤±r A Ú X ©O Š1Ú 5«Oé–„k•• Ï. •ØL, d• )º@˜ 1"Ù ‚5“ê•=„ 8 8 ÆSL§¥ yù«*:. .¾, 3 cù‡ !, ·‚ Ì‡8 Ø´)‰¯K, ´JÑ ¯Ke-Öö ÐÛ%, -yÖö‘X•r ¦•–UY8 ÆS. ·‚ffr (0.2.1.4) z¤ (0.2.3.4) {z/ª. ù?Ø „•´˜‡‚5•§ œ¹. 4· ‚¦‘JÂ, êþ25?n˜„‚5•§| (0.2.2.4) œ¹. Ù¢ (0.2.2.4) † (0.2.1.4) «OÃš ´õ A1. ·‚Œ±k˜1˜1 ?n•§| (0.2.2.4) ¥ z‡•§. •Ò´`, XJA1 = (a11 , . . . , a1n ) , A2 = (a21 , . . . , a2n ) Xd , @o$^(0.2.3.4) ª Lˆ•ª, •§| (0.2.2.4) 1 i 1ÒŒ± ¤ Ai X = bi . ‡•§| (0.2.2.4) ÃšÒ´2rù 1|Üå5, /¤Xe/ª:   A1 X = b1    A X = b 2 (0.2.3.5) 2   ··· ···     Am X = bm ù‡•§| mw,„k{z {/, Ï•§ z˜1Ñ´˜‡~ê ¤ 1•þ, Ú˜‡ú C | ¤ •þ X ƒ¦. JÑú ¦{Ïf X, 2r Ai ù 1•þUì•§|¥ ü¤ m , ·‚Œ±r (0.2.3.5) -# ¤e¡ /ª     A1 b1      A2   b2   . X =  .   .   .   .   .  Am bm (0.2.3.6) e5, • 2{zPÒ, ·‚  A1    A2   A= .  ,  ..  Am (0.2.3.7) u´, (0.2.3.6) C¤ (0.2.3.8) † (0.2.3.4) ˜   b1    b2   b=  ..   .  bm /ª: AX = b . •ØL, y3 A Ø´•k˜1, ´ m 1; ªm> b •Ø2´˜‡ê, ´k m 1 ˜‡ •þ. XJr A z˜1 Ai Ñ Ð«m, @o¢Sþ A Ò´d aij ù êü ¤ ˜‡ m 1!n à /, =   a11 a12 · · · · · · a1n    a21 a22 · · · · · · a2n   (0.2.3.9) A= . ..  ..  . ···  .. ··· .  am1 am2 · · · · · · amn 0.2 ‚5“êÆŸo 9 ù« àü ˜|ê ‰˜‡ N, ¿… Dƒ k¿Â $Ž(~X (0.2.3.6) ª¥Ð« ¦{)ƒ , •§‚;€ME˜‡¶câŠÒw k¿Â . ~X, (0.2.3.9) ª¤Ð« A Ò ¡• ˜‡Œ • m ¦ n Ý (matrix of size m by n), ½ö˜‡äk m 1 n Ý , q½ö¡•˜ ‡ m ¦ n Ý (m by n matrix). •uÝ A Ú•§| (0.2.2.4) w,éX, ·‚~~r A ¡••§| (0.2.2.4) XêÝ . ù ‡XêÝ ¥1 êþ, Ò´•§|¥•¹ •§êþ, •Ò´ù‡êiü¤ .¥1 êþ, TÝ ¥ êþKéA•§|¥¹kCþ êþ. w,, ·‚Œ±r˜‡k n ‡‹I 1•þÀ•˜‡ Œ • 1 ¦ n Ý , r˜‡k m ‡‹ I •þÀ•˜‡Œ • m ¦ 1 Ý . A ´‡Œ • m ¦ n Ý , éu?Û˜‡Œ • n ¦ 1 Ý X, ·‚Œ±½Â A Ú X ¦È. ù‡½Â•{•: Äk, 5½T¦{$Ž (J, Š â A Ú X Œ , 7L´ m ¦ 1 Ý , ½ö`‘k m 1 ˜‡ •þ. –uù‡ •þz˜1 ? êŠXÛ(½, ÒUìÝ A éA 1† •þ X ‰IþÈ (X (0.2.3.3) ª) ê5½Â. •Ò´`, A Ú X ‰¦{$ŽÑ5 (J, AT´d (0.2.3.5) Ú (0.2.3.6) ª(½ •þ b. ù , (0.2.3.8) ªÒŒ±wŠ˜‡dÝ ¦{Lˆ !† 5êi œ/ (0.2.1.1) aq ˜‡•§. •˜I‡3¿ Ò´, éu˜‡k m ‡•§!n ‡Cþ ‚5•§|5`, (0.2.3.8) ª ¥ A L« ´˜‡ m ¦ n Ý , X L«˜‡ n ¦ 1 Ý , §‚ƒ¦ (J‡¦ u‰½ m ¦ 1 Ý b. kvkÖö2ga': ùÐ ÛB ! y3·‚Œ±o(Ñ‚5“ê 1˜Ü©Ì‡SN . ù˜Ü©Ò´'u‚5•§|ÚÝ $ Ž Ü•£NX. Ù¥, Ý 9Ù$Ž´ù‡NX -‡|¤Ü©, •´?nÚ)û¯KØŒ½" ©Ûóä. ·‚ý m©ÆS‚5“ê žÿ, 1˜‡¯KÒ´‡ïÄXÛ|^ (0.2.3.8) ¥‰ Ñ {z/ª, rïÄ‚5•§| (0.2.2.4) ¯K=†•ïÄƒAXêÝ A ¯K. Œ±`, ï ÄÝ ÄÏÚnØå:, ÒAT3ùp. (0.2.4) þ¡·‚l‚5•§| Ý, 0 Ý Vg ) ¿Â. ·‚„`² , éu˜‡‰ ½ m ¦ n Ý A, XJ-Cþ X ?¿ n ¦ 1 Ý (½ö` n 1 •þ), @ooŒ±ÏL Ý ¦{ ˜‡ m ¦ 1 Ý Y = AX. • (½å„, ·‚b ‰½ Ý A ¥z‡ ˜Ñ y ê aij Ñ´¢ê8 R ¥ ƒ. - Rn×1 L«‹I Ü•¢ê n 1 •þ ¤ 8Ü. aq /, - Rm×1 L«‹I Ü•¢ê m 1 •þ ¤ 8Ü. @o, ÏLéA{K X 7→ Y := AX · ‚Œ± ˜‡N f = fA : Rn×1 −→ Rm×1 ; (0.2.4.1) X 7−→ Y = AX . ( ·‚r f ¤ fA , ¿g´rNù‡N ½Â•6u¤‰ A.) l/ªþw, ù‡N Lˆ ª f (X) = AX ÒÚ·‚ÙG ˜ ‚5¼ê f (x) = ax ˜— (ùp~ê a Ú gCþ x Ø”Ñ @• g¢ê8 R). • Bun) (0.2.4.1) ¥N AÛ¿Â, ·‚ m = n = 2 R ¥ ƒ@•´²¡† ‹IX¥²¡þ¤k: ‹I ¤ •þ f. ù f Œ±@•´‹I²¡ gC ˜‡N . Ý 2×1 A= ŠâÝ Ú •þƒm¦{ a c AÏœ¹5`²¯K. ·‚r 8Ü, •ØLy3r‹I ¤ A y3´ 2 ¦ 2 Ý , Ø” ! b d ½Â, XJ P ´‹I²¡¥‹I• (x, y) :, f (P ) L«: P 3 N 1"Ù ‚5“ê•=„ 10 f Šê (0.2.4.2) ‹I (x0 , y 0 ) AT´de¡ Lˆª‰Ñ ! 0 a b x x ax + by x =A = = . y y cx + dy y0 c d ”, @o f (P ) ù‡: |^ù‡úª, ÖöŒ±ÏLOŽ yù ˜^(Ø: éu²¡¥?Û ‚ n‡: P1 , P2 , P3 , Ï LN f Š^ƒ : P10 = f (P1 ), P20 = f (P2 ) Ú P30 = f (P3 ), 7,E´ ‚ . (ùp, XJn ‡:k ´-Ü :, Kg,/@•§‚´ ‚ .) Þ‡äN ~f5`, XJ ! 1 2 A= −1 1 Pi , i = 1, 2, 3 ‹I©O• P1 (1, 0), P2 (0, 1) Ú P3 (−1, 2), §‚Ñ3•§• x + y = 1 †‚þ. K f (P1 ) = (1, −1) , f (P2 ) = (2, 1) , f (P3 ) = (3, 3) . ùn‡:Ñ3•§• 2x − y = 3 †‚þ. Šâ±þAÛA5, ·‚Ï~r (0.2.4.1) ¥ N f = fA ¡•˜‡‚5N . o(þ¡ ?Ø, ·‚w : z ‰½ ˜‡ m ¦ n Ý A, §ÒŒ±û½Ñ˜‡l Rn×1 Rm×1 ‚5N . ·‚38 ÆS¥¬w , ?Ûü‡k•‘˜mƒm ?¿‚5N , Ÿþ Ñ´ÏL,‡Ý 5Uì (0.2.4.1) ªw« •ª5‰Ñ . ¤±, l‚5•§| nØ¥ø Ñ5 Ý nØ, •Œ±@•´3ïÄ,ãäkAÏ5Ÿ N , =¤¢ ‚5N . ·‚• , ïÄˆ«a. ¼êÚN Ù¢´êÆÆ‰ Š á5ƒ ˜. ¤±, ‚5N nØg,¤•‚5“ê ,˜Ü©Ø%SN. d , fâ·‚ÞÑ ‚5N ´3 Rn×1 ù 8Üƒmïá N . Uì·‚8 °( ½Â, Rn×1 ù«8Ü´˜«ä ( AÏ8Ü, \þ@«AÏ( ƒ ·‚S.¡ƒ•˜‡ •þ˜m. ‡?Ø˜‡N ´Ää “‚5”A , ÄkI‡‡¦3§ ½Â•ÚŠ•ƒ¥, ƒƒm Œ±?Ø¤¢ “ ‚”'X. ‡?Ø ‚'X, ¢SþÒ´‡¦^•þ˜mù‡Vg¤Dƒ ê ÆSº. Ø ƒcJ Rn×1 ù«8Ü, •þ˜m ¢~3êÆþÙ¢´š~š~2•/•3 . ù • ´‚5“ê•Ÿo¬¤•A ¤kp êÆnØƒÄ: -‡nd. Ïd, \/ÆSÚn )•þ˜m (k ŒU´' Ä– ~f) ±9§‚ƒm ‚5N , ´·‚ÆS‚5“ê ˜‡ Ä 8I. ±þ·‚J ‚5•§|!Ý nØ!•þ˜m±9‚5N ù ‚5“ê Ä SN. ,, ù „Ø´‚5“ê Ü. ØL, • \ SN·‚Ž3–Ööý k v ‚5“êÆS N ƒ 2?˜Ú?Ø. 0.3 ‚5“êNoÆ Ù¢)ö²~aú, !ØÆS•{ù‡¯~~¬‰<“/ þÆ”½“_ƒq_” aú. Nõõ uÆS <™7õuògC ÆS•{‰°8 ÎnÚo(, ØõuÆS < qú O<o (Ñ5 ¤¢ÆS•{‚ gC þž, oú õõ ¬YèØÑ. ¤±, kžÿ·‚¬~¦, Ä ¾o(ÆS•{ù‡¯œ´Äý k¿Â ? ØL, ·Ž·‚qØ Ø«@, ÆS¤1' âÑ <, õõ 3ÆS ÝÚÆS•{•¡k ˜ ½²w½Ø²w 5. 'X, – ·‚‡«@, ÆS¤1Ð ÓÆ, Ä þÆS ÝÑ´È4 , 7‡ •£^õ•´˜½ . ÒêÆÆ‰5`, êÆ¤1Ð <, Õág• S.Ñ' r, 0.3 ‚5“êNoÆ ‘ ¯K•' ŒþSK . k}ï° 11 . äN ÆS•{þ`, êÆÆ Ð <, A ´sLŒþžm}ïL )öú , l ˜Æ 5Æ•Ä, – ö•É öXŽ, •¦‚o(˜ ÆS ² !•{½ö 5¿¯‘, •,Ø„ Uéz‡ÆSös , l NþÀTéÉ ö+N„´kÃ . ¤±, é ‘§ ó, ·‚„´¤ • !˜e“‚5“êNoÆ” ¯K. S Äk, •Ä ÆS ‘§ ÓÆÄ Ê·5ïÆ•NŠ ÄkJÑ. þAT´ff?\ŒÆ ãÆS. k˜^'uŒÆ‘§Æ Ø êUâª°= , éýŒõêÓÆ5`, @ýé–k “Ç‘ ‘,ÆS´–'-‡ . ŒÆ ‘§ ´m˜ª!&Ä. ‘§, §Ä Ø¬É I[Ú˜ Œj•›, Ù8 •´-Æ Sö¦¯ä ™5ÕáïÄ!gÌM# Uå. Ïd, ŒÆ ‘§ •£—ÝéŒ, NõSN Æ SJÝ• p. , , ùoÈ— ‘§SN3‰Æ)žzn) žm¿Ø•. 3ù«œ¹e, XJÆ )vk3k² “‘+ekk˜‡ÌZ!-:SN •[ù)ÚÎn, ‡Ž³gÆ5 ˆÐ ÆS J, JÝ´š~Œ . …·‚ƒ&, ù«JÝØ´‚ÀJ˜ `D áU ŽÑ . ÃØ õo`D á, k “ùÇÚvk “ùÇ ØÓœ¹e, ÆS J åÑ¬' ²w. êÆa ‘§, cÙXd. ¤±, r·‚ 1˜^ïÆ` †x:, Ò´Z Z Ø‡Ã •‘í ! e¡·‚Ò‚5“êù€‘§ A:, !A: éù€‘ ÆS5¿¯‘j. ØL, éue¡ù Ü©SN, ïÆÐÆö3Äg Ö ÖžØ‡ Ö. ± XJÆS‚5“ê©ªú vk(J, @o •ØI‡ Ö !. Öözgaú ‚5“êš~JÆ žÿ, Œ±2}Áˆ£ Ö±eSN. 1. 5-“ê†AÛ éX. ±“ê °(5Úî>5\ éAÛy– n), Óž±AÛ†* 5n)“ê$ŽÚ“êVg AÛ¿Â, l \ n)“ê†AÛƒ9ƒ¤ ƒp'X. 2. 5¿òÄ–VgÚäN¢~!V)5 ½n(ØÚäNz )KA^;—(Ü. “êÆ n Ø°u, Ï~´˜ läN¢~¥o(¿Ä–Ñ5 Vg, ±9äkpÝV)5!°(5ÚÊ·5 ½n. n)Ä–VgÚ°(Ýº½n ,´p “ê‘§ Ä Ôö. ´, ·‚F"ÐÆöØ ‡#P: ?ÛÄ– VgÑ guŒþäN ¢~, ?Ûk¿Â ½nÑlØmŒþŒ±A^§ ¢S¯K. Ïd, ÐÆöA 5-n)~fÚA^, ;•3vkn)äN¢~ œ¹e H/ PÁ Ä–Vg, ïÆÏLŒþ¦^½n)û¢S¯K •ª5\ n)z˜^½n, Ø´==lÄ– nØÚÜ6ín ¡ PÁ½n. 3. ÐÆž5¿“l{üLÞ E,” ÌSì?L§, Ùö rz“òE,z8 {ü” g‘• {Ôö. kéõÐÆöú ‚5“êJÆ. (, ‚5“êp¡kØ ' E, Vg! nØÚ• {. ´, Ù¢§‚¿Ø´l•Ðå ÒÉ,JÃ . ()ö‡<@•, ?Û‰Æ uÐÑ´l{ü ¯KÑu, ˜Ú˜ÚLÞ E, ¯KÚnØNX .) ·‚ÆS˜‡# Vg, Œ±l§•{ü ~fm©n), , 3n)# ~fžØäÚÙG !•{ü ~f‰' , ù ~~Œ±•N ´k# N¬ÚÂ¼. Æ # nØ½•{, Ø”k¦^§ ¢S?n˜ {ü A^¯K, , Å ì]Ô•kJÝ ¯K, ù Œ±¦·‚3]ÔJK žÿéÄ nØÚ•{kv ÙGÝ, ƒ uO\ gC •£; Ú)KÔöƒ 2 ]ÔJK, Ïd¤õ ŒU5•¬O\. ùÙ¢´ ·^5š~2 ÆS•{. ùÒÐ'·‚ÆSf¨, Ï~Ñ´l”B .•f¨m©Æ, l† f ¨m©Æ, þ´öSžkÀJ<6•6' žmãÚm,´ã öS, Xd . ·‚l m© ÆS•Ä Žâ, ÆSOŽg,êƒm \~¦Ø. y3£Þc[Ž˜Ž, Ù¢·‚Ñ´•kÆ¬ ‡ êƒm \~¦Ø, , ?¿g,êƒm \~¦Ø•ØL´3‡ ê\~¦Ø$Ž{K Ä :þ, r§SF%/õ¢–AH ®. ùÙ¢ØÒ´·‚ÙG “l{üLÞ E,” ~fí ? Ù ¢ÆS?ÛwqE,p SN, •{þ• Ø ‡„lù 5Æ. ·‚ÆS˜¬SNk˜ãžm , BØA2÷vu•¬)û{ü ¯K. dž, Ž‡JpgC 12 1"Ù ‚5“ê•=„ UåŒ± öS)ûE, ¯K. ‚5“êk˜‡A:, •,kØ nØÚ•{•ªo(Ñ5 w E,½„¡, ´ÙÄ )KgŽš~{ü, Òó(˜@XÚ •{rE, ¯Kz8 { ü ¯K. ~X, ¦)˜„ ‚5•§|, ÙäNnØo(å5ŸÌw,' •. ´, ·‚[Ž˜e Ò¬uy, Ù¢Ø%gŽÒ´˜^: Ò´¦ å rÐ©‰ /G' E, •§|z¤/ª{ü •§| (Ï~Ò´•1 F/ •§|, éAuþn /$–´é /Ý •§|). ·‚ÆS Ý nØ½ö‚5C† nØ, •Ÿo‡séŒ °åù)Ý Ú‚5C† é zQ? ndé{ ü, Ï•·‚• ÃØéÝ „´‚5C†, é / Lˆªo´•{ü . ¤±, r˜„ !E, œ¹¦ŒU/=z¤é (½ö Cé ) œ¹, Ò´·‚ Ä 8IÚÄ üÑ. ‚5“ê °uÒ3u, §Œ±wŠ·‚ù«üÑ (´Œ1 , …Œ±wŠ·‚TüÑäNXÛ–1. ¤±, òE,z8 {ü, ùÙ¢´‚5“ê ˜‡-‡gŽ•{. F"ÆSöU ÅìN¬¿ Ýºù«gŽ. 4. X¿ Ä–g‘Uå. p “ê‘§´•8 êÆÄ:‡¦' p ;’ÆSO Ä :‘§. ù€‘òÅìÚ+ÆSöÆ äkpÝV)5!Ä–5!°(5ÚÊH5 “êVgÚ“ êÆ•{. “êÆ ˜‡wÍA , Ò´§•¹éõläN¢~½äNy–¥Ä–JõÑ 55 Æ, r§‚,uo(¤Ä– !äk˜„5 Vg, ¿²Lî— Ü6©ÛÚín Ñš~äkÊ ·5 ½n. 8 ÖöXJU ý N¬ “êÆ rŒ%å, 7,´®²ä v Ä–g‘ Uå. 3ˆ¤ù«Uå ?ƒc, ÓÆ‚Ø”²~]Ô˜eÔöÄ–g‘ SK, ÔögCÜ6ín Úî—êÆØy Uå. 1˜Ù 3 Ö Ý ÙSNƒc, ïÆk(@ ÖL 1.1 1.1.1 •§| Ð Ú‚5•§| Öcó. )‚5•§| C† (1.1.1) s{ `, y3·‚ÏL¢~5&?XÛ)‚5•§|. ±có¥J =Xe•§| •§| (2) •~,   3x + 2y + z = 39  (1.1.1.1) 2x + 3y + z = 34    x + 2y + 3z = 26 Ø• Öö´Ä®²gC}ÁL)ù‡•§|. ¢Sþ, •´3·IÜÇÇÉ2žÏÒ®²k<¦ )L§ , Ï•§Œ±„uÇÉ2žÏ ˜ úƒÜñ (c256–c152 c) 6> 5ÊÙŽâ6 (±e {¡5ÊÙ6) ˜Ö¥. TÖ8¤ ·Ik‹–ÜÇÐÏ êÆ¤J, Ù¥ 5ò1l · •§6† ˜Ù 1˜‡~KÒ´: 8kþÙn[, ¥Ù [, eÙ˜[, ¢n›ÊÌ; þÙ [, ¥Ùn[, eÙ˜[, ¢n›oÌ; þÙ˜[, ¥Ù [, eÙn[, ¢ ›8Ì. ¯þ!¥!eÙ¢˜[ÂÛ? ‰ : þÙ˜[, ÊÌ!o©Ìƒ˜, ¥Ù˜[, oÌ!o©Ìƒ˜, eÙ˜[, Ì!o©Ìƒ n. § ²L©ó )• x{ €È, ØJòùã©i=z¤ (1.1.1.1) ¥ x=9+ •§| (ë„ [34]), …·‚• 1 37 1 17 3 11 = , y =4+ = , z =2+ = . 4 4 4 4 4 4 @oþ¡ù‡)XÛâU¦Ñ5Q? y “ Í ¶ ê Æ ¤ [ î í # (1917–1988) k ) Q µ Ø `: “e r ' Ž â, d ¥ • Ù ´ Ê Ù”. 5ÊÙ6 Ø…ƒ?Ò3u, §Ø=‰Ñ •§| ), …¢Sþ•[)º ¦) •{. Ö¥ ©Ø2Kã, k, ÖöŒ±ë• [34]. x{óƒ, 5ÊÙ6¥ù) ¦)•§| (1.1.1.1) •{ ©¤XeÚ½: † ¢Sþ, 5ÊÙŽâ6Ö¥¤¢ “§”´•• ƒ'êâ ¤ “•§”Ú·‚y“êÆ¥ '~'X. 13 •§VgØ´˜è ¯. “•”´•êâ†m¿ü, Ù/• . 1˜Ù 14 Ý Ú‚5•§| 1. Äk, ò“þÙn[, ¥Ù [, eÙ˜[, ¢n›ÊÌ”˜é¥ êi‚m ¤˜ mÚ†>2X{ ›/ Ñ e5üé{¥ êi. u´ ˜‡êiL‚Xe 1 2 3 26 2. 3¥m˜ 1˜‡êiC¤ 0 ƒ 0. u´êiL‚¤• ,† ~KT 1 2 3 26 0 5 1 24 e5, ·‚3¥m˜ ê, , ±aqþ˜Ú½ iL‚¬C• 0 5 1 24 ê8) eZ 0 5 1 24 ƒ •{r†>˜ ¦ ¤kê, u´ ¥m˜ 1˜‡ 1˜‡êi•C¤ 3 2 1 39 1˜‡êi 0 e¡é˜‡š" •{Œ±ò†>˜ éA ˜ 0 0 36 99 ¦±¥m˜ 3 2 1 39 , ·‚2^Ó 0 4 8 39 3¥ 3 2 1 39 e5, ^m>˜ 1˜‡êi 3 (éA ©¥þÙ ƒc êiL‚C¤ 1 6 3 2 9 2 3 3 1 26 102 39 ; X^¥m˜ Øä/ ~m>˜ êC• 0, dž êiL‚¤• 3. 2 3 1 34 ,, ê, ^§ ¦±†>˜ z˜‡ êi2C¤ 0. ²Lù˜Úƒ ê 3 2 1 39 4. dž, • ˜eêiL‚éA ¹ÂÒŒ±• , † ^†>˜ ü‡êi 36 Ú 99 ‰˜‡ 99 11 Š (3 ©¥éAu“eÙƒ¢”). k z Šƒ , “ Ø{, 36 = 4 Ò´™•ê z \ ¥m˜ êiéA •§=Œ)Ñ y Š. • , ò y Ú z Š“\ •m>˜ , Œ± r x Š•)Ñ5. ±þ·‚ )‰ •§| (1.1.1.1) ¦)¯K. Ööú džAT?˜Úg•ŸoQ ? 3±þ~Kƒ , 5ÊÙ635•§6˜Ù¥qUY?Ø NõÓ a. ¯K, Ù¥ 9 ‚ 5•§|•õk 6 ‡™•ê 5 ‡•§, ¦)ù •§| •{ Ÿþ• Ü˜—. k ¢Ã ´, •,ù«•{•@„u¥I ©zP1, §•ª -.ú@ ¶i´“pdž {”. ‡¥ Ï™ù nóüŠéJù˜. •Nk˜:´, ŒU·I êÆkp‚3êÆgŽþ¿Ø-ÀÄ–/Jõo( ÑêÆnØ¿Ší2. ¤±, ·‚AT< k< Ô, @ýo(˜ec¡@‡~f¥Ñy êÆ. 1.1 )‚5•§| 15 ØJuy, ƒc·‚)Ñ•§| (1.1.1.1) '…´òÙ=z¤±en‡•§|¤    3x + 2y + z = 39  (1.1.1.2) 5y + z = 24    36z = 99 #•§| ù‡•§|ƒéu k •§| (1.1.1.1) ó¦)C N´éõ. • nØþ î>, ·‚AT¯ ˜é: d# •§| (1.1.1.2) ¦Ñ )•Ÿo˜½´ 5•§| )Q ? , , 3·‚l (1.1.1.2) ¤k)ƒ¥, ´Ä•¹ 5•§| (1.1.1.1) ¤k) vk?Û¢¦Q ? £‰ùü‡¯KÙ¢˜:ÑØJ. ·‚3dŽJ2Öö3¿“êÆD– ‡Ä:êÆ (•¡ XêÆ) ˜‡-‡£ÆgŽ: ·‚‡énØ!•{½öêÆ(ØŠo(ª °(•ãž, 7L¦ ^î>5‰ÃÜÂ Šó. 3k7‡ž, ATæ e½Â •ªïáå¤‡¦^ 5‰êÆŠóN X. y3·‚Ú\Xe½Â (ù´ Ö ©¥1˜g ª/ÑyêÆ½Â): ½ Â 1.1.2. •Ä˜‡‚5•§|   a11 x1 + · · · + a1n xn = b1     a x + · · · + a x = b 21 1 (1.1.2.1) 2n n 2   ··· ··· ··· ···     am1 x1 + · · · + amn xn = bm Ù¥ x1 , . . . , xn L«™•ê, aij Ú bi (é¤k i ∈ [[1, m]] Ú j ∈ [[1, n]]) L«®•~ê. •§| ˜ ‡ (½ö˜|) )´•˜‡ n kSê| (c1 , . . . , cn ) ¦ •§|¥z‡•§¥ ™•ê xi ©O± ci Š“\ƒ þU¦ ª¤á. •§| (1.1.2.1) ¤k) ¤ 8Ü¡•§ )8 (solution set, set of solutions). XJ, ˜‡•§|   a011 x1 + · · · + a01n xn = b01     a0 x + · · · + a0 x = b0 21 1 2n n 2 (1.1.2.2)   ··· ··· ··· ···     0 am1 x1 + · · · + a0mn xn = b0m ) 8 † (1.1.2.1) ) 8 ƒ Ó, · ‚ ` (1.1.2.1) Ú (1.1.2.2) ù ü ‡ • § | Ó ) (have the same solutions) ½ d (equivalent). †óƒ, ù´•: éu?Û˜‡ n kSê| (c1 , . . . , cn ), §´•§ | (1.1.2.1) ) …= §´•§| (1.1.2.2) ). éu‚5•§| ó, )Ó)•§|k˜«š~~„ •ª. • r§`˜Ù, ·‚ Xe½ Â: ½ Â 1.1.3. E•Ä (1.1.2.1) ¥ (1.1.3.1) ‚5•§|. α1 x1 + · · · + αn xn = β ´ ' u Ó ˜ | ™ • ê ˜ ‡ ‚ 5 • §. X J ò (1.1.2.1) ¥ z ‡ • § ¦ ± · ~ê, 2\ å5Œ± (1.1.3.1), K¡•§ (1.1.3.1) ´•§| (1.1.2.1) ¥ m ‡•§ ˜‡‚5|Ü (linear combination). †óƒ, ·‚` (1.1.3.1) ´•§| (1.1.2.1) ¥ˆ‡•§ ˜‡‚5|Ü, XJe¡^ ‡¤á: 1˜Ù 16 •3~ê λ1 , . . . , λm ¦ Š•'u™•ê xi Ý Ú‚5•§| Lˆª λ1 (a11 x1 + · · · + a1n xn ) + · · · + λm (am1 x1 + · · · + amn xn ) = α1 x1 + · · · + αn xn ¿… λ1 b1 + · · · + λm bm = β . w,, 3þã^‡¥·‚Œ±6žò™•ê xi mØ+, ••Ä~ê aij , bi , αj Ú β ƒm 'é. dž¤¢ (1.1.3.1) • (1.1.2.1) ¥ˆ•§ ‚5|Ü, ¢SþÒ´•kSê| (α1 , . . . , αn , β) Úe kSê| (a11 , . . . , a1n , b1 ) , . . . , (am1 , . . . , amn , bm ) ƒm•3“‚5|Ü” (1.1.3.2) 'X, =, •3~ê λ1 , . . . , λm ¦   λ1 a11 + λ2 a21 + · · · + λm am1 = α1        λ1 a12 + λ2 a22 + · · · + λm am2 = α2 ··· ··· ··· ··· ··· ···     λ1 a1n + λ2 a2n + · · · + λm amn = αn      λ1 b1 + λ2 b2 + · · · + λm bm = β þã^‡¤áž, ·‚•¡kSê| (α1 , . . . , αn , β) ´ (a11 , . . . , a1n , b1 ) , . . . , (am1 , . . . , amn , bm ) ù kSê| ˜‡‚5|Ü. w,, XJ˜‡‚5•§ Ò´˜‡•§|¥Ñy Ù¥˜‡•§, @o§7,´T•§| ¥ˆ•§ ˜‡‚5|Ü. (\U•ì½Âc[`²•Ÿoí ?)  2x − 3y − z = 1 ~ 1.1.4. žÖög1 y: ‚5•§ 4x + 3y − 7z = 11 ´•§| ˜‡‚5 −x + 6y − 2z = 4 |Ü, x + 3y − 3z = 5 ´Ó˜•§| ,˜‡‚5|Ü. g • K 1.1. •§| (1.1.1.2) ¥ z‡•§Ñ´•§| (1.1.1.1) ¥•§ ‚5|Ü, \U`²ù˜ :í ? ‡L5, •§| (1.1.1.1) ¥ z‡•§´ÄÑ´•§| (1.1.1.2) ¥•§ ‚5|ÜQ ? ½n 1.1.5: XJ•§| (1.1.2.2) ¥ z˜‡•§Ñ´•§| (1.1.2.1) ‚5|Ü, @o (1.1.2.1) )8•¹3 (1.1.2.2) )8ƒ¥. XJ‡L5•§| (1.1.2.1) ¥ z˜‡•§•Ñ´•§| (1.1.2.2) ‚5|Ü (dž·‚Œ± `ùü‡•§|p•‚5|Ü), @o (1.1.2.1) Ú (1.1.2.2) ùü‡•§|Ó). y². w,, ò˜| ª †mü>Óž¦±ƒÓ ~ê, , 2¦Ú\å5, ù ¤á. ¤±, Šâ‚5|Ü ½Â, ˜‡•§| )[ ´T•§|?¿‚5|Ü¤ Ò´½n1˜‡Øä y². 1 ‡(Ø‘ƒw,. ª•, •§ ). ù ¯çØ ? Š• Ö 1˜‡½n, § y²Ø,Xd{ü. ØL, ±eü:Š o(: 1˜, ½n 1.1.5 y²ƒ¤±Œ± Xd{' Ü6˜ß, Ø Ï•§¤^ n´š~f w ~£, „Ï•·‚ƒc®²‰ ¹Â°O ½Â. ²( ½ÂŒ±¦êÆØy{'² , •Œ± JøêÆín 5‰•â. ùATÒ´“e½Â”ù« öŠéuêÆ[‚ AÏ¿Âj. 1.1 )‚5•§| 17 ¯¢þ, Ø Ä Ü6{KÚ®• K ^‡ƒ , êÆy²Ün •â Ÿþ•kü«: Ê H É êÆún!3únÄ:þÜnïá ½Â. ·‚¤¢ fw ¼ ú@ ~£, 3êÆ+ •XJ(¢·^ {, Ù¢•ATÀ•Œ[%@¤á únNX ˜Ü©. (k, Öö ˜ eîAp 5AÛ 6¥ únÒŒ±N¬ù˜:.) ,, 3¢SöŠ¥·‚ØŒU[ I‡J ˆ únÚ½Â, •‡´ÏLúnÚ½Â®²y² ( (Ø (~X®• ½n), ·‚•Œ±Ú^ Š•y² •â. ¤±, ·‚3dJ2ÐÆêÆy² Öö5¿: ˜‡Ü‚ êÆy², Ø ÎÜÜ 65Æƒ , 7L3z‡ínÚ½Œ±lún (•)w,Ø _ún ~£)!®•K !®•½n (••)êÆ©z½ ‰Ö¥¡•“Ún”!“íØ”!“·K”ƒa (Ø) Ú½Âƒ¥é •â. 1 , êÆÙ¢´l{üÅÚuÐ E, . Z Ø‡@•{ü êÆ´Ã^ ! ƒ‡, NõA Ok^ ÀÜŒUå© ã½ö ŸSºÑš~{ü ! êÆÆSöAT5¿l{ü œ¹\ÃÅ Ú \/?1ïÄÚÆS, k éõ •£Ú² È\ƒ , ŒU 5 õE, ÀÜ•¬w { ü. ½n 1.1.5 wŠ·‚, ¦)˜‡‚5•§|Ú¦)?Û˜‡†ƒp•‚5|Ü •§|´ d . 3 (1.1.1) ˜ã¥·‚w , XJl•Ð‰½ •§|·‚U é ˜‡†ƒp•‚5|Ü! /ªþ••{ü •§|, @o)•§| ¯KÒé¯)û . ¤±, e5·‚?Ø˜eN )†‰½‚5•§|p•‚5|Ü #•§|. ½ Â 1.1.6. éu˜‡‚5•§|, ˜‡Ð ˜: †•§|¥,ü1 1. •§ C† (elementary transformation) ´•±en«öŠƒ ˜; 2. ò•§|¥,˜1 •§ü>Óž¦±,‡š"~ê; 3. ò•§|¥,˜1 •§¦±˜‡~ê, , • \ , ˜1 «O, ·‚~r±þn«C†©O¡•1ã!1 . aÚ1naÐ C†. Ï~·‚r•§| 1 i 1P• Li . XJ´ †•§| 1 i 1Ú1 j 1, ·‚8 ± Li ↔ Lj IPù«Ð C†. XJ´ò1 i 1¦±˜‡~ê λ, ·‚òTöŠP• Li → λLi . XJ´ò1 i 1 ¦±~ê k 2\ 1 j 1, ·‚æ^PÒ Lj → kLi + Lj 5P¹. 'uÐ C†·‚1˜‡{ü * ´: Ún 1.1.7: ± (S) IP,‚5•§|. b (S 0 ). K: (S) ²L˜gÐ C†ƒ # ‚5•§|P• 1. Œ±ÏL˜gÐ C†ò (S 0 ) C£ (S). ¯¢þ, XJl (S) (S 0 ) ¤²{ @gÐ 0 1ã (½1 a!1na), @oò (S ) C£ (S) Ð C†•´Ó aO. C†´ 2. •§| (S) Ú (S 0 ) p•‚5|Ü. AO/, ùü‡•§|Ó). y². ©n«a. y3e¡ Ð (ØY C†?1?Ø=Œ. [!3‰Öö. (½‘,ùÇL§¥)º.) ±¤. íØ 1.1.8: e,‚5•§| (S) ²Lk•õgÐ (S 0 ) ´Ó)•§|. C†ƒ # ‚5•§| (S 0 ), K (S) Ú 1˜Ù 18 Ý Ú‚5•§| g • K 1.2. y²: b ‚5•§| (T ) ¥ z‡•§Ñ´‚5•§| (S) ¥•§ ‚5|Ü. XJ ˜‡‚5•§´ (T ) ¥•§ ‚5|Ü, @oT•§•˜½´ (S) ¥•§ ‚5|Ü. (ù‡(ØŒ±n)•: “‚5|Ü”ù«öŠŒ±EÜ —— ‚5|Üƒ‚5|ÜE•‚5| Ü.) ddy²: e‚5•§| (S) ²{k•õgÐ C†ƒ ‚5•§| (S 0 ), K (S) Ú (S 0 ) p•‚5|Ü. (ò5Öö¬w , ù‡(Ø‡L5•é. =, XJü‡Œ ƒÓ (••§êþƒÓ, ™ •êêþ•ƒÓ) ‚5•§| (S) Ú (S 0 ) p•‚5|Ü, @o˜½Œ±ÏLk•õgÐ C†r (S) C¤ (S 0 ).) 1.1.2 XêÝ ÚÝ ž { y3·‚®²• : ?1Ð C†Ø¬UC˜‡‚5•§| )8. u´˜‡¦)‚5•§| Ä üÑÒg, ) : ‰½ ˜‡‚5•§|ƒ , Œ±²L‡E Ð C†òÙC¤˜‡/ ª••{ü ‚5•§|, , 2±•† •{éÑ). ¢Sþ, 3 (1.1.1) ã?ØL 5Ê Ù6Ö¥@‡~KÙ¢Ò´^ù«g´5) . @p ?ØÙ¢„wŠ·‚, ù‡¦)L§¥·‚ ‰1öŠ é–Ù¢•k•§|Ñy ~ê (•) Ò†> Úm> ). ¤±“L™•ê i13 ¥mÚ½¥ vk7‡Ñy. ò•§|™•ê XêJ Ñ5, U 5 üÙ•ª ¤˜‡Ý/ êiL, Ò ¤¢ XêÝ . 2ò•§|¥ Òm> ~êV\?5, Š••m>˜ , ù •Œ˜: Ý , ¡•O2Ý . (ƒ5`, ·‚‰ÑXe½Â: ½ Â 1.1.9. •Ä/X (1.1.2.1) ‚5•§|   a11 x1 + · · · + a1n xn = b1     a x + · · · + a x = b 21 1 2n n 2   ··· ··· ··· ···     am1 x1 + · · · + amn xn = bm ù‡•§| XêÝ / (2 ±)Ò) (coefficient matrix) ´•d aij ù  a11   a21 A :=   ..  . am1 (1.1.9.1) ò aij ¡• A 1 (i, j) ·‚~¬ò (1.1.9.1) a12 a22 ··· am2 ··· ··· .. . ··· ··· .. . ··· ··· ~êUìXe•ª à/ü ¤Ý/  a1n  a2n  ..   .  amn ƒ (element)!Xê (coefficient) ½©þ (entry). ¤e¡˜ Ñ /ª±B! •¡˜m: A = aij 1≤i≤m ½ A = (aij )m×n ½ö•{Ñ/, A = (aij ) . 1≤j≤n ò•§|¥ (1.1.9.2) Òm> ~ê‘|¤˜ V\ þ¡XêÝ A m>. ù   a11 a12 · · · · · · a1n b1    a21 a22 · · · · · · a2n b2   (A ; b) :=  . .. ..  .. ..  . .  .. ··· . .  am1 am2 · · · · · · amn bm /¤ êL 1.1 )‚5•§| 19 ¡••§| (1.1.2.1) O2Ý (augmented matrix). ,, ·‚•Œ±Øl?Û‚5•§|Ñu, † ò/X (1.1.9.1) êiL‚¡•˜‡Ý (matrix)‡ . kž• ²(˜‡Ý Œ , Œ±` (1.1.9.1) ª¥ Ý ´˜‡ m × n Ý (m ¦ n Ý ). e m = n, Œ¡ƒ•˜‡ n • (square matrix of order n). e n = 1 (=Ý •k˜ ), K Œ¡TÝ •˜‡ Ý (column matrix) ½ •þ (column vector). aq/, eÝ •k˜1 (= m = 1), Œ¡ƒ•˜‡1Ý (row matrix) ½1•þ (row vector). 3c¡ Ý PÒ¥, ·‚æ^ )Ò´ )Ò, •kéõÖKæ^•)Ò. ùXá‡<U Ð. 3êÆ¹Âþ·‚ØAƒ±«O. …, kž• PÒØÚþe©¥ Ù¦PÒ )ÀúØ·, ·‚ŒU¬‘žŠâäNœ¹C† )Ò½•)Ò ÀJ. {óƒ, (1.1.9.1) ¥ Ý kž•¬ ¤   a11 a12 · · · · · · a1n    a21 a22 · · · · · · a2n   A :=  . ..  .. ..  . .  .. ··· .  am1 am2 · · · · · · amn k Ý Vgƒ , ·‚Œ±éÝ ½ Â 1.1.10. éu˜‡Ý ˜: 1. †Ý ,ü1 , ˜‡Ð •§ 15½ÂÐ 1C† (elementary row transformation) ´•±en«öŠƒ ˜; ·‚± Li ↔ Lj IPé†1 i 1Ú1 j 1 2. òÝ C†. öŠ. ,˜1¦±˜‡š"~ê; ·‚± Li → λLi IP “ò1 i 1¦±~ê λ”. 3. òÝ ,˜1 •§¦±˜‡~ê, , \ , ˜1 ·‚± Lj → kLi + Lj 5L« “ò1 i 1¦±~ê k , ; \ 1j1 ”. aq/, Œ±½ÂÝ Ð C† (elementary column transformation). •[ Öö. †ƒc1C† PÒ©OéA C†PÒ•: Ci ↔ Cj ; Ci → λCi ; (ùpæî1 C ´Ï•§´=©üc column Cj → kCi + Cj . Äi1.) (1.1.11) k ±þ½Âƒ , ·‚£Þ5w (1.1.1) ˜ã¥?Ø Ù6•·‚£ã •{ ý“ ! Äk, ·‚ò•§| (1.1.1.1) O2Ý J Ñ5, = (1.1.11.1)  3  (A ; b) = 2 1 (ÁŽ, XJØ´¥I Ø´Ú·‚y3 “ { ‡Ý 2 3 2 1 1 3  39  34 26 ©iÖ S.´lþ ŸÃ í?) £ã·‚3‰  3  Ù¥ A = 2 1 •§|¦)L§. ·‚uy: 5Ê 2 3 2    1 39    , b = 1 34 3 26 e!lm–†, @o5ÊÙ6ù (matrix) ù‡c•@´d=IêÆ[ James Joseph Sylvester (1814–1897) u 1850 cc ¦^ Ý . {J 1˜Ù 20 e55ÊÙ6• ·‚X{éO2Ý  3  0 0 (A ; b ) = 0 0 (1.1.11.2) 2 5 0 1 1 36 ‰Ð  39  24 96 C†, • Ý  3  0 Ù¥ A = 0 0 2 5 0 Ý Ú‚5•§|    39 1  0   1  , b = 34 99 36 ù‡#Ý éA ‚5•§|Ò´ (1.1.1.2) ¥ •§|. •Ò´`, Uì½Â 1.1.9 `{, (1.1.11.2) ´‚5•§| (1.1.1.2) O2Ý . •§| (1.1.1.2) ´˜‡˜úŒ±wÑXÛ¦) •§|. ¤± –d, ÏLÐ 1C†5z{•§| 8IÐÚ¢y . ^·‚y3 L§Xe: Šóo(, 5ÊÙ6¤£ã l (1.1.11.1) =z¤ (1.1.11.2) ¤²{ Ð 1C†     3 2 1 39 3 2 1 39    L2 →3L2   2 3 1 34 −−−−−→ 6 9 3 102 1 2 3 26 1 2 3 26     3 2 1 39 3 2 1 39 L →L −2L1    L3 →3L3  −−2−−−2−−−→ 0 5 1 24 −−−−−→ 0 5 1 24 3 6 9 78 1 2 3 26     3 2 1 39 3 2 1 39 L3 →L3 −L1   L3 →5L3   −− −−−−−→ 0 5 1 24 −− −−−→ 0 5 1 24  0 4 8 39 0 20 40 195   3 2 1 39 L →L −4L2   −−3−−−3−−−→ 0 5 1 24   0 0 36 99 (1.1.11.3) ±þ·‚ò˜ ¯ØÓ. ˜Ú/ Ñ' AÏ Ý ƒ\þ•µŠ•IP, XJI‡)º•Ÿo± (1.1.11.2) •O2Ý †T•§| XêÝ /GAÏk'. •§|w ¡·‚¬)º§‚•Û† {ü, ƒ&éõÖöÑU5¿ ½ Â 1.1.12. A • n • , Ù ¥ 1 (i, j) © þ P • aij . ò i = j ž é A a11 , a22 , . . . , ann ¡•• A é (diagonal element, diagonal entry). Ý ù ©þ XJ A é e • Ý © þ Ü • 0, = i > j ž o k aij = 0, K ¡ A • þ n (upper triangular matrix). (5¿: é 9é þ• Ý ƒŒ±• 0 •Œ±Ø• 0.) XJ A •þn triangular matrix). †þn kî‚en , ¿…§ é • Ü• 0, K¡ A ´˜‡î‚þn (strictly upper ƒé Vgken (lower triangular matrix). aq/, †î‚þn (strictly lower triangular matrix). žÖögCòùü«Ý °(½Â ƒé Ñ. • £ã•B, ·‚Œ±Ž–k˜^l• †þ me !BL• ¤ké †‚, ¡ •T• Ìé ‚ (principal diagonal) ½{¡é ‚ (diagonal). ¤±, þn Ò´š" ƒ• U÷XÌé ‚½ö3Ìé ‚±þÑy • , î‚þn Ò´š" ƒ•Uî‚ uÌé ‚±þ • . (1.1.13) éNó(Ñ±e(Ø: XJ˜‡‚5•§| XêÝ ´˜‡é Üš" þn 1.1 )‚5•§| 21 , =•§|/X (1.1.13.1)   a11 x1 + a12 x2 + ··· + a1n xn = b1       a22 x2 + ··· + a2n xn = b2   ··· ······ ···     an−1, n−1 xn−1 + an−1, n xn = bn−1      ann xn = bn Ù¥ a11 , a22 , . . . , ann þØ• 0 K•§|k•˜), ¿…Œ±ÏLXe•{¦): kl• ˜‡•§)Ñ xn \ ê1 ‡•§Œ)Ñ bn−1 − an−1, n xn xn−1 = . an−1, n−1 , ò xn−1 Ú xn Š“\ n , òÙ“ Š xn = abnn ê1n‡•§Œ) xn−2 = bn−2 − an−1, n−1 xn−1 − an−2, n xn . an−2, n−2 •gaí, •ª=Œ)Ñ Ü™•ê. † (1.1.13.1) ƒq q ‡ØÓ œ¹„k±eü«:   a11 x1 + a12 x2 + ··· + a1m xm · · · + a1n xn = b1       a22 x2 + ··· + a2m xm · · · + a2n xn = b2   ··· ······ · · ·· · · · · · ···  (1.1.13.2)    am−1, m−1 xm−1 + am−1, m xm · · · + am−1, n xn = bm−1      amm xm · · · + amn xn = bm Ù¥ m < n , a11 , a22 , . . . , amm þØ• 0 ½  a11 x1 + a12 x2 + ··· + a1n xn = b1       a22 x2 + ··· + a2n xn = b2       ··· ······ ···       an−1, n−1 xn−1 + an−1, n xn = bn−1 (1.1.13.3) ann xn = bn                   an+1, n xn = bn+1 .. . .. . amn xn = bm Ù¥ m > n , a11 , a22 , . . . , ann þØ• 0 •§| (1.1.13.2) ){Ù¢Ú (1.1.13.1) aq. •˜ «O3udž•§ êþ uCþ ê þ. dž, ·‚Œ±-õ{Ñ5 Cþ xm+1 , · · · , xn ©OÕá/? ˜‡ëêŠ t1 , · · · , tn−m , , xm = bm − am+1 t1 − · · · − an tn−m . amm 1˜Ù 22 Ý Ú‚5•§| e52Úƒcaq, ò xm , xm+1 , · · · , xn Š“\c˜‡•§Œ±)Ñ xm−1 . •daí. dud ž xm+1 , · · · , xn Œ±?¿ Š, ¤± (1.1.13.2) )8¢Sþ´¹k n − m ‡ëê Ã¡õ‡) ¤. –u•§| (1.1.13.3), § •e¡'u xn ù˜‡CþkØŽ˜‡•§ å, =, ann xn = bn , an+1, n xn , · · · · · · , amn xn = bm . ù •§ƒmkŒU )gñ, =, ŒUk– ü‡ØÓ •§)Ñ xn ØƒÓ. ù«œ¹e , (1.1.13.3) •ÒÃ). XJ´þ¡ù •§ƒmvkgñ, §‚ Ó‰Ñ )Ò´ xn = bn /ann . òù ‡ xn Š“\ (1.1.13.3) ¥•c¡ •§, ÒŒ±•g2)Ñ xn−1 , · · · , x1 ù Cþ Š. ¤±, (1.1.13.3) ù‡•§|3k) œ¹e•Ú (1.1.13.1) ){éƒC. ±þ?Ø •{¡•£“{ (back substitution). 5¿, XêÝ ´ n • ž, ù«•{Œ ¢y cJ´XêÝ •þn ¿…Ìé ‚þ ƒ a11 , . . . , ann Üš". éuXêÝ Ø ´• œ/, •I‡kaq ^‡. £“{ ·^Û•½¦·‚?˜Ú&?^Ð C†•{)‚5•§| £ (1.1.11.3) ¥ Ñ Ð C†[!·‚Œ±uy, ‡Ž1ƒk ,‚5•§| O2Ý ), ·‚F"±e^‡ ÷v: (a) Ý 1 11 – •¹˜‡š" Eâ[!. /z{˜‡Ý (~X5g ˜), Œ±òÝ 1 1 11 ƒ. ed^‡¤á, ÏLeZg1ãÐ C† (= †Ý , ˜ ƒN •˜‡š" ƒ. †óƒ, Ø”b a11 = 6 0. 1 ˜„œ¹e, e1˜ •¹õ‡š" ƒ, KŒ±?¿]À˜‡Š•Ì (pivot). “Ì ”ù‡c ==´lŽ{£ã•B Ý5¡¢˜ A½ ƒ, Ï~´duù ƒÀ ¤ƒ =Œ¦ Ž{k –1. Ì ÀJÏ~Ø´•˜ , •‡U¦Ž{k ?1, ŒUØ•˜‡ ƒŒ À•Ì . XJ,Ž{©¤eZÚ½, kž•¬ éz‡Ú½!ØTÚ½ Ì . éu·‚ùp?Ø ± Ð 1C†z{Ý Ž{, 1˜g Ì (nØþ`) Œ±lÝ 1˜ š" ¥‘¿À . Š âÌ À , ·‚Œ±rÌ ¤3 1¡•˜‡Ì1 (pivotal row). (Ï•Ì ÀJŒ±‘Ž{ ‰1ö Ì*UÐØÓ, ¤±Ì1½Xd.) ~X, 3 (1.1.11.3) üŽL§¥, ·‚À 1˜‡Ì Ò´Ý 1 (1, 1) ƒ a11 = 3. (ù ´·‚ƒcòÙ\µ Ï.) ¦^ù‡Ì , ²LeZg (3 (1.1.11.3) ¥´ 4 g)Ð 1C†Œ     3 2 1 39 3 2 1 39     {, ƒ u`•Ð Xê òÝ 2 3 1 34 z• 0 5 1 24. =w†> 3 × 3 fÝ 1 2 3 26 0 4 8 39     3 2 1 3 2 1     Ý A = 2 3 1 =z¤ 0 5 1. –dŒ±Ž ¤ Ý A z{ 1˜ ã. 1 2 3 0 4 8 ^‡ (a) ¤áž, ·‚®²• ˜½Œ±ÏLk•õgÐ 1C†¦Ý 1 1 •k˜‡ š" ƒ, …T ƒ u1 1 1 1 1, = a11 . ( ,, ¢yù‡8I äN•YŒUkéõ «.) 3ù‡ ã Ý Cz¥ a11 w,å ÕA Š^, ·‚Ï~Ò±§Š•1˜‡Ì . Šâc¡?ØL ¢~, e5·‚Œ±Ø2'%Ý 1 1 1Ú1 1 u Ý me (1Ú ê8Ñ~ 1 ‡ ) fÝ 5UYþ˜ ãÓ ‚qF" (b) 3me E,– ù‡fÝ ¥, 1 1 •¹˜‡š" ƒ. (Ù¢éA ŒÝ 12 Ø , ´é•e ! öŠ. ¤±, dž· •þà ƒ¤ ) 1.1 )‚5•§| 23 †ƒcaq/, ^‡ (b) 2g¤áž·‚Œ±-Eþ˜ ã öŠ. ØL, dž·‚´é•Ð O2Ý ('X´ m × (n + 1) Ý ) C/ ˜‡fÝ (Œ ´ (m − 1) × n) 5‰1Ð 1C †. éuù‡fÝ ó, ·‚2l§ 1 1 À ˜‡Ì , òÙÏLÐ 1C†˜u1 1 1 1 1, , |^ù‡Ì UYÐ 1C†òfÝ 1 1 •C¤Ì e• • 0 /G. ,, Xd‡E, ·‚F" e5z‡ ã¥† (b) ƒéA ^‡ÑU¤á. ù {, bX· ‚öŠ é–5gu˜‡‚5•§| O2Ý , @o•ª¢y JÒ´ ˜‡aq (1.1.13.1) •§|. Ï•ù‡•§| Ì‡A:´lþ–ez˜1 •§¥¢SÑy C êþÅì~ , †–• ˜‡•§¥•Ñy˜‡C (l Œ±† )Ñ), ¤±þ¡•ŸŒØ ù@•{ (ÃØ´ é˜‡‚5•§|¢–„´é˜‡Ý ¢–) Ø”¡ƒ•ž { (elimination). (1.1.14) þ˜ŒãÌ‡?Ø ž {Ž‡ˆ ýÏ •Ð JI‡Ÿo^‡. )¹w,ØŒU? ? ¡·‚%¿. Ø÷vcã^‡ œ¹Ù¢:• ´. ¤±·‚I‡UY?Øž {Ã{^| ¤ýÏ8IžNo•. ·‚Q²JL, l{ü E,!läN Ä–´Š J† ˜«ÆS•{. ¤±y3·‚kl ˜‡{ü äN ~fm©. b       0 0 1 1 1 0 1 1 1 0 1 0 1 2 3  0 1 2 3 1       (1.1.14.1) (A; b) =   , b=   Ù¥ A =  2 0 1 3 5  0 1 3 5 2 0 XJš‡ò±þÝ 1 4 0 3 1 4 7 3 À•,‡‚5•§| O2Ý , @oT•§|AT´   0 · x1 + 1 · x2 + 1 · x3 + 1 · x4 = 0     0 · x + 1 · x + 2 · x + 3 · x = 1 1 (1.1.14.2) 7 2 3 4   0 · x1 + 1 · x2 + 3 · x3 + 5 · x4 = 2     0 · x1 + 1 · x2 + 4 · x3 + 7 · x4 = 3 y3, O2Ý (1.1.14.1) ²wØ÷vc˜ãJ ^‡ (a), § 1˜ ƒ Ü• 0. XJ• ÄƒA ‚5•§|, ·‚uyù¿Ø´˜‡<s ¯œ (‡ ŒU´‡Ð¯), Ï•ù¦ •§| (1.1.14.2) •,¶ÂþV 4 ‡C , x1 3z‡•§¥ XêÑ´ 0, ùƒ u` x1 vkÑ y, •§|é§vk?Û å, ù‡™•êŒ±?¿/gd/ Š. ·‚¦)•§| (1.1.14.2) žŒ ±ò x1 Ñ, •w•{ 3 ‡C . lÝ Ý`, ùƒ ur (1.1.14.1) ¥ Ý 1 1 Ñ K (5¿§ ù˜Ú·‚• Ñ˜ , Ø´”ž { ¤,˜ ãƒ @ , Óž Ñ˜1Ú˜ ±BUYž ). , ·‚‡‰ Ò´é{e Ý 2–1ž {. ò (1.1.14.1) ¥Ý 1˜ ! ƒ , •{ Ý ´     1 1 1 0 1 1 1 1 2 3 1 1 2 3     (1.1.14.3) (A0 ; b) =   Ù¥ A0 =   1 3 5 2 1 3 5 1 4 7 3 1 4 7 éù‡# (1.1.14.4) Ý ·‚Œ±^Ð 1C† (äNXÛöŠžÖög1MÖ) òÙC¤   1 1 1 0 0 1 2 1     0 2 4 2 0 3 6 3 1˜Ù 24 e‡‹l © Ý ) (A ; b) ŸoCz, ·‚Œ±w  0 0   0 0 (1.1.14.5) 1 0 0 0 1 1 2 3 Ý Ú‚5•§| ±þöŠÓžò (A ; b) C¤  0 1   2 3 1 2 4 6 =, † 3 (1.1.14.4) ¥Ý •†>¡E Ñ ˜ 0 BŒ• ©Ý (A ; b) =z¤Ÿo f . ùp né{ü, •†>˜ ƒ ´ 0 ž, ?ÛÐ 1C†ÑÃ{UCù˜ •0 yG. ¤±‘žVþ½ Kù˜ 0 Ø¬k?Û¢Ÿ5 K•. y3·‚Œ±o(˜ež {Ã{‰1žkÛ©O. Ù¢<Kp•I‡ èi: Ñ! Ï•ž {Ã{¢–¿›XÃ{é Ì , •Ò´`Ý •†>˜ ƒ • 0. dž† Ñù˜ é•{ Ý 2•Ä–1ž {. (·‚2grN, ù˜Ú·‚• Ñ˜ , Ø´Ó ž Ñ˜1. Óž Ñ˜1Ú˜ öŠ´3 ¤ ,‡ ã ž {, e5£½81 me fÝ ž‡‰ ¯œ.) XJ Ñ 1 1 E,Ã{¢–ž 0 =Œ. (¬k<¯XJ˜†éØ š" ±þ·‚ ¤ éÝ (1.1.14.1) ¢–ž  0 0  (A ; b) =  0 0 Xc¤ã, e¡·‚rmÃ> Ý 11 {No•Q ? No•í ?) Ú1 1 1) Ñ Ý 1 1 1 1 1 2 3 4 { 1 3 5 7 X Ñ˜ 1˜‡ ã, =¢y   0 0 0 1    −→  0 2 (= (1.1.14.5) ¥Ý ÑK, •Šâme 1 0 0 0 0 3 E ! Xd˜† 1 1 2 3 1 2 4 6 XeÝ ,˜ Ø • C/:  0 1   2 3 1 2  Ú1 1  1 (ƒ u (1.1.14.4) ¥ 1 2 1   3 × 3 fÝ 2 4 2 5ÀJ Y ž ö 3 6 3 ) Š. N´w , A‡{ü Ð  1C†Œ±¢y 0 0   0 0 XJd•¤ Ý ®²2•Ã{é ØÂÃ ? í, 1 0 0 0 1 1 2 3 1 2 4 6   0 0 0 1    −→  0 2 3 0 1 0 0 0 1 1 0 0 1 2 0 0  0 1   0 0 †>n Úþ>ü1 (žg•: •Ÿodžń Úü1 ?) Ñ š" . ùžÿ„U‰ŸoQ ? —— WWZúÒ1 . Ñž . AT2¯Öö˜(, •§| (1.1.14.2) ¯¢þ, y3·‚† •Ð )y3s )XÛ(½í? •§| (1.1.14.2) †±e•§|Ó):   0 · x1 + 1 · x2 + 1 · x3 + 1 · x4 = 0     0 · x + 0 · x + 1 · x + 2 · x = 1 1 2 3 4   0 · x1 + 0 · x2 + 0 · x3 + 0 · x4 = 0     0 · x1 + 0 · x2 + 0 · x3 + 0 · x4 = 0 Ñ, ·‚uy ´0 „ 1.1 )‚5•§| w,, ùp• ‡ ü1 Ñd•§|) (1.1.14.6) 25 •§Œ± ÑØO. •ª·‚•‡¦)•§|  0 · x + 1 · x + 1 · x + 1 · x = 0 1 2 3 4 0 · x 1 + 0 · x 2 + 1 · x 3 + 2 · x 4 = 1 ˜„/ª, Ù¥˜«•ª´ù :   x1 = t     x = s − 1 2 Ù¥ t , s Ñ´Œ±?¿   x = 1 − 2s 3     x4 = s Š ëê. 5¿, 4 ‡™•ê¥Ø ƒc·‚• x1 Œ±‘¿ Š, „k x4 Œ±ku x2 Ú x3 ? ˜‡Š, , x2 Ú x3 Š¬‘ x4 Š (½. Ù¢ùp•´˜«£“, •ØLy3 x4 Ø´±äN(½ êŠ/ª“£ , ´±Œ±‘¿ Š ëê/ª. ù«y–éukÃ¡õ) ‚5•§|5`, Ù ¢´˜«š~;. y–. ••[ nØo(ò3e˜ !‰Ñ. d•·‚6…UeØL. nþ¤ã, Ã{–1ž ž, %g±e•ü=Œ: ‘" Ñ2 Ñ, ˜ Øv2˜ . %YœByÌ , ž £“= (. 3UYnØ&?ƒc, 4·‚2ïÄü‡äN~f, Ù¥•)˜‡•§êþÚCþêþØƒÓ ~f. ~ 1.1.15. ¦)‚5•§|   x1 + x2 − 3x4 − x5      x1 − x2 + 2x3 − x4  4x1 − 2x2 + 6x3 + 3x4 − 4x5      2x + 4x − 2x + 4x − 7x 1 2 3 4 5 (äNL§lÑ.) Ù¥˜«Lã (‰Y   x1 = 65 − t + 76 s      5 5   x2 = − 6 + t + 6 s x3 = t     x4 = 23 + 13 s      x5 = s (ž¹¢gdCþ†Ì ˜ ~ 1.1.16. ¦)‚5•§| •ª´ù = −2 = 1 = 7 = 1 : Ù¥ t , s Ñ´Œ±?¿ 'X !)   x1 + 2x2 + 3x3 + 4x4      x1 + 2x2 − 5x4   2x1 + 4x2 − 3x3 − 19x4     3x + 6x − 3x − 24x 1 2 3 4 Š ëê. = −3 = 1 = 6 = 6 1˜Ù 26 (äNL§lÑ.) ²Lž {öŠ, ØJuyd•§|Ã). @o, Öösú |, ž {(åž Ý kŸowÍA:Q ? Ý Ú‚5•§| éuÃ) ‚5•§ k<rk) ‚5•§|¡•´ƒN (consistent), ÄK¡•´ØƒN (inconsistent). !¤?Ø ~fÚnØ, Ù¢ÑŒ7X·‚3có¥J ˜‡¯K: éu‰½ ˜‡‚ 5•§|, XÛÏLÙXêÝ Ú Òm> ~êû½•§| )8´Äš˜Ú´Ä•k•˜). {ü/`, ·‚y3• Œ±ÏLž {éXêÝ ÚO2Ý ‰C†5 ù‡¯K ‰Y. –u3k) œ¹eXÛ°(/‰Ñ˜„) Lˆª, ·‚òé¯3e˜ !?Ø. ˜„ ó, Ø²Lž { (½ö`Ð 1C†) † l © ‚5•§|O2Ý m´éJ äÙ)85 XÛ . ØLk‡4à œ¹Š ˜J, =àg‚5•§| (system of homogeneous linear equations) œ¹. ùp, ˜‡‚5•§¡•´àg (homogeneous) XJÙ¥ Ò m> (= Ø¹™•ê ˜>) ~ê‘• 0, =/X α1 x1 + · · · + αn xn = 0 ‚5•§ (Ù¥ αi •~ê). ˜‡‚5•§|¡•´àg , XJÙ¥Ñy z‡‚5•§Ñ´à g . éuàg‚5•§|5`, -¤k™•ê u 0 w,Œ± ˜‡). ù‡)¡•") (zero solution) ½ö²…) (trivial solution)† . Ù¦ )g,Œ±¡•š") (nonzero solution) ½š²… ) (nontrivial solution). w,, éuàg‚5•§|5`, 'uÙš") &Eâ´ý k¿Â & E. 1.1.3 ^1 F/(½)85 ·‚3có¥ÒQJL, ˜„5`˜‡‚5•§| )85 kn«ŒU œ¹: ˜´Š vk), ´k…=k•˜ ), n´k) )Ø•˜. þ˜ !·‚w , ÏLÝ ž {r‚5 •§| O2Ý z¤AÏ “/G•Ð” fƒ BØJ äùn«œ¹´=˜«Ñy. ! ¥·‚•ù@nØ2Ö¿˜ [!. k5o(˜eÝ ž {U “/G•Ð” Ý Ä¾Ńo‡Ð{. ½ Â 1.1.17. ·‚¡˜‡Ý •1 F/ (row echelon form), XJ§÷v±eü‡^‡: 1. z‡š"1 (=– ¹k˜‡š" •. •Ò´`, "1•U uÝ éuz‡š"1 ÙÌ (pivot) ∗ ƒ 1) Ñ3?Û •.Ü. ó, l†–m1˜‡Ñy 2. ¤k š"1•Ä3S, z‡š"1 ‚‚m .§ éu‚5•§|, XJÙO2Ý ´1 š" ÄXê "1 (=¤k ƒ u0 1) þ ƒ¡•ÙÄXê (leading coefficient) ½ö ˜Úþ¡š"1 ÄXê F/, ·‚Œ±`T•§|´1 ˜ƒ' F/. o´î • •\²(¤•, <‚kžrÏLÐ 1C†ò‚5•§|½öÝ z¤1 F/ L§¡ •pdž { (Gaussian elimination). c[• §1.1.2 !¥?Ø ž •{ÚL§ (AO´ (1.1.13) Ú (1.1.14) üŒã), ØJuy· ‚Ù¢®²y² e¡ ½n: ½n 1.1.18: ?Û˜‡Ý (½‚5•§|) ÑŒ±ÏLeZgÐ † “²…”ù‡c3êÆþvkž@ƒ¿, ∗ “Ì §k 1C†z¤1 F/. •´k:“” ´Þ”!“ 5 Ø¤õÅ”½ö“ ØLŠ” ”ù‡c•C ÑºÇq k:p, [ŽƒeØJuy, §zgÑyž ¹ÂÙ¢ÑØgñ. ÖŒU¬ ‡¦z‡š"1 ÄXêÑ u 1. ¿›. 1.1 )‚5•§| 27 y². A •?¿‰½ Ý , ÙŒ • m × n. XJ A z˜ Ñ Üd 0 |¤, =, A ƒ Ü•", @o A ®²´1 F/ (dž•‡‰˜g²… Ð C†, 'Xò1 1 1¦± 1). ¤ ±·‚Œ±b A – k˜ •¹š" ƒ. XJ A 1 1 •¹š" ƒ, @od·‚ƒc ?ØŒ•, ²L˜ Ð 1C†Œ±ò A C ¤Xe/ª:   a11 ∗ · · · ∗     0  A=   ..   . A0 0 Ù¥ a11 6= 0 (XJI‡, ·‚„Œ±õ‰˜gÐ C†¦ a11 = 1), (Ò ∗ 3ùpL«˜ ØI ‡'% Ý ƒ (§‚Œ±´ 0 •Œ±Ø´ 0), A0 ´' A 111 ˜‡fÝ . džN ´• , •‡2é A e¡ m − 1 1‰Ð 1C†¦ A0 C¤1 F/, @o A •ÒC¤ 1 F/. ¤±, ¯KÒ8(•r ê8• Ý A0 z¤1 F/. XJ A 1 1 ƒ Ü´ 0, @o A /X   0  .  . A= B  . 0 Ù¥ B ´í A 1 1 fÝ . dž, •‡ÏLÐ 1C†r B C¤1 F/, Ý A •ÒC¤ 1 F/. Ïd, ù«œ¹e¯KE,8(•r˜‡ ê8î‚~ Ý z¤1 F/. ÏLé A ê‰8B{, ·‚Òy² A ˜½Œ±ÏLÐ 1C†z•1 F/. e¡˜^(Ø•éN´d1 F/ “ý-{ô” : ·K 1.1.19: b (S) ´1 F/ ‚5•§|, (A ; b) ´§ O2Ý (½=, •§| (S) 9 ™•ê k n ‡). r • A š"1ê8. 1. Ø . XêÝ Akn ª r ≤ n o¤á. 2. ‚5•§| (S) k) …= ê8ÚXêÝ ¥š"1 O2Ý (A ; b) ¥š"1 ê8ƒ . 3. b ‚5•§| (S) k). @oÙ)Ø•˜ uÙ ê8, = r < n. ê8•´ r, =, O2Ý ¿©7‡^‡´Ý A ¥š"1 ¥š"1 ê8î‚ …, džØ r ‡š"1 ÄXê¤3 , •{ n − r ‡ éA CþÑŒ±‘¿À ? ¿Š. ¡ù Cþ••§| (S) gdCþ (free variable). gdCþ ê8 (d?= n − r) • ¡••§| (S) gdÝ (degree of freedom). ¤kgdCþ Š e5´˜^mq²… ½ƒ , æ^£“{=Œ•˜(½Ù¦Cþ Ù¢›©k^ íØ 1.1.20: XJ˜‡àg‚5•§| ½kš"). Š. (Ø: ™•êê8î‚Œu•§ê8, @oTàg‚5•§|˜ y². b m ´•§| •§‡ê, n ••§| Cþ‡ê. kò•§|^pdž {z¤1 F/. Ù¥š"1 ‡ê r w, ` m < n, ¤± r < n. (Ød·K 1.1.19 (3) á . u u m. Ï•b ^‡ 1˜Ù 28 ±þ·‚)º XÛÏL1 ){2‰˜ `z. F/û½‚5•§| )85 . • Ý Ú‚5•§| , ·‚é1 F/•§| ±ƒc?ØL •§| (1.1.1.2) •~. ž·‚ 1 F/ (½þn /) XêÝ ƒ m©æ^£“{¦)ˆ‡™•ê Š. y3·‚„Ž2CB˜:. (l,«¿Âþ`, J¦CB´< a©²?Ú rŒíÄåƒ˜.) ·‚Ž, ‡´XêÝ f2Ð˜:, •NŸoÑØ^‰ÒŒ± é •§| ) . ~X, bX•§| XêÝ ´ù‡ f:  3  0 0 @oÃØ Òm> XJk< UŽŽXù 0 5 0  0  0 36 ~ê‘´Ÿo, •§| ¦)Ñ¬š~{ü. Ï•dž   3x = b1   5y = b2    36z = b3 ÜB, s´Äú •§|ÃšÒ´ k ØU= ? ´ÄŽŸ¯: „Œ±2B˜:í ?  1  •¤ 0 0 0 1 0  0  0 ù 1 ˜¢y Q? `Ø½„ýk<£‰: Œ±. Ø ´ , XJ•§| 2 ¡Š•§| ! XêÝ ƒ&·, ŽŽ´˜½‡k . ½ Â 1.1.21. · ‚ ¡ ˜ ‡ Ý • • { 1 reduced form), XJ§÷v±e¤k^‡: 1. TÝ •1 -<u•!{ü §Ñ F / (reduced row echelon form) ½ { ¡ 1 • { / (row ÄXêþ• 1. 3. éuz‡š"1 ÄXê (Šâþ˜^‡§7L e˜‡‚5•§| )$ u (1.1.22) éu®‰ •§| (1.1.22.1) {ü F/. 2. z‡š"1 žÖöá= ê8QŒ± u , @âý O2Ý u 1), §¤3 Ù¦ ƒ Ü• 0. ´1•{/, ·‚Œ±`T•§|´1•{/. egCM°¥2y ê8, qŒ±Œu 1•{/Ý ê8 ! ~f ! ž5¿, 1•{/ ‚5•§|·‚NoòÙz•1•{/Q ? 5w‡äN   x1 + 2x2 − x3 + x4 − x5 − x6 = b1       x3 − 2x5 + 3x6 = b2   x5 − 2x6 = b3     0= b4      0= b5 Ý ~fj. • 1 Xe 1.1 )‚5•§| T•§| O2Ý 29 •  1        (1.1.22.2) 2 −1 1 1 −1 0 −2 1 −1 3 −2 0 0  b1  b2   b3    b4  b5 ùp·‚ ½Ý ˜x? ƒÑ´ 0. 8 ·‚¬²~æ^ù ÎÒ ½, žÖö3¿. XJ b4 = b5 = 0, KÝ (1.1.22.2) Ú •§| (1.1.22.1) Ñ´1 F/. Šâ·K 1.1.19, •§| (1.1.22.1) ˜½k), …k 3 ‡gdCþ ! y3·‚5`², ( b4 = b5 = 0 ž) Œ±ÏLüŒÚ½òÝ (1.1.22.2) z¤1•{/. Äk, ·‚ÏL•g‰1 L1 → L1 + L3 , L2 → L2 + 2L3 Œ± Ý   1 2 −1 1 0 −3 b1 + b3    1 0 0 −1 b2 + 2b3     1 −2 b3      0 0   0 0 e5, ÏL L1 → L1 + L2 ù‡1C†Œ±ò±þÝ z•Xe1•{/Ý   1 2 0 1 0 −4 b1 + b2 + 3b3    1 0 0 −1 b2 + 2b3      (1.1.22.3) 1 −2 b3     0 0   0 0 †±þÝ (1.1.22.4) éA •§|=•   x1 + 2x2 + x4       x3   : − 4x6 = b1 + b2 + 3b3 − x6 = b2 + 2b3 x5 − 2x6 = b3          0= 0 0= 0 Ø š"1ÄXê¤3 , Ù{ éA Cþ• x2 , x4 Ú x6 . ddØJ •, •§| (1.1.22.4) ) (•Ò´ b4 = b5 = 0 ž (1.1.22.1) )) Œ±Lˆ•   x1 = −2t − s + 4w + b1 + b2 + 3b3       x2 = t     x = w + b + 2b 3 2 3 Ù¥ t , s , w Ñ´Œ±?¿ Š ëê.   x = s 4      x5 = 2w + b3     x = w 6 5¿, dg¦)•§Ù¢ØI‡?Û£“Ú½, † éz‡š"1£‘=Œ. 1˜Ù 30 Ý Ú‚5•§| y3·‚b b4 Ú b5 Ø • 0. dž ,•§| (1.1.22.1) Ã). ØL, e••ÄÝ ·‚E,Œ±òÙz•1•{/. Äk, ·‚Ø”b½ b4 = 1 … b5 = 0. (eØ,, ÏL•õügÐ 1C†=ŒòÝ z¤÷vd^‡ f.) ¤±džÝ (1.1.22.2) Ò´   1 2 −1 1 −1 −1 b1    1 0 −2 3 b2     1 −2 b3      0 1  0 0 (1.1.22.2), (1.1.22.2) e5, ·‚Œ±©¤ 3 Úò±þÝ z•1•{/. 1˜Ú, ÏL‰1 L1 → L1 − b1 L4 , L2 → L2 − b2 L4 Ú L3 → L3 − b3 L4 Œ±   1 2 −1 1 −1 −1 0    1 0 −2 3 0    1 −2 0     0 1  0 0 1 Ú, ‰1 L1 → L1 + L3 , L2 → L2 + 2L3 =Œ  1 2 −1 1 0 −3   1 0 0 −1   1 −2   0  0 1nÚ, 2‰1 L1 → L1 + L2 =         (1.1.22.5) ù Ò . c[• Ž‡ 1•{/Ý 1 2 0 1 0 1 0 0 1 . XJØO• ˜ −4 −1 −2 0 0  0  0  0   1 0  0  0  0   1 0 , ù‡Ý Ù¢Ú (1.1.22.3) ¥ Ý ´˜ þ¡~f¥ ½n 1.1.23: ?Û˜‡Ý öŠ•{, ØJo(Ñe¡ (Ø: (½‚5•§|) ÑŒ±ÏLÐ 1C†z¤1•{/. y². Šâ½n 1.1.18, Ø” TÝ (½•§|) ®² z¤1 F/. e,˜š"1 ÄXê• λ (Šâ½Â 1.1.17 1 1 ^, λ 6= 0), òT1¦± λ−1 =Œ¦ÄXêC¤ 1. l•e¡ š"1m©, Åg¦ê¡ š"1•þ‰ž {, Œ±¦z‡Ì ¤3 þ• • 0. Šâ1 F/½Â ¥ 1 2 ‡^‡, Ì e• ƒ•7,Ñ´ 0. ¤±, •ªÝ (½•§|) z•1•{/. <‚~~rÏLÐ 1C†ò‚5•§|½öÝ (Gauss–Jordan elimination, Gauss–Jordan reduction)‡ . ‡ ùp z¤1•{/ L§¡•pd–e ž { e (Jordan) ´ I/nÿþ“ Wilhelm Jordan (1842–1899), ¦†·‚8 ¤Æe IO/ 9 e Ø´ Ó˜<. @˜ e ´{IêÆ[ Camille Jordan (1838–1922). 3ŒÆêÆ;’ ‘§ ¥, ù êÆ[e „¬Ï“e 4-‚½n”!“e ÿÝ”!“e –â ½n” õg³¡. 1.1 )‚5•§| 31 No ? þ¡ù‡½nkvk-Öö Ñ˜«X x¸ ‰Žaú ? s˜ú25, uy˜‡ {‰¾,´ý ! J džsý Œ±››4gC ,¯í ? ,, ,¯ƒ{·‚„´‡‰¢¯ . Äk, 4·‚x-/a ˜e•Œ c êÆ[‚. XJ `J¦CB´<a©²?Ú ˜«rŒíÄå, @oÆ¬± (Ün •ª CBK´<a?Ú ˜‘-‡I“. é (Ün CBå»¿ ¬‰¯<, ù ´êÆ[‚¤l¯ ’!•Œ q" óŠ. ¦‚Ø==‚ù°óŠ [ •, …„³/ù°óŠ5 “˜©Œ ˜©Â¼” @«34Ú÷v. Š•,˜‡•\Ö¢ ¯œ, 4·‚{üo(˜e1•{/ •§|'ƒc˜„ 1 F/ •§|•?˜Ú `:´Ÿo. =, éu1•{/ ‚5•§|, 3gdCþ (ÙÀ •ª®3· K 1.1.19 (3) ¥ ²) ?¿ ½ˆg Šƒ , Ù¦Cþ Š•‡3z‡š"1¥£‘=Œ . –d, ·‚é‚5•§| ?ØÒ‡w˜ãá . ŠO Öö, s„k•{ ¯Kí ? XJk {, sØ”kÕág•˜e. k˜ ¯K ‰Y•N3·‚™5 ÆSL§¥¬g,2ÑY¡5. 1.1.4 SK S K 1.1.1. ^ž {¦)e ‚5•§|:   − x3 + x4 = −3  3x1     2x1 − x2 + x3 − x4 = 1 1.  − 3x4 = 2   2x1 − x2    2x + 2x − 2x + 5x = −6 1 2 3 4   x1 + 2x2 − 3x4 + 2x5 = 1      x1 − x2 − 3x3 + x4 − 3x5 = 2 2.  2x1 − 3x2 + 4x3 − 5x4 + 2x5 = 7      9x − 9x + 6x − 16x + 2x = 25 1 2 3 4 5   5x1 + 5x2 + 2x3 = 2       3x + 2x2 + x3 − x4 = 1   1 3. x1 + 2x2 + 3x3 − x4 = 1     2x1 + 3x2 + x3 + x4 = 1      2x + 2x + 2x − x = 1 1 2 3 4 S K 1.1.2. ^ž {¦)e ‚5•§|:   3x1 + 4x2 − 5x3 + 7x4 = 0      4x1 + 11x2 − 13x3 + 16x4 = 0 1.  7x1 − 2x2 + x3 + 3x4 = 0      2x − 3x + 3x − 2x = 0 1 2 3 4   x1 − 2x2 + 3x3 − 4x4 = 4      x1 + 3x2 + x4 = 1 2.  x2 − x3 + x4 = −3      − 7x + 3x + x = −3 2 3 4 1˜Ù 32 Ý Ú‚5•§|    2x1 − x2 + x3 − 3x4 = 4     2x1 + x2 − x3 + x4 = 1 3.  5x1 + x2 − x3 + 2x4 = −1      3x − 2x + 2x − 3x = 2 1 2 3 4   x1 + 2x2 + x3 − x4 + x5 = −1       x + 3x2 + 5x3 − 4x4 = 1   1 4. x1 + 3x2 + 2x3 − 2x4 + x5 = −1     x1 − 4x2 + x3 + x4 − x5 = 3      x − 2x + x − x − x = 3 1 2 3 4 5 S K 1.1.3. )  a x + a x = 0 11 1 12 2 aij , 1 ≤ i, j ≤ 2 • K ¥~ê. y²: àg‚5•§| a21 x1 + a22 x2 = 0 kš" ¿©7‡^‡´ a11 a22 − a12 a21 6= 0. S K 1.1.4. äe àg‚5•§|´Äkš"):   x1 − x2 = 0      x2 − x3 = 0 1.  x3 − x4 = 0      −x + x4 = 0 1   4x1 + x2 − 3x3 + 6x4 = 0      2x1 + 3x2 − x3 + 5x4 = 0 2.  3x1 − x2 + 2x3 − 7x4 = 0      x − 2x + 4x − 7x = 0 1 2 3 4   − 3x4 = 0  x1 + 3x2     x2 − x3 + x4 = 0 3.    − 7x2 + 3x3 + x4 = 0    x − 2x + 3x − 4x = 0 1 2 3 4   x1 − 2x2 + x3 − x4 + x5 = 0      2x1 + x2 − x3 + 2x4 − 3x5 = 0 4.  3x1 − 2x2 − x3 + x4 − 2x5 = 0      10x − 7x + 5x − 6x − 3x = 0 1 2 3 4 5 S K 1.1.5. b ü‡Cþ x, y ÷v,«¼ê'X y = f (x), ¿…®•±eéA'X: f (1) = 2 , f (2) = 7 , f (3) = 16 , f (4) = 29 . Á¯ y ´ÄŒU´ x S K 1.1.6. 3²¡† g¼ê ? e´, Á¦Ñ÷v‡¦ ‹IX¥, b l1 : x + y = 1 ; g¼ê, eÄ, ž)º•ŸoØŒU. kn^†‚ l1 , l2 , l3 l2 : 3x − y = 1 ; •§Xe: l3 : 4x − 10y = −3 1.1 )‚5•§| 33 1. þãn^†‚´Äkú 2. : ? kõ {UC†‚ l3 :. •§¥ x ½ y Ù¥ a •ëê. ?Ø a XÛ S K 1.1.7. 3²¡† ú :? Xê, †‚ l4 Šž, n^†‚ l1 , l2 , l3 kú ‹IX¥, b kn^†‚ l1 , l2 , l3 l1 : 2x − y = 1 ; l2 : −x + y = −2 ; Ù¥ a •ëê. ?Ø a XÛ •§, ¦ l1 , l2 , l4 n^†‚vkú :. •§Xe: l3 : 4x − ay = −1 Šž, n^†‚ l1 , l2 , l3 kú :. S K 1.1.8. •Äàg‚5•§|    2x + y + z  ax − z    −x + 3z (½Ù¥ëê a ∈ K   ax + x2 + x3   1 1. x1 + ax2 + x3    x1 + x2 + ax3 k) = 0 = 0 = z‡‚5•§|, ?Øëê a, b = a = a2 = 4 = 3 = 4 = 1 = a = 3 = b a1 , · · · , a5 ∈ K. y²: •§|   x1 − x2       x2 − x3   x3 − x4     x4 − x5      −x + x5 1 ¿©7‡^‡´ ŸoŠž•§|k), ¿3k 1   x1 + x2 + x3 + x4 + x5      3x1 + 2x2 + x3 + x4 − 3x5 3.  x2 + 2x3 + 2x4 + 6x5      5x1 + 4x2 + 3x3 + 3x4 − x5 S K 1.1.10. 0 Š, ¦T•§|kš"), ¿¦ÑÙ¤k). S K 1.1.9. a, b ∈ K. ée )ž¦Ñ•§| ¤k).   ax + x2 + x3   1 2. x1 + bx2 + x3    x1 + 2bx2 + x3 = P5 i=1 ai = 0. 3k) = a1 = a2 = a3 = a4 = a5 œ¹e, ¦ÑÙÏ). 1˜Ù 34 S K 1.1.11. ¦±eàg‚5•§| Ï):   x1 + x2       x2 + x3      x3 + x4  ··· ··· ··· ··· ··· ··· ··· ···       xn−1 + xn     x + xn 1 S K 1.1.12. = 0 = 0 = 0 Ý Ú‚5•§| ··· ··· = 0 = 0 a, b1 , · · · , bn ∈ K. ¦e¡‚5•§| Ï):   x2 + x3 + x4 + · · · + xn−1 + xn = b1       x1 + ax3 + ax4 + · · · + axn−1 + axn = b2      x1 + ax2 + ax4 + · · · + axn−1 + axn = b3  ······ ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ···       x1 + ax2 + ax3 + ax4 + · · · + + axn = bn−1      x + ax + ax + ax + · · · + ax = bn 1 2 3 4 n−1 S K 1.1.13. ‰½~ê a1 , · · · , an ∈ K. ) ‚5•§| Ï): xij , 1 ≤ i, j ≤ n ´ n2 ‡™•ê. ¦±e (n4 ‡•§ Ù¥ i, j, k, l ©O ai al xjk − aj ak xil = 0 , 1.2 Ý ¤ H 1, 2, · · · , n . Žâ ·‚c¡Ú\Ý ´• 9Ïn)‚5•§| nØ, Ì‡¦^Ý 5‰Ð 1C†. !· ‚òrÝ Š•Õá ïÄé–, •Ý ½ÂÜn $Ž. ·‚¬w , 3· ^‡eÝ $Ž •U¦ŒUõ/÷vÚê˜ {K. ,, ·‚•¬lÝ $Ž5Æ¥w ˜ êi-.wØ y–. !¥·‚± K 5L«Eê• C ˜‡f• (N¹ § A.1 !), •Ò´`, K Š• C f8´' u\~¦ØoK$Ž´µ4 , =, •‡ a, b ∈ K, @o a + b, a − b, ab Ú ab−1 (ab−1 •3 b 6= 0 ž •Ä) Eáu K. XJÖö„Ø ÙG• Vg, Œ±kb K ´knê8 Q, ¢ê8 R ½Eê8 C nöƒ˜. 1.2.1 Ý Ä (1.2.1) é?¿ $Ž ê m Ú n, ·‚P Mm×n (K) := A = (aij )1≤i≤m éz‡ i Ú j þk aij ∈ K . 1≤j≤n =, Mm×n (K) ´z‡XêÑ5gu K ¤k m × n Ý ¤ 8Ü. ±eÑ´˜ Ò: Mn (K) := Mn×n (K) ; K n×1 = Mn×1 (K) ; K 1×n = M1×n (K) . =, Mn (K) ´Xê …ü •Ý• n ~^ {zP Ü5gu K n • N, K n×1 ½ K 1×n K©O´Xê Ü5gu K ¿ •þ½1•þ N. ·‚Ø¬·^1•þÚ •þž, •~~Ø«© K n×1 1.2 Ý Žâ 35 Ú K 1×n , ò§‚Ú˜P• K n . Ù¥ ƒŠâ1©•B5À 1•þ½ •þ /ª, ¿…Œ± <Ú/¡• K þ (ü ) •Ý• n •þ† (vector of length n over K) ½ö n ‘•þ (n-dimensional vector over K). e α = (a1 , . . . , an ) Ú β = (b1 , . . . , bn ) Ñ´ K þ n ‘•þ, = α, β ∈ K n , §‚ Ú† AT ´ÖöÙG . Ù½Âª• α + β := (a1 + b1 , . . . , an + bn ) α − β := (a1 − b1 , . . . , an − bn ) . •Ò´`, ƒÓü •Ý (1½ ) •þƒm \~{ÒUì‹I©þ3ƒéA ˜‰\~{ n 5½Â. , , e α = (a1 , . . . , an ) ∈ K c ∈ K, KŒ±½Â~ê c †•þ αƒm ¦{ c.α = c.(a1 , . . . , an ) := (ca1 , . . . , can ) . ù‡¦{¥·‚¿Ø3¿~ê c Ú•þ α †müS. Ïd•Œ± ¤ α.c = (a1 c, . . . , an c). ~êÚ •þƒm ù«$Ž¡•ê¦ (scalar multiplication). – éu n = 2 Ú n = 3 œ¹, Ööéu•þ\~{±9•þ ~ê (=ê¦$Ž (J) AT®²' ÙG¿…k Ôn½AÛþ a*. 1•þÚ •þ•ØL´Ý A~. ¤±éN´Ž ±þ$ŽŒ±í2 ˜„Ý œ¹. e A, B ∈ Mm×n (K), Ù¥ A 1 (i, j) ƒP• aij , B 1 (i, j) ƒP• bij , K A † B Ú (sum) † (difference) ©O½Â• A + B = (aij + bij ) , A − B = (aij − bij ) . =, Ý A + B 1 (i, j) ƒ´ aij + bij , Ý A − B 1 (i, j) e c ∈ K, A = (aij ) ∈ Mm×n (K), Kê¦ (scalar multiplication) cA ƒ´ aij − bij . ½Â• cA = (caij ) . =, Ý cA 1 (i, j) ƒ´ caij . Ï•~{Œ±@•´\þ (−1) , ¤±§Œ± @•´\{Úê¦û)Ñ5 $Ž. ò\{Úê¦·Üå5, Œ± ‚5|Ü Vg. O(5`, e A1 , . . . , Ar ∈ Mm×n (K), Ké ?¿~ê c1 , . . . , cr ∈ K, UìLˆª c1 A1 + · · · + cr Ar Œ± Mm×n (K) ¥˜‡ ƒ, ¡ƒ• A1 , . . . , Ar ù|Ý ˜‡‚5|Ü (linear combination), ½ö• rNXê 5 , ¡•§‚ ˜‡ K-‚5|Ü. AO/, é?¿k•õ‡1•þ½ •þ, ·‚k‚5|Ü Vg. …ù‡Vg3ƒc ½Â 1.1.3 ¥¢Ÿþ®²ÑyL . ~ 1.2.2. ‰‚5|Ü´d®‰ Ý ½•þ )•õÝ ü‡•þ e1 = (1, 0) Ú e2 = (0, 1) ƒ , ÏL e1 Ú e2 ¤k•þ, Ï• ½•þ -‡•ª. ~X, 3 R2 ¥ ½ ¤k¢Xê‚5|ÜŒ± R2 ¥ (x, y) = (x, 0) + (0, y) = x.e1 + y.e2 . † ·‚kž¬^“ü •Ý”ù‡ ¡5“O“•Ý”ù‡{¡, cÙ´3˜ AOI‡c[O „` |Ü. 3Ôn½A ÛÆ¥, <‚Ï~•¬r“•þ”˜cn)•˜«Qk••qkŒ þ, aqu“k•‚ã”. Ö¥, ·‚ …^“AÛ• þ”½ö“¥þ”5¡ @«¿Âe •þ. 'u¥þ,k˜‡Ýþ¿Âe “•Ý” (•~¡•“ •”) Vg, =, ±kS¢ê q | (a1 , · · · , an ) •‹I ¥þ•Ý´ a21 + · · · + a2n . ù‡•ÝVgw,†·‚ùp¤¢ “ü •Ý” ØÎ. ¤±, òü •Ý{¡••Ýq ¬ )˜ ÜÂ. 3Ð, 3¢SA^¥, ·‚A o´Œ±lþe©¥«©Ñ·‚¦^ “•Ý”˜c ´=«¹Â. ¤±ÖöŒŒØ7•dÅ(. 36 aq/, XJ38Ü M2 (R) ¥ ½Ý ! 1 0 0 E11 = , E12 = 0 0 0 K M2 (R) ¥?Û˜‡Ý (1.2.3) b ! 1 , E21 = 0 0 1 0 0 Ñ´ E11 , E12 , E21 Ú E22 ùo‡Ý A ∈ Mm×n (K). ò A 1˜Ù Ý 0 0 ! ! , E22 = 0 1 Ú‚5•§| , ‚5|Ü. ¤k À• •þ8Ü K m×1 ¥     a1n a11      a2n   a21     c1 (A) =  .  , · · · , cn (A) =  .  .  ..   ..  amn am1 ƒ, ©OP• · ‚ ¡ c1 (A), . . . , cn (A) ù | • þ • Ý A • þ | (system of column vectors of A), ò c1 (A), . . . , cn (A) ù | • þ ¤ k K-‚ 5 | Ü ˜ 3 ˜ å ¤ ˜ ‡ 8 Ü, P • CK (A) (½ • { ü /, C (A)), ¡• A ˜m (column space). aq/, ·‚Œ±½Â A 1•þ| (system of row vectors of A) r1 (A), . . . , rm (A), Ù¥ r1 (A) = (a11 , a12 , . . . , a1n ) , · · · , rm (A) = (am1 , am2 , · · · amn ) ∈ K 1×n . ^ RK (A) (½•{ü/, R(A)) 5L«ù|1•þ ¤k K-‚5|Ü ¤ 8Ü, ¡• A 1˜m (row space). • •B8 3Ý 1Ú ƒm=†À , ·‚½Â A ∈ Mm×n (K) =˜ (transpose) •X e•ª n × m Ý AT : Ý AT 1 (j, i) ƒ uÝ A 1 (i, j) ƒ. •Ò´`, AT ´˜‡ n × m Ý , § 1˜ Ò´ 5Ý A 1˜1Uì •þ •ªLˆÑ5 (½ö/ –/`¤=L5), AT Ù¦ •±aq •ª(½. ¤±, AT 1 Ò´ A 1 1çX Ñ 5, Xd . w,, XJ Ñî•Úç•Ö •ª O {, =˜Ý AT (½1) •þ|Œ± @•Ú 5Ý A 1 (½ ) •þ|´˜ . k ÖŒUæ^Ù¦ PÒL«Ý =˜, ~X [22] ˜Ö¥ AT P• A0 . T XJ˜‡Ý A ÷v A = A , K¡ A •é¡Ý (symmetric matrix). XJÝ A ÷ v A = −AT , K ¡ A • ‡ é ¡ Ý (anti-symmetric matrix) ½ é ¡ Ý (skew-symmetric matrix). Šâ½Â, é¡Ý ½‡é¡Ý Ñ7L´• . e5·‚•ÄÝ ÚÝ ƒm´Äk¦{. éuü‡Ó Œ Ý A = (aij ) ∈ Mm×n (K) Ú B = (bij ) ∈ Mm×n (K), Œ [ , Œ ± r 1 (i, j) ƒ • aij bij Ý Š A Ú B ˜ «“¦ È”. ù « ¦ È ¡ • Hadamard ¦ È ‡ § (Hadamard product) ½ö Schur ¦È (Schur product). ù«¦È3˜ A½|Ü•k˜½ ^?. ¢‚y², ù«¦{½ÂªÄ„´w ”Å ª •, §˜•¡Ø2”•þ \{Úê¦@ k† Ôn½AÛ¿Â, ,˜•¡•ØU¿©/NyÝ nØ %å, Ã{ )š~´L êÆ #SN. ¤±, êÆ[‚¿vkæ^þãù«•,{ü Ø ¢^ •ª5½ÂÝ ¦{. @oý •¢^•k¿Â Ý ¦{TXÛ½ÂQ ? ù‡¯K ‰Y¿Ø›©w,. 3ùp, ·‚6žl˜‡ (ŒUØ›© Ÿ ) ý¡5)ºù‡¯K. (8 ÆS ‚5N Vg , ÖöØ ”2£Þg•ù‡¯K.) ‡ Jacques Salomon Hadamard (1865–1963), {IêÆ[, cÙ±y² § Issai Schur (1875–1941), Ï•¦Ñ)3 k<@•¦´ IêÆ[. I, ‘Äz‰@•¦´ ƒê½n ª¶. IêÆ[. Ï•¦ ˜)Œõ3 IóŠ)¹, • 1.2 Ý Žâ 37 3 ¥ Æ Ô n ½ ‹ I A Û ¥, Ö ö Œ U „ L • þ † • þ ƒ m I þ È (• ¡ ê þ È) (scalar product) ½SÈ (inner product) q½ö:È (dot product). ù«$Ž éü‡ü •ÝƒÓ 1• þ (½ •þ) 5‰1, $Ž (J¿Ø´˜‡•þ, ´˜‡ê. äN5`, e α = (a1 , . . . , an ), β = (b1 , . . . , bn ), @o§‚ IþÈ α · β ½Â• α · β = a1 b1 + · · · + an bn . 3ù‡½Âª¥, † α Ú β ˜Ø¬UC$Ž (J. Ø †ÔnÚAÛ éXƒ , l‚5 •§ Ý5w, ù‡½Â3/ªþŒ±‰·‚"À‚5•§ ,˜«ú1. ¯¢þ, éu˜‡/X a1 x1 + · · · + an xn = b ‚ 5 • §, Ò † > Œ ± @ • ´ ± ~ ê • ‹ I • þ α = (a1 , . . . , an ) Ú ± C þ • ‹ I • þ X = (x1 , . . . , xn ) ƒm IþÈ. •N´• wÑ~þÚCþ / O, <‚Ž r~ê•þ ¤1 f, rCþ ¤ •þ ¤ f. ù ·‚ÒŠâIþÈ ½Â ˜«†>˜˜˜‡1•þ!m>˜˜˜ ‡ƒÓü •Ý •þ ¦{$Ž, Ù$Ž(J´˜‡ê. †óƒ, 3ü •ÝƒÓ œ¹e, ˜ 13†˜ 3mŒ±¢–˜«¦{ ˜‡ê. ìù‡g´ í2˜e, XJ†>˜˜ m 1, m>E•˜˜˜ , @o (3z‡1Ñ†T ü •ÝƒÓž) ·‚Œ±4†>z˜1ÑÚm> ‰¦{, ù Œ± m ‡ê. ù m ‡êg,/ü ¤ m 1, 3˜å ¤˜‡ü •Ý• m •þ. æ^Ý •ª)ºù‡öŠ, Œ±`XJ A ´‡ m × n Ý , X ´ n × 1 Ý (½= n ‘ •þ), @o¦È$Ž AX Œ±½Â, ¿…Ù$Ž(J´˜‡ m × 1 Ý (½= m ‘ •þ). e5, XJ3Ý A ∈ Mm×n (K) m>˜˜˜‡ n × p Ý (=l†–m Ñ p ‡ü • Ý• n •þ), @o A †m>Ý z˜ •g‰c¡¤` ¦{, ©O ˜‡ m ‘ •þ. ù p ‡$Ž(J†mümÒ ˜‡ m × p Ý . ¯¢y², ùÒ´·‚éÝ ½Â¦{ (g´. 3c© (0.2.3) ˜Œã¥·‚®²w Uì ù«¦{Œ±† ò˜‡‚5•§| ¤Ý ¦{ /ª. éu·‚fuy Ý ¦{ ó, ùŒ U´Ù¢^5 1˜‡~y. o(±þ?Ø·‚‰Xe½Â: ½ Â 1.2.4. A ∈ Mm×n (K), B ∈ Mn×p (K). ½ÂÝ ¦È!¦ÈÝ (matrix product, product matrix) AB •Xe•ª(½ m × p Ý : éu?¿ i ∈ [[1, m]] Ú?¿ j ∈ [[1, p]], AB 1 (i, j) ƒdLˆª ai1 b1j + ai2 b2j + · · · + ain bnj ‰ Ñ. Ï • K ' u \ { Ú ¦ { $ Ž µ 4, ¤ ± ¦ È Ý AB ¤ k X ê • á u K. • Ò ´ `, Mm×n (K) ¥Ý Ú Mn×p (K) ¥Ý ƒ¦ (JÑáu Mm×p (K). 5P 1.2.5. 3½Â 1.2.4 ¥, ·‚½ÂÝ ¦È AB ž‡¦†>Ý A ê8 um>Ý B 1ê8. ù‡^‡Ø ÷vž, ·‚`Ý ¦È AB vk¿Â!Ø•3½öÃ{½Â. âd·‚é NúyÝ ¦{Úê ¦{1˜:wÍØÓ: Lˆª AB ˜„‡•6uÝ A Ú B †m^S. Ï~±e 3 «œ¹ÑŒUÑy (žÖö}Á•z«œ¹éÑäN ¢~): 1. ¦ÈÝ AB k¿Â, BA vk¿Â, ½ö‡L5. 2. ¦ÈÝ AB Ú BA Ñk¿Â, ´ùü‡¦ÈÝ ´Œ ØÓ Ý . 3. ¦ÈÝ AB Ú BA Ñk¿Â, …ùü‡¦ÈÝ Œ ƒÓ, ´ AB 6= BA. ( éu1 3 «œ¹, A Ú B ˜½´Œ ƒÓ • .) 1˜Ù 38 Šâ±þy–, ·‚`Ý ¦{Øä Ú BA Ñk¿Â, •kŒU AB 6= BA. ˜„/, XJ A Ú B ´Œ ƒÓ • • AB = BA. †Æ (commutativity), =˜„ , ·‚¡ A Ú B Œ Ý Ú‚5•§| ó, =¦Ý ¦È AB † (commute with each other) ´ (1.2.6) y33Ú\˜ AÏÝ ½Â. ù Ý éuc¡®²0 Ý $Ž5`ˆüXAÏ Ú. Äk, ·‚± Om×n L«Xê Ü´ 0 m × n Ý , ¿¡ƒ•Œ • m × n "Ý (zero matrix). ùp PÒ¥ O ´=©i1, æ^ù‡i1ŠPÒ Ì‡nd´§Úêi 0 • ”. ¢S þ, éõÖ•^ 0m×n 5L«ù‡Ý , •Ò´`Zy^çÚoN êi 0 “Oi1 O. ·‚kž• ¬æ^ù«PÒ. ùp eI m × n ´ûrN¤ 9 "Ý Œ XÛ. XJù‡&Eéþe© 5`¿Ø-‡, ½ölþe©¥éN´• , @o·‚•Œ±† ±i1 O ½çÚoN " 0 5L «"Ý . ØL, 8 ·‚•~^ ˜«‰{ (•´êÆ 2• É ‰{) ´Œ±† æ^êi 0 5L«"Ý . –ÖöÙG Ý $Ž¿…U n)"Ý ý ^åƒ , F"U ·Aù«•, Ø î> ÃúŒä ·^PÒ‰{. w,, éuÝ \{ ó, "Ý ˆüXêi 0 3ê \{¥¤ˆü Ú. =, (1.2.6.1) Om×n + A = A + Om×n = A ¡˜‡ n 0. ·‚~r n • é (1.2.6.2) é¤k A ∈ Mm×n (K) þ¤á. D •é Ý (diagonal matrix), XJÝ Ý P¤Xe/ª:   λ1   ..  D= .   λn D ¥é ƒ Ù¦ ƒÑ´ (Ù¥Ý ˜x?@• ƒÑ´ 0.) é Ñ u1 n é P• In , ¡ƒ• n ü Ý ½ð Ý (identity matrix of order n). k Ö (~X [22]) ±i1 E “Oùp I 5L«ü Ý (l ·‚ In P¤ En ). éu?¿ m × n Ý A Ú?¿ n × p Ý B, (1.2.6.2) ¥Ý D OŽ AD Ú DB Œ±uy, AD 1 j u A 1 j ¦± λj , DB 1 i 1 u λi ¦± B 1 i 1. AO/, (1.2.6.3) AIn = A , In B = B é¤k A ∈ Mm×n (K) Ú¤k B ∈ Mn×p (K) þ¤á. ù`²éuÝ ¦{ ó, ü Ý In ˆü Ú XÓêi 1 3ê ¦{¥ Ú. ,, •ØJ y: "Ý 3Ý ¦{¥ Š^†êi 0 3ê ¦{¥ Šâq, =, éu¤k A ∈ Mm×n (K) þk AOn×p = Om×p , Op×m A = Op×n . éu?¿ λ ∈ K, Ý λIn ´é Ñ uλ n é . ù Ý ¡•XþÝ þÝ (scalar matrix). ØJ y, é?Û A ∈ Mm×n (K) Ú B ∈ Mn×p (K), þk A(λIn ) = λA = (λA)In , (λIn )B = λB = In (λB) . ·‚òÝ ƒ Ñ u 0 ½ 1 Ý ¡•"˜Ý (zero-one matrix). "Ý ,´"˜Ý . , , éu?¿ k ∈ [[1, m]] Ú l ∈ [[1, n]], P (1.2.6.4) (m×n) Ekl := k…=k1 (k, l) ½ê ƒš" m × n "˜Ý . Úü Ý 1.2 Ý Žâ 39 (m×n) •Ò´`, Ekl ù‡ m × n Ý ¥•˜ š" ƒ u1 (k, l) , ¿…T? ƒ u 1. ·‚¡ (m×n) Ekl ù Ý •Œ • m × n Ä "˜Ý (basic zero-one matrices). Ý Œ Ø7‡² (m×n) (rN½ölþe©† Œ•ž, ·‚~ò Ekl {P• Ekl . Ä "˜Ý ˜‡-‡Š^´§‚ ‚5|ÜŒ±‰Ñ¤kƒÓŒ Ý . =, éu?¿ A = (aij ) ∈ Mm×n (K), ·‚k A = a11 E11 + a12 E12 + · · · + a1n E1n +a21 E21 + a22 E22 · · · + a2n E2n (1.2.6.5) +········· +am1 Em1 + am2 Em2 + · · · + amn Emn é?¿ C ∈ Ms×m (K). ÏLOŽŒ±uy 11 ··· ··· 1l 0 0 ··· ··· 0 0 0 0 ··· ··· 0 0 c1k c2k .. . 1k , Ù§ 111 ··· .. . 0 .. . ··· .. . 0 .. . ··· bl2 ··· .. . 0 ··· 0 .. . ··· ··· ··· .. . 0 blp 0 .. . ··· 0 ··· 0    CEkl =    (1.2.6.6) =, Ý CEkl 1l •C  0  ..  .   0   1 k 1  bl1   0  .  .  . 1n1 0 (1.2.6.7) csk ··· 1n 0 0 ··· ··· 0 0 0 0 ··· ··· 0 0      Ñ´". aq/, éu?¿ B ∈ Mn×p (K), OŽŒ• =, Ý Ekl B 1 k 1• B 1 l 1, Ù§1Ñ´". ±þ (1.2.6.6) ˜‡A~´  E (s×n) (s×m) (m×n) il (1.2.6.8) Eij Ekl = Os×n Ù¥ i ∈ [[1, s]], j, k ∈ [[1, m]], l ∈ [[1, n]]. •NùpŠ J˜eêÆþ˜‡~^         = Ekl B .      ej=k e j 6= k PÒ. éuü‡^±ŠeI  1 e i = j δij = 0 e i 6= j (1.2.6.9)  i1 i, j, ·‚P ù‡PÒ¡• Kronecker delta ÎÒ (Kronecker delta symbol)¶ æ^ù‡PÒŒ±ò (1.2.6.8) U (s×m) (m×n) (s×n) Eij Ekl = δjk Eil . ·K 1.2.7: b A, B, C Ñ´Xê5gu K Œ ¦ƒ' Ý $Žk¿Â. K: ¶ d¶¡5 u Ý IêÆ[ Leopold Kronecker (1823–1891) • , λ ∈ K. ±eØä¥ob½ A, B, C äkÜ· 6¼. 1˜Ù 40 1. ¦{†\{ƒm © Ý Ú‚5•§| Æ (distributive law): (A + B)C = AC + BC, C(A + B) = CA + CB. 2. λ(AB) = (λA)B = A(λB). 3. (A + B)T = AT + B T , (λA)T = λAT , (AB)T = B T AT . y². y²3‰Öö. (½ö‘,Ç‘ž ã.) í Ø 1.2.8: b A1 , . . . , Ar Ú B, C • Œ λ1 , . . . , λr ∈ K. K: Ü· Ý ¦ ±eØä¥ Šâ (1.2.6.5), (1.2.6.6) Ú (1.2.6.7), éN´líØ 1.2.8 XeíØ: Ý $ Ž k ¿ Â. 1. (λ1 A1 + · · · + λr Ar )B = λ1 A1 B + · · · + λr Ar B. 2. C(λ1 A1 + · · · + λr Ar ) = λ1 CA1 + · · · + λr CAr . y². nÜ·K 1.2.7 íØ 1.2.9: b Ý (1) Ú (2), ±{ü 8B{Œ . ¦È AB k¿Â. K: 1. AB z˜1Ñ´ B 1•þ|ƒ‚5|Ü. †óƒ, AB óƒ, ¦ÈÝ 1˜m•¹um>Ý 1˜mƒS. z˜1Ñ3 B 2. AB z˜ Ñ´ A •þ|ƒ‚5|Ü. †óƒ, AB óƒ, ¦ÈÝ ˜m•¹u†>Ý ˜mƒS. z˜ 'u1˜mÚ ˜m 1˜m¥. 2† Ñ3 A ˜m¥. 2† ½ÂŒ£w (1.2.3). y². ùp•‰Ñ (1) y², (2) y²3‰ÖööS. ·‚ò¦ê¡˜‡¯¢ (§Úg•K 1.2 (Ø Ÿþ´˜ ): XJ1•þ| β1 , · · · , βs ¥ z˜‡Ñ´1•þ| α1 , · · · , αn ‚5|Ü, 1•þ γ ´ β1 , · · · , βs ‚5|Ü, @o γ • ´ α1 , · · · , αn ‚5|Ü. 3e¡ ?Ø¥, éu˜‡Ý M , ·‚^ r1 (M ), r2 (M ), · · · 5L« M 1 1 1, 1 2 1 . A ´ m × n Ý . c¡·‚J , A Œ± ¤Ä "˜Ý Eij , 1 ≤ i ≤ m, 1 ≤ j ≤ n ‚5 |Ü. ŠâíØ 1.2.8 Œ•, AB ´ Eij B ù Ý ‚5|Ü. XJ·‚* AB ,˜1, 'X1 1 1 r1 (AB), @oN´wÑ§´ Eij B ù Ý 1 1 1 r1 (Ei jB) ƒ‚5|Ü. y3·‚òc¡J ¯¢Aûeãœ¹: γ = r1 (AB) , α1 = r1 (B) , · · · , αn = rn (B) , β1 , · · · , βs • r1 (Eij ), 1 ≤ i ≤ m, 1 ≤ j ≤ n ù 1•þ . ù · ‚ • , ‡ y ² r1 (AB) ´ r1 (B), · · · , rn (B) ‚ 5 | Ü, • I ‡ y z ˜ ‡ r1 (Eij ) Ñ ´ r1 (B), · · · , rn (B) ‚5|Ü. • ùéØä´w, , Ï•Šâ (1.2.6.7) ª OŽ(J, r1 (Eij B) ‡o• 0 ‡oÒ´ rj (B). ·K 1.2.9 ¢Sþ•Œ±éN´/ÏL©¬Ý $ŽwÑ, ù«•{·‚ƒ Ô˜˜Ä˜t˜ej. Ý ¦ÈØ7a, †1m ‡Ó•. eò(J1ƒ', m ˜m A1. ¬J9. 1.2 Ý Žâ 41 cüé¢Ý ƒ¦{L †>Ý z˜1†m>Ý 1˜m ƒm>Ý ƒ1˜m•• . z˜ ü •ÝƒÓ. üé¢¦ÈÝ ƒ (1.2.10) c¡·‚w , Ý m \{$ŽÙ¢Úê \{A ÷vƒÓ 5Ÿ, Ý ¦{•,˜ „Øä †Æ (ë„5P 1.2.5), §Ú\{ƒmE,”ê˜ äk© Æ (X·K 1.2.7 (1) ¤ ã). , , Ý Úêƒm ¦{•Úê†êƒm ¦{$Ž5KA Ã . ¤±·‚Œ±½Â• õ‘ª. äN5`, b A ∈ Mn (K). éz‡ ê r, ·‚^48/•ª½Â A r g˜ (r-th power) Xe: A1 := A , (1.2.10.1) 2Ö¿˜^ r > 1 ž , Ar := Ar−1 A . ½: A0 = I n . (1.2.10.2) ù , XJ f (X) = cr X r + cr−1 X r−1 + · · · + c1 X + c0 •˜‡'uC X õ‘ªk , Ù¥Xê c0 , . . . , cr þ5gu K, @o·‚Œ±ò X ±Ý \” f Lˆª ˜‡½Â: A ““ f (A) := cr Ar + cr−1 Ar−1 + · · · + c1 A + c0 In . (1.2.10.3) 5¿, ùp~ê‘ c0 3Ý “\žC¤ c0 In , ÄK†c¡ Ý ØUƒ\. ù Jr c0 w¤ c0 X 0 , 2ò X O†• n • A žÒŠâ (1.2.10.2) c0 In . e5·‚y²Ý ¦{ ‰ Ün53u, X ˜^-‡5Ÿ. ·K 1.2.11: Ý ¦{÷v(ÜÆ (associativity). =, éu?¿ A ∈ Ms×m (K), B ∈ Mm×n (K), C ∈ Mn×p (K), þk (AB)C = A(BC). y². Šâ (1.2.6.5), Ý A oŒ± ¤Ä "˜Ý ƒ‚5|Ü. ŠâíØ 1.2.8, “‚5|Ü”†“Ý ¦{”ùü«öŠŒ±Ø«©‰1^S ƒÓ (J. ¤±, XJ A = λ1 E1 + · · · + λN EN , Ù ¥ E1 , . . . , EN þ•Ä "˜Ý , λ1 , . . . , λN þ• K ¥~ê, K (AB)C = λ1 (E1 B)C + λ2 (E2 B)C + · · · + λN (EN B)C A(BC) = λ1 E1 (BC) + λ2 E2 (BC) + · · · + λN EN (BC) . ¤±, ‡ y (AB)C = A(BC), •I‡ yéz‡ Ei þk (Ei B)C = Ei (BC). ùƒ –yƒ(Ø, ·‚•I‡•Ä A ´Ä "˜Ý œ¹. Ó n, Ý B Ú C •Œ± b •Ä "˜Ý . ¤±·‚Ø” (s×m) Ù¥ i, j, k, l, u, v þ•T (s×m) Eij –d·K (n×p) , C = Euv A = Eij , B = Ekl Š‰ŒS •I. Šâ (1.2.6.8), ÏL¦^ Kronecker delta ÎÒŒ (m×n) Ekl (m×n) (s×p) (n×p) Euv = δjk δlu Eiv (s×m) = Eij (m×n) Ekl (n×p) Euv . y. k ÖöŒ±6ž±•Èƒ u`, éu¤ •ªn)“õ‘ª” Vg. •\°( ½Âò3 (2.3.22) ˜ã20 . 1˜Ù 42 Ý Ú‚5•§| ±þy²•{Ø„ ´•{'² y²•{. ·‚@•, |^“‚5”A ò˜„¤á ( Øƒy²z8 ˜ •{üÝ œ¹, ù«gŽAT´‚5“ê¯K -‡)KgŽ. ¤±, ±þ y{Š Öö[%N¬. ·K 1.2.11 – „k1 ^5Æ (ë„N¹ § A.3). «{' y². ØLù«y²I‡ÖökÆS˜eë\ÎÒ Σ ·K 1.2.11 ƒy{ . 3–y ª¥, ÃØ†>„´m>, •ª$Ž(JÑ´ s × p Ý ‡éz‡ i ∈ [[1, s]] Úz‡ l ∈ [[1, p]] 5 y: Ý (AB)C Ú A(BC) 1 (i, l) ƒƒ . ¤±• . –¦ (AB)C 1 (i, l) ƒ, k‡ŽÑÝ AB 1 i 1. éu?¿ k ∈ [[1, n]], AB Pm (i, k) ƒ• j=1 aij bjk . (ùp, ·‚^‘eI i1 a L«Ý A ¥éA ˜ Ý ƒ. éu B ½æâqPÒ.) ù , (AB)C 1 (i, l) ƒdeª‰Ñ n X ©ÛŒ•, A(BC) j=1 1 (i, l) m X ƒdeª‰Ñ n X aij j=1 Öö•‡• K y. 1.2.2 ë\ÎÒ Σ Œ_Ý ÚÐ ¹ÂÚÄ 1   m X  aij bjk  ckl . k=1 †±þaq ¦ ! bjk ckl . k=1 5ŸBŒ²x±þü‡ (wqE, ) ¦Úªƒ . ¤±· Ý (1.2.12) ·‚Q3 (0.2.3) ˜ãJL, Ú\ Ý ¦{ƒ , ‚5•§|   a11 x1 + · · · + a1n xn = b1     a x + · · · + a x = b 21 1 2n n 2   ··· ··· ··· ···     am1 x1 + · · · + amn xn = bm =ŒU ¤Ý /ª AX = b, Ù¥  a11   a21 A=  ..  . am1 a12 a22 ··· am2 ··· ··· .. . ··· ··· ··· ··· ···  a1n  a2n  ..   , .  amn   x1    x2   X=  ..  ,  .  xn   b1    b2   b=  ..   .  bm ·‚q• , n ü Ý In Œ±÷v In X = X. ¤±, XJ•3˜‡ n × m Ý P ¦ P A = In (ùƒ u` A 'uÝ ¦{Œ±3†>k˜‡_), @oŠâÝ ¦{ (ÜÆ (·K 1.2.11) Œ• (1.2.12.1) AX = b =⇒ P (AX) = P b ⇐⇒ (P A)X = P b ⇐⇒ In X = P b ⇐⇒ X = P b . ¤±3ù«^‡e, •§| AX = b )XJ•3, @o7,•k•˜ ), = X = P b. 1.2 Ý Žâ ùpÖöI‡3¿, Ï•Ý AP = Im . ~X, e 43 ¦{vk  1  A = 0 2 †Æ, ==l P A = In ù‡^‡˜„  0  1 , 3 P = 1 0 0 1 ó´Ã{íÑ ! 0 0 K P A = I2 AP 6= I3 . ØL, Û ˜‡¯œ´, XJ A (Ú P ) ´ n • , ·‚ò¬y² P A = In Ú AP = In ùü ‡^‡Œ±pƒíÑ (ë„·K 1.2.15 ÚíØ 1.2.29). , , ÖöI‡3¿ ´, 3 (1.2.12.1) ª¥, ·‚l AX = b íÑ X = P b, ‡L5 í K™7¤á. ù´Ï•3 A Ø´• ž, kŒU AP 6= Im . ¤±˜„5`, X = P b ž, ·‚OŽ AX = A(P b) = (AP )b (JkŒUØ u Im b = b. †óƒ, ·‚• •3 P ¦ P A = In ž, ¿ØUä½ AX = b ˜½k), •U3b §k) cJe, ä½Ù)•U´ X = P b. ,, éõÖöŒU¬kù ¦¯: ÷v P A = In Ý P ŒUØ•˜, 'Xkü‡Ý P1 , P2 ÷v P1 A = In = P2 A, @o P1 b Ú P2 b Ø´kŒUØƒÓí ? c¡ ín`², AX = b k)ž, Ù)7, u P1 b, •7, u P2 b. ùJ Ú)•3ž •˜5vkgñí ? ùý ´˜‡š~š~Ð ¯K ! ·‚AT•Ž ù‡¯K Öö:˜‡ŒŒ 7 ! ¯¢ þ, Ö ö ¦ ¾ Ã š Ò ´, ² ² k P1 b Ú P2 b ü « L ˆ ª, • Ÿ o k ) ž ) ´ • ˜ Q ? Ù ¢ ‰ Y • é { ü, Ò ´ · ‚ ~ ¦ 8 ~ ¦, ¯¢þ AX = b k ) … P1 , P2 ü ‡ Ø Ó Ý ÷ v P1 A = P2 A = In ž, ·‚¿vk•{U y² P1 b 6= P2 b. ƒ‡/, Šâ·‚þ© ín, AX = b k)ž, § )•U´ P1 b, ••U´ P2 b. Ïd, dœ/e˜½k P1 b = P2 b. ¤±ùÚ·‚ƒcØ y (3)•3 cJe) ) •˜5vkgñ ! ØL, Š 5¿ ´, XJ AX = b vk), @o P1 b Ú P2 b ÑØ´ AX = b ). dž (kŒU P1 b 6= P2 b. ´·‚c¡•` 3 AX = b k) ž7, P1 b = P2 b. ¤±·‚c¡ {E,vk†. XJÖöEk¦¾ {, Ø”6ž#Pþ¡ nØØy, 5we¡ äN~f. ·‚   ! ! 1 0 1 0 0 1 0 0   A = 0 1 , P1 = , P2 = . 0 1 0 2 3 −1 2 3 † ŽŒ• P1 A=  I2 = P2 A. 0   XJ b = 1, @oŒ±† )•§ AX = b • § )´•˜ , = X = 01 . dž2† 3 ŽŒ• P1 b = P2 b = 01 . (k, Öö„Œ± éù‡~f5 Ž: éu?Û÷v P A = I2 0 Ý P , þk P b = 1 .)   0   ´, X J b = 1, @ o Œ ± † ) • § AX = b • § Ã ), dž2† ŽŒ• 1 P1 b = 01 , P2 b = 02 . ½ Â 1.2.13. A ∈ Mm×n (K). XJ•3 P ∈ Mn×m (K) ¦ P A = In , ·‚¡ A 3 K þ´†Œ _ (left invertible) ½ö A 3 K þk†_ (left inverse), ¿…¡ P ´ A 3 K þ ˜‡†_. aq/, XJ•3 Q ∈ Mn×m (K) ¦ AQ = Im , ·‚¡ A 3 K þ´mŒ_ (right invertible) ½ö A 3 K þkm_ (right inverse), ¿…¡ Q ´ A 3 K þ ˜‡m_. XJ A 3 K þQk†_qkm_, ·‚¡ A 3 K þ´Œ_ (invertible) . 1˜Ù 44 Ý Ú‚5•§| 5P 1.2.14. Uì·‚ ½, K ´ C ˜‡f8. ?ÛÝ A ∈ Mm×n (K) ,•áu Mm×n (C). ¤±, A ∈ Mm×n (K) 3 K þ†Œ_ (½mŒ_!Œ_) ž, g, A •3 C þ†Œ_ (½mŒ_! Œ_). ¯¢þ, •‡ A Xê Ü5gu K, @o §3 C þ†Œ_ (½mŒ_!Œ_) ž, 7,• 3 K þ†Œ_ (½mŒ_!Œ_). ¤±, ¢Sþ3½Â 1.2.13 ¥·‚•Œ±Ø7rN“3 K þ”ù˜ •›. • `²ù˜:, ·‚±Ïém_•~. ¦ A m_¢Sþƒ u¦) m ‡± A •Xê ‚ 5•§|, ù •§| O2Ý Xê• Ü5gu K. (žÖög•: ù m ‡•§|Ä¾´= Q ?) ·‚• , ‡(½˜‡/X AX = b ‚5•§|´Äk) (Ù¥ •þ b •b z‡©þ Ñ5gu K), Œ±ÏLpd–e ž {òO2Ý (A; b) z•1•{/ (A0 ; b0 ) 5 ä. 3ù‡ öŠL§¥, Ï•ž { z˜ÚÑ• 9 K ¥ ƒ \~¦Ø$Ž, ¤±•ª 1•{/Ý (A0 ; b0 ) •´Xê Üáu K Ý . ˜ 1•{/Ý (A0 ; b0 ), ä´Äk)•I‡w A0 Ú (A0 ; b0 ) š"1ê8´ÄƒÓ, ù‡¯œ3 K ¥•ÄÚ3 C ¥•Ä‰Y´˜ . ¤±, A 3 K þkm_Ú3 C þkm_´ d . ·K 1.2.15: b Ý A ∈ Mm×n (K) 3 K þŒ_. K•3•˜˜‡Ý A−1 ∈ Mn×m (K) ¦ A−1 A = In … AA−1 = Im . dž A ?Û†_Ú?Ûm_Ñ udÝ A−1 . A−1 ¡• A TÝ Ý _ (Ý ) (inverse, inverse matrix). A−1 w,•3 K þŒ_, Ù_Ý y². ? (ÜÆ, A =• (A−1 )−1 = A. ˜‡†_ P ∈ Mn×m (K) Ú A ˜‡m_ Q ∈ Mn×m (K). Šâ½ÂÚÝ ¦{ P = P Im = P (AQ) = (P A)Q = In Q = Q . ù`² A ?¿˜‡†_Ú?¿˜‡m_Ñ7Lƒ . Ïd, A ¤k†_Ñƒ ƒ , ¿…Ù†_ um_. ¤±, -Ý A−1 • A ˜‡†_=Œ. ±eü‡Ún Ún 1.2.16: , ¤km_•Ñ y²3‰ÖööS. A ∈ Mm×n (K), B ∈ Mn×p , P ∈ Mn×m (K). K: 1. P ´ A ˜‡† (m) _ …= P T ´ AT ˜‡m (†) _. Ïd, A † (m) Œ_ AT m (†) Œ_. …, A Œ_ž, (A−1 )T = (AT )−1 . …= 2. e A, B þk† (m) _, K AB k† (m) _. e A, B þŒ_, K AB •Œ_¿… (AB)−1 = B −1 A−1 . Ún 1.2.17: b A, B •Œ Ü· Ý ¦ Ý ¦È AB k¿Â. 1. e AB k†_, K B k†_. 2. e AB km_, K A km_. íØ 1.2.18: ä¤á: P ∈ Mm (K), A ∈ Mm×n (K), Q ∈ Mn (K). b P Ú Q þ3 K þŒ_. K±eØ 1. P A †Œ_ …= A †Œ_, …= AQ †Œ_, …= P AQ †Œ_. 2. P A mŒ_ …= A mŒ_, …= AQ mŒ_, …= P AQ mŒ_. 1.2 Ý Žâ 45 y². ·‚•yc˜(Ø, ˜(Ø y²3‰Öö. Š â Ú n 1.2.17 (1), P A † Œ _ ž, A • † Œ _. ‡ ƒ, Ï • ® b P Œ _, ¤ ± Š â · K 1.2.15, •3Ý P −1 ∈ Mm (K) ¦ P −1 P = Im . u´ A = Im A = P −1 (P A). 2¦^˜gÚ n 1.2.17 (1) (ØŒ•, A †Œ_ž P A •†Œ_. ± A0 = AQ “Oþ¡ A Œ•, A0 = AQ †Œ_ …= P A0 = P AQ †Œ_. •I2y A †Œ_ …= AQ †Œ_. ¯¢þ, e P0 ∈ Mn×m (K) ´ A ˜‡†_, @o −1 Q P0 ´ AQ ˜‡†_, ù´Ï• (Q−1 P0 )AQ = Q−1 (P0 A)Q = Q−1 In Q = Q−1 Q = In . ¤±, A †Œ_ž AQ •†Œ_. Ï• A = AQQ−1 , ¤±aq ½†Œ_. Øy•{Œ±y² AQ †Œ_ž A 3UY?1nØ&?ƒc, 4·‚kÏLA‡~fw˜eÝ †_½m_ OŽ•{. ·‚= Þ¦m_ ~f. e–¦Ý A †_, Œ±ÏLÚn 1.2.16 (1) r¯K=z•¦ AT m_, , ¤ m_ =˜= A †_.   1 0 1   ~ 1.2.19. A = −1 1 1. Á¦ A ˜‡m_. 2 −1 1 ·‚I‡Ïé˜‡ 3 × 3 Ý Q ¦ AQ = I3 . •d, ·‚©ÛÝ Q ƒI‡÷v ¿© 7‡^‡, ÏLù ^‡éÑ Q ¤k ˜ ƒ, l (½ Q. XJù ^‡Ã{ Ü ÷v, K ·‚äó Q  Ø•3, = Avkm_. x1 y1 z1   Q = x2 y2 z2 . K ª AQ = I3 ¢Sþ due¡n‡ ª: x3 y3 z3 (1.2.19.1)     1 x1     A x2  = 0 , 0 x3     0 y1     A y2  = 1 , 0 y3 ‡¦)1˜‡•§|·‚æ^pd–e ž  1 0   −1 1 2 −1 (1.2.19.2)  1 0 1  −−−−−−−→  0 1 2 L3 →L3 +L2 0 0 1 ¤±ù     0 z1     A z2  = 0 . 1 z3 {5éÙO2Ý ?1Ð 1C†:    1 1 1 0 1 1  L2 →L2 +L1   1 0  −−−−−−−−→  0 1 2 1  L3 →L3 −2L1 1 0 0 −1 −1 −2    1 1 0 0 2  L1 →L1 −L3   1  −−−−−−−−→  0 1 0 3  L2 →L2 −2L3 −1 0 0 1 −1 Œ±¦Ñ (x1 , x2 , x3 )T = (2, 3, −1)T . e5• ¦) (1.2.19.1) ¥ 1 ‡•§|, ØJuy, •‡‰1Ó ˜X Ð 1C†= T T Œ¦ (y1 , y2 , y3 ) = (−1, −1, 1) . Ó /, éu (1.2.19.1) ¥ 1n‡•§|, „´•I‡‰1Ó ˜X Ð 1C†=Œ¦ (z1 , z2 , z3 )T = (−1, −2, 1)T . u´·‚• A •3m_, …Tm _•U´   2 −1 −1   (1.2.19.3) Q =  3 −1 −2 −1 1 1 1˜Ù 46 Ý Ú‚5•§| ·‚ ¯Ky3)û . Œ´, kvkÖöú fâ •{k:æ†Q ? Q,3¦) (1.2.19.1) ¥n‡•§|žÑ´²{ƒÓ ˜X Ð 1C†, @o·‚UØUrn‡•§|Ü¿3˜å) Q ? Ù¢, Ï•©ªÑ´•‰1C†, ·‚3O2Ý ¥õO\˜ {, Ù¢ JÒƒ uÓž •Ä õ‡•§|. ¤±, XJ·‚•Ðl•Œ “O2Ý ”   1 0 1 1 0 0    −1 1 1 0 1 0  2 −1 1 0 0 1 Ñu, , •g‰1 (1.2.19.2) ¥ Ñ5 Ð 1C†, K•ª (JÒ´   1 0 0 2 −1 −1    0 1 0 3 −1 −2  0 0 1 −1 1 1 . •Ò´´`, XJ·‚rÝ A Ú ü Ý I3 Ü¿å5 ¤˜‡ 3 × 6 ••/Ý (A .. I3 ), @ o, ·‚uyÏLÐ 1C†Œ±ò†> Ý A z¤ü Ý I3 ž, m> Ý I3 Ò‘ƒC ¤ ÷v AQ = I3 ù‡^‡ @‡Ý Q, •Ò´Ý A m_. ùÙ¢Ò´˜«š~k ¦Ý _ •{. ù«•{¤á nØ•âÒ´: ˜X Ð 1C†ò A z¤ I3 ž, pd–e ž {L² 5m>Ý I3 n ÒC¤ (1.2.19.1) ¥n‡•§| ), •Ò´‡¦ m_ Q n ‡ .   1 0 1 0   ~ 1.2.20. B = −1 1 1 1 . Á¦ B ˜‡m_. 2 −1 1 −1 ·‚÷^~ 1.2.19 ¥ )Kg´, •Ä“O2Ý ”   1 0 1 0 1 0 0   (1.2.20.1)  −1 1 1 1 0 1 0  2 −1 1 −1 0 0 1 E” (1.2.19.2) ¥¤ (1.2.20.1) ¥ ‡Ý (1.2.20.2) Ú½öŠ, Œ±ò±þ“O2Ý ”¥J‚†> ¬C¤   1 0 0 0 2 −1 −1    0 1 0 1 3 −1 −2  0 0 1 0 −1 1 1 Ý B z¤1•{/, Óž ± (1.2.20.2) ª¥J‚†>Ý •XêÝ , ±J‚m>Ý n‡ ©OŠ•~ê‘ 5•§|, §‚ˆg )Ò‰Ñ B m_ (Š• 4 × 3 Ý ) n‡ . ¤±, ?Û/X   2 −1 −1 3 − t −1 − s −2 − w      −1 1 1  t Ý Ñ´ B 2 s m_, Ù¥ t, s, w ´Œ±?¿ Š   1 −1 2 0 1 −1   C= . Á¦ C ˜‡m_. 1 1 1 0 1 −1 w ëê. n‡‚ 1.2 Ý Žâ UìÓ±þ˜ 47 g´, •Ä“O2Ý  1  0    1 0 (1.2.20.3) E” (1.2.19.2) ¥¤ (1.2.20.1) ¥ ‡Ý ” −1 1 1 1 2 −1 1 −1 1 0 0 0 0 1 0 0 Ú½öŠ, Œ±ò±þ“O2Ý ¬C¤  1  0    0 0 (1.2.20.4) 0 1 0 0 0 0 1 0 2 −1 −1 0 3 −1 −2 −1 0 0 1 0  0 0    0  1 ”¥J‚†> −1 1 1 0 Ý C z¤1•{/, Óž  0 0    0  1 ± (1.2.20.4) ª¥J‚†>Ý •XêÝ , ±J‚m>Ý o‡ ©OŠ•~ê‘ 5•§|, §‚Ò´ C m_ (Š• 3 × 4 Ý ) o‡ I‡÷v •§|. ùp 1 ‡•§|w,Ã). ¤±Ý C vkm_. o‡‚ ‡Ú1o ØJwÑ, ±þ~ 1.2.19 Ú 1.2.20 ¥¦Ý m_ •{ ŸþÒ´ò¤‰Ý ŠA‡‚5• §| Ó XêÝ , , ¦^pd–e ž {òTÝ =z•1•{/¿¦éA•§| ), ± ¤ )Š• Ò ¤¦ m_. (edL§¥, •§|Ã), K`²m_Ø•3.) ù«•{• ¡•pd–e { (Gauss–Jordan method). ù«•{w,Ø==·û¦Ý m_, •Œ±^ 5¦)•˜„ Ý •§, ~X, ÖöŒ±gC}Á¦Ñ÷v     1 1 1 0 1     −1 1 1 X =  0 1 −1 0 2 −1 1 3×2 Ý X. ! a b (1.2.21) éu˜‡ Ý A= , ·‚r det(A) := ad − bc ù‡êŠ¡•´ A 1 ª c d (determinant). •˜„ 1 ªnØ·‚ò31oÙ2•[Øã. ùp·‚•Ñ, |^1 ªŒ±é •B/ ä˜‡ • ÛžŒ_, ¿…3Œ_ž¦Ñ_Ý . ¯¢þ, XJ1 ª det(A) u 0, Kàg‚5•§| AX = 0 7kš") (žÖög1 yd(Ø). (ùp X = (x1 , x2 )T ´ü‡Cþ ¤ •þ.) Ïd, džÝ A 7½ØŒ_ (ë„ (1.2.12)). ! ! d −b a b 1 XJ1 ª det(A) 6= 0, ÖöÏLOŽŒ± y: Ý _ ad−bc −c a ´ A = c d Ý . ±eyü½NŒ±•ÏÖöPÁ Ý ¦_úª: é é 1 y3·‚=£nØ‘K ¿˜^{ü Ún. ü>CÎÒ, ÷åêéN. ªŠŠ©1, ¦_Up . &?. ·‚òXÃy²: Œ_ Ý 7,Ñ´• . •d, kžÖö5 1˜Ù 48 Ún 1.2.22: Ý Ú‚5•§| A ∈ Mm×n (K). 1. e A †Œ_, Kéu?Û •þ b ∈ K m×1 , ‚5•§| AX = b •õ•k˜‡). 2. e A mŒ_, Kéu?Û •þ b ∈ K m×1 , ‚5•§| AX = b – 3. e A Œ_, Kéu?Û •þ b ∈ K m×1 , ‚5•§| AX = b k…•k˜‡). y². (1) ù´ (1.2.12) ˜ã®²?ØL (Ø. (2) e Q ∈ Mn×m (K) ´ A ˜‡m_, K X = Qb ´ AX = b (3) nÜ (1) Ú (2) =Œ. 8 ·‚¬w k˜‡). , Ún 1.2.22 ¥n^(Ø ˜‡). _·K•Ñ¤á. Ún 1.2.23: b A ∈ Mm×n (K) •1•{/Ý , r • A š"1ê8. K A †Œ_ …= r = n, A mŒ_ …= m = r. ¤±, A Œ_ …= m = r = n, …= A • n ü Ý . y². bX A †Œ_, KŠâÚn 1.2.22 (1) Œ•àg‚5•§| AX = 0m×1 vkš"). Šâ· K 1.1.19, ù«œ¹•k3 r = n žâ¬Ñy. ‡ƒ, XJ r = n, @oŠ•1•{/ Ý , A •þ¡ r 1´ÄXê• 1 š"1, ¿…ù ÄXêˆgÓâØÓ . Ï• r = n, ¤±¢Sþ A þ¡ r 1 ¤ Ò´ r ü Ý Ir A e¡ m − r 1Ñ´"1 (Ï• r ´š"1ê8). ¤± A /GX I0r (ùp 0 ´Œ • (m − r) × r "Ý ). džN´ y A =˜ AT Ò´ A ˜‡†_. ùÒy² 1˜‡(Ø. éu1 ^(Ø, k m = r. ùžéu?¿ b ∈ K m×1 , O2Ý (A; b) •´Tk m 1 1 F / Ý , Ù ¥ š " 1 ê 8 w , • ´ m. ¤ ± Š â · K 1.1.19, é u ? ¿ b ∈ K m×1 , • § | AX = b ok). AO/, éu     1 0   . 0  ..  T    b1 = (1, 0, . . . , 0)T =   ..  , · · · , bm = (0, . . . , 0, 1) =   . 0 0 1 ·‚Œ±©Oé •þ X1 , . . . , Xm ∈ K n×1 ¦ éz‡ j ∈ [[1, m]] þk AXj = bj . dž, e- Q •1 j • Xj n × m Ý , K AQ = Im , = Q • A ˜‡m_. ‡ƒ, b A km_. ŠâÚn 1.2.22 (2) Œ•, éu¤k b ∈ K m×1 , •§| AX = b Ñk). džXJ r < m, éN´w éu b = (0, 0, . . . , 0, 1)T 5`, AX = b vk). ù˜gñL²7L r = m. Ún–d y. Ún 1.2.23 `², 1•{/ Œ_Ý 7L´• , ·‚y3I‡Ú\˜‡•-‡ Œ_• X . . • òÓ (Øí2 ½ Â 1.2.24. XJ P ´d n ü In ²L˜g1Ð C†½ö Ð P ´˜‡ n Ð Ý (elementary matrix). ŠâÐ C† n«ØÓa., ·‚Œ±rÐ Ý •©•na: C† š1•{/ Ý Ý , ·‚¡ 1. 1ãÐ Ý , •¡•Ð ˜†Ý (elementary permutation matrix) ½é†Ý sition matrix), ´• †ü Ý ,ü1 (½,ü ) ¤ Ý . (transpo- 1.2 Ý Žâ 49 XJ´p† In 1 i 1Ú1 j 1, ¤ Ý = 1i        1i1          1 j 1      1j  1 .. .             =: Pn (i, j)           1 0 1 1 .. . 1 1 0 1 .. . 1 (ùpÝ ˜x?@• ƒÑ´ 0, é ‚þ ØJwÑ, † In Ú1 j (J•´þ¡ù‡Ý XJÝ 3þe©²wŒ•, ·‚Œ±rPÒ Pn (i, j) { 2. 1 aÐ Ý XJ´ò In 1i ´•òü Ý ¤ ŽÑÒ“L ,˜1 (½,˜ 1 i 1¦±š"~ê λ, ¤ Ý ŽÑ ƒ@•´ 1. ) . • P (i, j). ) ¦±˜‡š"~ê¤ Ý . ´ 1i   1 ..       1 i 1      .        =: Pn (λ · i)      1 λ 1 .. . 1 w,, r In 1 i {P• P (λ · i). 3. 1naÐ . Ý XJ´ò In ¦± λ ´•òü Ý •´±þÝ . 3Ø¬Úå¹Â·Ïž, ·‚~r Pn (λ · i) ,˜1 (½ ) ¦±˜‡~ê\ 1 i 1¦±~ê c \ 1 j 1, ¤ 1i     1i1      1 j 1    Ý ,˜1 (½ ´ 1j  1 ..        =: Pn (c · i , j)       . 1 .. c . 1 .. . 1 )¤ Ý 1˜Ù 50 w,, ò In 1j ¦± c \ Öö„I‡3¿, ±þ·‚Ö Pn (c · i, j) fXe 1i ¤ •´±þÝ 1j  œ¹5 . XJ i > j, K 1i  1 ..    1 j 1      1i1    Ú‚5•§| . •ªÚì i < j Pn (c · i, j) Ý        =: Pn (c · i , j)       . 1 c .. . 1 .. . 1 3Ø¬Úå¹Â·Ïž, ·‚Œ±r Pn (c · i , j) {P• P (c · i , j). 5¿, ·‚•Œ±ò Pn (c · i, j) Š In ‚•ò Pn (c · i, j) UP• Pn (i, j · c). 5¿˜‡{ü 1 ( ) C†UCÝ up† A 1 i, j 1¤ 2. ¦ÈÝ Pm (λ · i)A ¦± λ ¤ Ý . uò A 1 i 1¦± λ ¤ 3. ¦ÈÝ Pm (c · i, j)A APn (i, j · c) uò A ÖöŒ±† ¤ Ý . Xk7‡, · aÐ Ý •é¡Ý L§, ¢Sþƒ . 1na u3† (m) >¦±Ð c, λ ∈ K, λ 6= 0. Kéu?¿ A ∈ Mm×n (K) ±eØä¤á: 1. ¦ÈÝ Pm (i, j)A i, j ¤ Ý . y². ÏLOŽ 1i =˜E•Ð Ý . ¯¢þ, 1ãÚ1 =˜ u Pn (c · j, i). ±e·KwŠ·‚, ÏLÐ L§. ·K 1.2.26: ¦± c \ ¯¢: ·K 1.2.25: Ð Ý Ð Ý Pn (c · i, j) Ý 1j Ý ; ¦ÈÝ APn (i, j) up† A 1 ; ¦ÈÝ APn (λ · i) uò A 1i ; ¦ÈÝ APn (c · i, j) = Ý uò A 1 i 1¦± c \ 1 j 1¤ 1 j ¦± c \ 1 i ¤ Ý . Ý y=Œ. ÏLOŽy²±e(Ø, ½öÏL3·K 1.2.26 ¥ A •· Ð Ý Pn (λ · i) Ý y ². ·K 1.2.27: Ð Ý ÑŒ_, ¿…Ù_Ý E•Ð Ý . ¯¢þ, 1ãÐ Ý Pn (i, j) _Ý uÙg ; 1 aÐ −1 Pn (λ · i); 1naÐ Ý Pn (c · i , j) _Ý • Pn ((−c) · i , j). Šâ·K 1.2.26, ·‚Œ±rƒc ½n 1.1.23 =ã•Xe/ª: ½n 1.2.28: é?ÛÝ A ∈ Mm×n (K), o•3 Mm (K) ¥ Pr Pr−1 · · · P1 A •1•{/. nÜƒc ˜X _Ý (Ø·‚y3Œ±”´ ±eíØ: k•õ‡Ð Ý P1 , . . . , P r ¦ • 1.2 Ý Žâ íØ 1.2.29: 51 A ∈ Mm×n (K). 1. e A Œ_, K7, m = n. =, Œ_Ý 2. e®• A • n (a) A Œ± • , Ke ¤k•õ‡Ð 7,´• •ã d: Ý ¦È. . (b) A Œ_. (c) •3k•õ‡Ð Ý P1 , . . . , P r ¦ Pr Pr−1 · · · P1 A = In . (c ) •3k•õ‡Ð Ý Q1 , . . . , Qs ¦ AQ1 · · · Qs−1 Qs = In . (d) ²Lk•õgÐ 1C†Œ±ò A z•ü Ý In . (d0 ) ²Lk•õgÐ C†Œ±ò A z•ü Ý In . 0 (e) A †Œ_. (f) A mŒ_. y². Šâ½n 1.2.28, •3Ð Ý P1 , . . . , Pr ¦ A0 := Pr · · · P1 A •1•{/Ý . Ï•Ð Ý ÑŒ_ (·K 1.2.27), ¤±, ÏLõg¦^íØ 1.2.18 Œ±• , A Œ_ …= A0 Œ_. Ú n 1.2.23 `² A0 Œ_ž7L´• , = m = n. ¤±Øä (1) –d y. y3b ®• A • n • . Kþ¡J 1•{/Ý A0 •´ n • . ù Ún 1.2.23 L²: A0 †Œ_ …= §mŒ_, …= §Œ_, …= A0 = In . (ÜíØ 1.2.18 Œ•, ù ^‡©O du (b), (e), (f) Ú (c). ¤±·‚• (b), (c), (e), (f) ù äó Ñ d. (c) Ú (d) 0 0 d5 u·K 1.2.26. (aq/, (c ) Ú (d ) d.) d , Pr · · · P1 A = In ž, ·‚k A = P1−1 · · · Pr−1 . Ï•Ð Ý _Ý •Ñ´Ð Ý (·K 1.2.27), ¤±ùy² (c)⇒(a). Ï•Œ_Ý ¦ÈEŒ_ (Ún 1.2.16 (2)), ¤± (a)⇒(b). –d·‚y² (a), (b), (c), (d), (e), (f) Ü d. • , e A = Et · · · E2 E1 , Ù¥ Ei þ•Ð Ý , @o s = t, Qi = Ei−1 Œ AQ1 · · · Qs = In . ¤± (a)⇒(c0 ). ÖöŒ±gy (c0 ) ⇒(a). –d¤k(Ø y. žÖö3¿: íØ 1.2.29 (2) ¿vk`é?¿Ý d, •´`éu• 5`, ùnö d. 1.2.3 Ý 5`“†Œ_”!“mŒ_”Ú“Œ_”n‡^‡ d ·‚• , éu?¿‚5•§| AX = b, oŒ±ÏLÐ 1C†òÙO2Ý (A; b) z•˜‡ 1•{/ Ý (A0 ; b0 ), •§| AX = b Ú A0 X = b0 Ó). •Ò´`, éu¦)™•þ X ù‡¯ K5`, (A; b) Ú (A0 ; b0 ) ùü‡Ý ´“ ” (d§‚û½Ñ X ¤kŒU Š´ƒÓ ). Ù ¢3)¹¥·‚•~‘ ù œ¹: éuü«ØÓ ¯Ô, ŒUéu, A½ ¯K½ö3, A½ |Üe, §‚¤å Š^ŒU´ƒÓ . 3êÆþ, ‡£ãØÓ ¯Ô éA½¯K¤Ly “ ”y–, Ï~·‚´ÏL¦^“ d'X”ù‡Vg5ˆ 8 . ·‚k°(/½ÂÝ ƒm n«“ d'X”. ½ Â 1.2.30. A, B ∈ Mm×n (K). ·‚¡ A Ú B 3 K þ1 d (row equivalent over K), XJÏLk•õg K þ Ð 1C† Œ±ò A C• B. (¤¢Ð C†´ K þ , ´•ÙéA Ð Ý ´Xê Ü5gu K Ý . •Ò´`, XJ´1 a½ö1naÐ C†, KÝ 1¤¦ ~ê gu K.) Šâ·K 1.2.26, ù•ƒ u`•3k•õ‡ m Ð Ý P1 , . . . , Pr ¦ B = Pr · · · P1 A. 2ŠâíØ 1.2.29 (2), ùq du`, •3 m Œ_Ý P ¦ B = P A. 1˜Ù 52 Ý Ú‚5•§| aq/, ·‚¡ A Ú B 3 K þ d (column equivalent over K), XJÏLk•õg K þ C†Œ±ò A C• B. Šâ·K 1.2.26 ÚíØ 1.2.29 (2), ù•ƒ ue¡ü«`{¥ z˜ Ð ‡: • •3k•õ‡ n Ð • •3 n Q¦ Œ_Ý ·K 1.2.31: Ý Q1 , . . . , Qs ¦ B = AQ1 · · · Qs . B = AQ. A, B, C ∈ Mm×n (K). 1. A Ú A g 1 (½ d. ) 2. e A Ú B 1 (½ ) d, K B •Ú A 1 (½ ) 3. e A Ú B 1 (½ ) d d, @o A Ú C 1 (½ B Ú C 1 (½ ) d. d. ) y². 3‰ÖööS. 5P 1.2.32. ˜„/, b S ´˜‡š˜8Ü, 2b éu S ¥¤k ƒ·‚Œ±?Ø§‚ƒm ,«(½ 'X R. ùp, ¤¢“(½ 'X”´•éu?¿ü‡ (ƒÓ½ØÓ ) ƒ a, b ∈ S, ‡ o a Ú b äk'X R, ‡o a Ú b Øäk'X R, …ü«œ¹•k˜«Œ±u). XJü‡ (ƒÓ ½ØÓ ) ƒ a, b ∈ S ƒmäk'X R, ·‚æ^PÒ a ∼ b. XJe¡n‡^‡¤á, ·‚Ò` R ´8Ü S þ ˜‡ d'X (equivalence relation): 1. g‡5!‡ 5 (reflexivity): é?¿ a ∈ S þk a ∼ a. 2. é¡5 (symmetry): é?¿ a, b ∈ S, e a ∼ b K½k b ∼ a. 3. D45 (transitivity): é?¿ a, b, c ∈ S, e a ∼ b … b ∼ c, K a ∼ c. Šâù‡½Â, ·K 1.2.31 d'X. ·K 1.2.33: A, B ∈ Mm×n (K). K±e^‡ 1. ²Lk•õg K þ 2. •3Œ_Ý Ð A†C 1 d”Ú“ “1 d”Ñ´8Ü Mm×n (K) þ d: 1C†Úk•õg K þ P ∈ Mm (K) ÚŒ_Ý 3. •3 C ∈ Mm×n (K) ¦ ±þˆ Ÿþ3`: 3 K þ Ð C†Œ±ò A C¤ B. Q ∈ Mn (K) ¦ d, B † C B = P AQ. d. d^‡¤áž, ·‚¡ A Ú B 3 K þƒ-½ d (equivalent)∗∗ . y². 3‰ÖööS. N ´ w Ñ, X J A Ú B 1 d ½ d, K A Ú B ƒ -. , , (3 K þ)“ƒ -”• ´ 8 Ü Mm×n (K) þ d'X. ! ! 1 0 0 0 g • K 1.3. A= ,B= . y²: A Ú B ƒ-, §‚QØ1 d•Ø d. 0 0 0 1 g • K 1.4. A • n • . y²e (1) A Œ_; (2) A 1 duü Ý ∗∗ d? æ^ equivalent ˜c ,˜‡c5€Èd? ^‡ d: In ; (3) A duü Ý In ; (4) A Úü , ~€È•“ d”. ØL, Ï•“ d”ù‡c3êÆpÑy equivalent, =“ƒ-”, ŒV¿g´Œ±“ƒp-ˆ”j. ªÇ p Ý In ƒ-. , ¤±¥© á¥ 1.2 Ý Žâ 53 ±e(ØÃš´½n 1.1.23 †˜«•ª•ã˜e: ·K 1.2.34: ?Û˜‡Ý Ñ1 du,‡1•{/ Ý . ·‚Œ±ò˜‡Ý B ¡•´ •{/ Ý , XJ=˜Ý ·K 1.2.34 •k'u d ‡ : ?Û˜‡Ý Ñ du,‡ ½n 1.2.35: éu?¿ A ∈ Mm×n (K), o•3šK A0 := Ir 0(m−r)×r B T ´1•{/ •{/ Ý . ê r ≤ min{m, n} ¦ ! 0r×(n−r) Ý . w,, A ƒ-u±eÝ 0(m−r)×(n−r) ùp, XJ r = m, K@•±þÝ {¥ A0 /X (Ir 0). aq/, XJ r = n K@• A0 = 0 (e r = m = n, K A = Ir . e r = 0, K@• A0 = 0m×n .) Ir 0 . y². Šâ·K 1.2.34, •3˜‡1•{/ Ý C ¦ A † C 1 d. b r • C š"1ê 0 þ. Šâ·K 1.2.33 ¥ ^‡ (3), •I‡2y² C du½n¥‰Ñ A =Œ. ¯¢þ, ÏL C r ‡Ì ¤3 5Š1naÐ C†Œ±ò C ¥ Ù¦ ÜC•". , ÏLé†˜ 0 ˜ (ù Ñ´1ãÐ C†) =ŒòÝ N • A /G. 8 ·‚¬y², ½n 1.2.35 ¥ ê r ´dÝ A •˜û½ . Uì8 ò¬‰Ñ ½Â, ù‡ê r Ò uÝ A •. ù , ½n 1.2.35 ¥ Ý A0 •´d A •˜(½ , ·‚¡ƒ• A ƒ-IO/ (canonical equivalent form). 1.2.4 ©¬Ý ŒUéõÖö®²aú , Ý ¦{´˜«OŽ¹•„¡ $Ž, cÙ´Ý 1Ú žÿ. ØL, éu/G' {ü Ý 5`, éõœ¹e¯K´Œ± {z . ·‚kw˜‡{ü ê8 õ ~f. (1.2.36) •Ä˜‡Xe5Ÿ m×n Ý (1.2.36.1) A= Ir 0 ! 0 0 Ù¥ Ir ´˜‡ u A †þ r ü Ý , A Ù¦ ˜ ƒÑ´ 0. 3±þ {¥, ·‚ ¢Sþ¦^ “Ý ©¬” •{5{zPÒ. =¦ A 1Ú ê8ŒU¤Zþ , 3þ¡ “©¬ PÒ”¥, ·‚r A ©¤ 2 × 2 = 4 ‡«¬, z‡«¬S ƒÜ¿Ö ¤˜‡ƒé{ü fÝ . ù ‰ Ð?´ŸoQ ? bX·‚ ˜‡ n × p Ý B, ÏLOŽŒ±uy¦ÈÝ C := AB fXe:  b11  b  21  .  .  .  C= br1  0  .  .  . ··· ··· .. . 0 ··· ··· ··· .. . b1s b2s .. . brs 0 .. . 0 b1,s+1 b2,s+1 .. . br,s+1 0 .. . 0 ··· ··· .. . ··· ··· .. . ···  b1p  b2p  ..   .   brp    0 ..   .  0 54 1˜Ù Ý Ú‚5•§| ùp s Œ±´ [[1, p]] ƒ¥?À ˜‡ê. ØJuy, ¦+ C Ú A ¢SŒ ŒUØÓ (Ï n Ú p ŒUØÓ), ·‚EŒ±ò C ” (1.2.36.1) ª@ y©• 2 × 2 = 4 ‡«¬, l ò C = AB ÑX e/ª ! B11 B12 (1.2.36.2) C= , 0 0   b11 · · · b1s  . ..  ..  . Ù¥ B11 =  r × s fÝ , B12 ´ B ¥ umþ . .  ´Ý B ¥ u†þ  . br1 · · · brs r × (p − s) fÝ . ¢Sþ, ·‚Œ±r B •‰aq ©, ¦ B /X ! B11 B12 (1.2.36.3) B= B21 B22 Ù¥ B21 ´ B †e ? (n − r) × r fÝ , B22 ´ B me ? (n − r) × (p − s) fÝ [* (1.2.36.1)–(1.2.36.3) Œ±uy, (1.2.36.2) ª (Jƒ u` ! ! ! Ir 0 B11 B12 B11 B12 (1.2.36.4) · = 0 0 B21 B22 0 0 . c 5¿, fâ·‚ (1.2.36.4) ª L§´, kUìÏ~ Ý ¦{OŽÑ C = AB äN ƒ, , ò ª AB = C ¥ n‡Ý ©O ©• 2 × 2 = 4 ‡«¬5Ö , ù Ò (1.2.36.4). ,˜•¡, b ·‚kò A Ú B Uì (1.2.36.1) Ú (1.2.36.2) ¥ •ªy©¤ 2 × 2 “©¬Ý ”, , ò (1.2.36.4) ª †>/ªþ ¤ü‡ 2 × 2 Ý ¦{, @o$Ž (Jw,• u (1.2.36.4) ª m>. ù`²ÃØÝ A Ú B ©ý¢Œ (=Ø‰ ©žw 1Ú ê8) kõŒ, ò §‚ ©¤©¬Ý ƒ , OŽ¦È AB ž/ªþ•‡OŽü‡ 2 × 2 Ý ¦{. ±þù‡{ü ~fÙ¢Œ±í2•˜„ ©¬Ý ¦{$Ž{K. 8 Öö¬uy, Ý ©¬5$Ž´˜«š~~^ Ý $ŽE|, §3NõnØ¯K ©Û¥Œ±užãŒŠ^. (1.2.37) y3·‚5°(Qã˜e©¬Ý b é ê m Ú n ©O‰½ ˜«  m = m + · · · + m 1 r (1.2.37.1) n = n1 + · · · + ns VgÚ§‚ Ä $Ž{K. © (partition), =Xe˜«Lˆª Ù¥ r , s ∈ N∗ , mi , nj ∈ N∗ . ·‚@• (1.2.37.1) ù| ©‰ ¤k m × n Ý ˜‡ © ª (partitioning pattern). •Ò´`, ·‚Œ±Uì m Ú n ±þ ©5ò?¿ m × n Ý A ©• r × s ‡fÝ Auv , Ù¥ u H [[1, r]] ¥ ê, v H [[1, s]] ¥ ê, z‡fÝ Auv éA Œ • mu × nv . =, A Œ± ¤   A11 · · · A1s  . ..  ..  .  (1.2.37.2) . .   . Ar1 · · · Ars • rN A 3þ¡ {¥‘k©¬ &E, ·‚6žò (1.2.37.2) ¥ Ý P• A (= A \• µ), ¿òù«‘k©¬&E Ý ¡•©¬Ý (partitioned matrix, block matrix). Ù¥ fÝ Auv ¡•©¬Ý A 1 (u , v) ˜«¬ (block). 1.2 Ý Žâ 55 (1.2.38) y3b m × n Ý ÑUì (1.2.37.1) ¥ © ª ¤©¬Ý . ®²‰½ A, A0 ∈ 0 0 Mm×n (K), b éA ©¬Ý ¤ A = (Auv ) , A = (Auv ). ·‚Œ± ( ž/) æ^PÒ 5 L« A Ú A0 ©¬\{ (block addition), ÙäN½Âª•  A11 + A011 · · ·  .. .. A A0 := (Auv + A0uv ) =  . .  0 Ar1 + Ar1 · · · =, A Ú A0 ¬Ò´ A Ú A ©¬\{$Ž(J´˜‡äkƒÓ 0 1 (u, v) © ª  A1s + A01s  ..  .  Ars + A0rs ©¬Ý , TÝ 1 (u, v) « ˜«¬éAƒ\ƒ(J. N´wÑ A A0 = A + A0 . (1.2.38.1) •Ò´`, ©¬Ý A Ú A0 Uì©¬\{ •ª‰$Ž, Ù(J ukò A, A0 ŠÊÏ (Ø¹ ©&E ) Ý ‰\{$Ž, , 2±ƒÓ © ª ©•©¬Ý . ½ö`, ©¬\{ ( JXJØ•Ä©¬ J {, Ò uUÊÏ\{ (J. XJ λ ´‡~ê, ·‚Œ±½Â§Ú©¬Ý A ƒm ©¬ê¦ (blockwise scalar multiplication), T$Ž PÒ6žæ^ , ÙäN½ÂªXe:   λA11 · · · λA1s  . ..  ..  . λ A := (λAuv ) =  . .   . λAr1 · · · λArs Ó ØJwÑ (1.2.38.2) λ A = λA . •Ò´`, ~ê λ Ú©¬Ý A Uì©¬ê¦ •ª‰$Ž, Ù(J ukò λ Ú A UìÊÏ (Ø¹ ©&E ) Ý ê¦‰$Ž, , 2±ƒÓ © ª ©•©¬Ý . ½ö`, ©¬ê¦ (JXJØ•Ä©¬ J {, Ò uUÊÏê¦ (J. e5·‚?Ø©¬Ý ƒm ¦{. ·‚b A ∈ Mm×n (K) EU (1.2.37.1) ¥ © ª ©•©¬Ý A. B ∈ Mn×p (K). ·‚ò B UìXe © ª ©•©¬Ý B:  n = n + · · · + n 1 s (1.2.38.3) Ù¥ s , t ∈ N∗ , ni , pj ∈ N∗ . p = p1 + · · · + pt 5¿, ùp·‚‡¦ n 3 (1.2.38.3) Ú (1.2.37.1) ¥ ©•ªƒÓ ! (žÖöÖ o.) ·‚y3½Â©¬Ý A Ú B ƒm ©¬¦{ (block product) Xe: A B := Cuw 1≤u≤r , Ù¥ Cuw = 1≤w≤t s X Aul Blw . l=1 •Ò´  (1.2.38.4) A11  .  .  . Ar1 ··· .. . ···   A1s B11  ..    .. .   . Ars Bs1 ··· .. . ···   B1t C11  ..   : =  .. .   . Bst Cr1 Ù¥ Cuw = s X l=1 ··· .. . ··· Aul Blw .  C1t ..   .  Crt ãƒ g••Ÿ 1˜Ù 56 (ùp·‚æ^ 5L«©¬¦{•´ ž .) ƒéu\{Úê¦5`, ©¬¦{ÚÊÏÝ Ý Ú‚5•§| ¦{ d5y²å5q Ñ‡æ†˜::. ·K 1.2.39: b A ∈ Mm×n (K) Ú B ∈ Mn×p (K) ©O± (1.2.37.1) Ú (1.2.38.3) •² ©¬, ò¦ÈÝ AB UìƒA ©¬ ª©¬, =U  m = m + · · · + m 1 r (1.2.39.1) p = p1 + · · · + pt é¦È AB 5©¬. ©¬¦{$Ž(J, K A B = AB . •Ò´`, A Ú B 2ò¤ (JU (1.2.39.1) 5 © 5 ©¬Ý . ½ö`, ©¬¦{ J {, Ò uUÊÏ¦{ (J. ©¬ ª ukOŽÊÏÝ ¦{ AB (JXJØ•Ä©¬ (JP• C . ÑK©¬&E, òù y². X (1.2.38.4) ª@„ò©¬¦{$Ž A B Ý P• C. K C Ú AB ˜ Ñ´ m × p Ý . ·‚•‡y², é?¿ i ∈ [[1, m]] Ú?¿ j ∈ [[1, p]], C 1 (i, j) ƒ u AB 1 (i, j) ƒ. UÊÏÝ ¦{ ½Â, ö u Pn ©, w, k=1 aik bkj . Šâ n = n1 + · · · + ns ù« n X aik bkj = aik bkj + = s X N2 X Nl X Ns X aik bkj + · · · + k=N1 +1 k=N0 +1 k=1 (1.2.39.2) N1 X aik bkj k=Ns−1 +1 aik bkj l=1 k=Nl−1 +1 Ù¥ N0 = 0 , N1 = n1 , N2 = n1 + n2 , · · · , Ns = n1 + · · · + ns = n (1.2.39.3) y35wÝ C 1(i, j) ƒ cij ´Ÿo. 5¿ C Ò´ (1.2.38.4) ªm> ©¬Ý , ·‚ k‡é (i, j) ù‡ ˜3=‡fÝ Cuw ƒS. w,, •3•˜ u ∈ [[1, r]] Ú w ∈ [[1, t]] ¦ m1 + · · · + mu−1 < i ≤ m1 + · · · + mu−1 + mu , p1 + · · · + pw−1 < j ≤ p1 + · · · + pw−1 + pw . (ùp, XJ u = 1, K ÑØ ª m1 + · · · + mu−1 < i ¤–\ •›. aq/, XJ w = 1, K ÑØ ª p1 + · · · + pw−1 < j ¤–\ •›.) †óƒ, ·‚ C 1 (i, j) ˜ ƒ u©¬Ý C 1 (u, w) ˜«¬ Cuw ƒS. 2 cij ù‡ ƒ3fÝ Cuw ƒ¥ ƒé ˜´1 (i0 , j0 ) ˜, =, i0 = i − (m1 + · · · + mu−1 ) , j0 = j − (p1 + · · · + pw−1 ) . (XJ u = 1, K@• i0 = i. aq/, XJ w = 1, K@• j0 = j.) Ps Ï• Cuw = l=1 Aul Blw , ¤±Š• Cuw 1 (i0 , j0 ) ˜ s X (1.2.39.4) ƒ, cij u Aul Blw i j 0 0 l=1 ùp, é?Û1ê8Ø u i0 ê8Ø u j0 Ý G, ·‚^ Gi0 j0 L« G 1 (i0 , j0 ) ƒ. 5¿ Ý Aul Œ ´ mu × nl , § 1 i0 1Ù¢ÒéAu A 1 i = i0 + (m1 + · · · + mu−1 ) 1 1 l ã (ù˜ã •Ý• nl ), = Aul 1 i0 1 = (ai, n1 +···+nl−1 +1 , · · · , ai, n1 +···+nl ) = aik N l−1 +1≤k≤Nl . 1.2 Ý Žâ 57 aq/, Ý Blw 1 l ã (ù˜ã Blw Œ ´ nl × p w , § •Ý•´ nl ), = 1 j0 Ù¢ÒéAu B 1 j = j0 + (p1 + · · · + pw−1 ) 1 j0 1 = (b n1 +···+nl−1 +1 , j , · · · , bn1 +···+nl , j ) = bkj N l−1 +1≤k≤Nl ¤± (1.2.39.4) ¥ ¦ÚªÙ¢Ò u (1.2.39.2) ¥1 (Ø: ©¬¦{¤ Ý A B ?¿ ˜ ƒ Ku´ y. g • K 1.5. éu·K 1.2.39, ´ÄŒ±”Ý 8 Ä "˜Ý œ¹5y²˜„œ¹ ? (1.2.40) ·‚„Œ±é©¬Ý ¦{ . ‡ Òmà Lˆª. Ïd·‚y² Ž‡ uÊÏÝ ¦È AB 3ƒA ˜ ƒ. · (ÜÆ (·K 1.2.11) 1˜«y²@ ?1=˜öŠ, =, e  A11 · · ·  . ..  A =  .. . Ar1 · · ·  A1s ..   .  Ars  AT11  . T . A :=   . AT1s  ATr1 ..   .  ATrs , ÏLz KÙ=˜• ¢Sþ, Ý =˜ù«öŠ•Œ±ÚÝ 3‰Öög1 y. ©¬ ··· .. . ··· öŠ †k ^S, = A T = AT . ù‡(Ø·‚ –d·‚ `² ©¬Ý $Ž5ÆÚÊÏÝ $Ž5Æ ŸÃ , ¤±·‚± •Ø 23PÒþ?1«©, =©¬Ý ØI‡2±\µ /ªL«, ©¬Ý ƒm \{!ê¦ÚÝ ¦{•Ñ† UìÊÏÝ ƒA$Ž5Ö =Œ. 5wA«•~„ (1.2.41) © ª. A ∈ Mm×n (K), B ∈ Mn×p (K). XJ^ α1 , . . . , αn 5L« A     a11 a1n  .   .   .  .  α1 =   .  , · · · , αn =  .  am1 amn KŒ±ò A U •þ|, = ©¬• A = (α1 , . . . , αn ) . dž/ªþ A ¤•äk n × 1 ‡«¬ ¦. dd·‚ ©¬Ý  (1.2.41.1) b11  .  AB = (α1 , . . . , αn )  .. bn1 aq/, e± β1 , . . . , βn L« B ··· .. . ··· . dž§Œ±†  b1p ..   . = bnp n X Ú B ∈ Mn×p (K) U©¬¦{ƒ αi bi1 , · · · , i=1 1•þ|, = β1 = (b11 , . . . , b1p ) , · · · , βn = (bn1 , . . . , bnp ) n X i=1 ! αi bip . 1˜Ù 58 Ý Ú‚5•§| K B U1©¬•   β1  .  .  B=  .  βn dž/ªþ B ¤•äk 1 × n ‡«¬ ©¬Ý . dž§Œ±† Ú A ∈ Mm×n (K) U©¬¦{ƒ ¦. dd·‚       Pn β1 a11 · · · a1n j=1 a1j βj      . .. ..  ..  .   .   . (1.2.41.2) AB =  . . .  ·  ..  =    . Pn βn am1 · · · amn j=1 amj βj XJc[* (1.2.41.1) Ú (1.2.41.2) ùü‡úª, ÖöATØJuy, ùü‡ªf¢Sþ‰ƒ c·‚„ íØ 1.2.9 Jø # y². •(ƒ/`, ·‚w : • AB 1 j ƒ. Á • AB 1 i 1´ B ƒ. XJÓžò A U (1.2.41.3) ˆ‡ ƒ‚5|Ü, T‚5|Ü¥Ñy êiXêT•Ý B 1j ˆ‡1ƒ‚5|Ü, T‚5|Ü¥Ñy êiXêT•Ý A 1i1 ©¬, B U1©¬, ·‚„Œ±   β1  .   AB = α1 , . . . , αn  ..   = α1 β1 + · · · + αn βn . βn ØL, ùpÖöI‡3¿, z‡ αj βj Ñ´ m × p Ý , Ø´ê ! Ø• kvkÖö3¿ : XJ·‚ò A U1©¬, ò B U   A1  .  .  A=  .  , B = (B1 , . . . , Bp ) Am ©¬, = Ù¥ A1 , . . . , Am • A 1•þ|, B1 , . . . , Bp • B •þ|, @o# ©¬•ªeE,Œ±ò A Ú B ‰©¬¦{. dž (J´     A1 A1 B1 · · · A1 Bp  .   ..  ..  ..  B1 , . . . , Bp =  ... (1.2.41.4) AB =  . .     Am Am B 1 · · · Am B p ù‡ªfÙ¢Ø ´Ý ¦{ ½Âí ? Ø U1ÚU ‰ ©, ·‚„²~ŠâÝ ‡†dk' -‡½Â. ½ Â 1.2.42. A•n • 3é , XJ•3˜« ©  A11 A12  A22  0 A= . ..  .  . . 0 0 ‚NC ª¦ ··· ··· .. . ···  A1r  A2r  ..   .  Arr m5‰· ©. ±e´˜ 1.2 Ý Žâ 59 Ù¥ A11 , . . . , Arr ù Ý ¬Ñ´• , …§‚e• Ý ƒ Ü´ 0, @o·‚¡ A ´˜‡O þn (quasi-upper triangular matrix) ½©¬þn (block upper triangular matrix). aq/Œ ±½ÂOen (quasi-lower triangular matrix) ½©¬en (block lower triangular matrix). XJ•3˜« © ª¦   A11   A22     A= ..  .   Arr Ù ¥ A11 , . . . , Arr ù Ý ¬ Ñ ´ • , … Ý ˜ x ? ƒ Ü ´ 0, K ¡ A ´ ˜ ‡ O é (quasi-diagonal matrix) ½ö©¬é (block diagonal matrix). kž• ! Ö¡˜m, Œ±òþ ¡ ©¬é ¤ A = diag(A11 , . . . , Arr ). 5¿, 3½Â 1.2.42 ¥, ÷Xé ÒØUŽŠOé (1.2.43) Oé ´ © ªƒÓ ‚ Ý ¬ A11 , . . . , Arr Ñ‡¦´•   1 2 3 0 0   2 1 1 0 0   0 0 0 1 −1     0 0 0 −1 2   0 0 0 −1 −3 . Ïd, Ý . ƒm $Žéõžÿ”é Ý   A11   A22     , A= ..  .   Arr Oé ˜ •B. ~X, e  B11  B22  B= ..  .        Brr ,K  A11 B11   AB =     A22 B22 ..      . Arr Brr d , ØJy²: ±þOé Ò‰Ñ A _Ý A Œ_ …= z‡Ý ¬ A11 , . . . , Arr ÑŒ_,  −1  A11   A−1   22 −1   A = ..  .   −1 Arr . ~ 1.2.44. y35•Ä˜‡©¤ 2 × 2 = 4 ‡«¬ Ú s := n − r …^‡¤áž • . ©¬Ý M = A C ! 0 , Ù¥ A, B ©O• r B 1˜Ù 60 ·‚5y²: M Œ_ Äk, Ï•´• 3˜‡Ú M Œ …= A Ú B þŒ_, , ¤± M Œ_ ƒÓ ©¬Ý du M km_ (íØ 1.2.29 (2)). ! X1 Y1 N= ¦ X2 Y2 A C (1.2.44.1) OŽ˜eùp ©¬Ý 0 B ! X1 X2 Y1 Y2 ! = Ir 0 0 Is = Ir 0 = Ú‚5•§| A−1 −B −1 CA−1 M km_ ! 0 . B −1 ¿g´`• ! ¦ÈŒ± AX1 CX1 + BX2 (1.2.44.2) …3^‡¤áž M −1 Ý AY1 CY1 + BY2 ! 0 Is ! (ùp·‚J2Öö5¿, ±þ¦È OŽL§¥I‡AO %, Ï•©¬Ý ¥ Ý ¬˜„5` ØU”ê˜ ?¿ ††m^S, ¤±3‰¦{$Žž˜½‡©˜Ù=‡3†>=‡3m>. žÖ ög1 y: •, C, Y1 , X2 ùA‡Ý Ø˜½´• , þ¡Ñy CX1 + BX2 ƒa LˆªÑ ´k¿Â .) ±þ©¬Ý m ª`² AX1 = Ir , AY1 = 0 , CX1 + BX2 = 0 , CY1 + BY2 = Is . d AX1 = Ir Œ• A km_. Ï•®• A ´• , ¤±ù`² A Œ_ (íØ 1.2.29 (2)), … X1 = A−1 . u ´ 2 ( Ü ª AY1 = 0 Œ • Y1 = 0. 2 l ª CY1 + BY2 = Is = • B Œ _, … B −1 = Y2 . • , d CX1 + BX2 = 0 Œ X2 = −B −1 CX1 = −B −1 CA−1 . ¤±·‚y² ¤I (Ø. y²(åƒ ! ·‚JÆÖöé'˜eþ¡ XJ ù‡Ý a c Œ_ 0 b ~fÚÊÏ 2 × 2 êiÝ œ¹ ∈ M2 (K), = a, b, c ∈ K Ñ ´ ê, @ o (1.2.21) ˜ ã ? Ø ¿ © 7 ‡ ^ ‡ ´ ab − 0c = ab 6= 0, ! ^‡¤ážÙ_Ý ÉÓƒ?. ( J w Š · ‚, ! b 0 1 = • ab −c a a−1 0 . l/ªþw, • ù‡(JÚþ¡©¬Ý M _Ý f ˜—. ¢S −1 −1 −b ca b þùpkü‡-‡ «O: 1˜, éuêiÝ ó, ab ù‡¦Èo´k¿Â , A, B þ•Ý ž, AB ù‡¦ÈØ˜½´k¿Â (þ¡·‚¿ØI‡•vkb½ r = s = n − r). , , éuê 1 œ¹ ó, ab c Ú b−1 ca−1 •´Ó˜‡ê ü«ØÓ { ®, ùü« {Ñk¿Â. A, B, C −1 þ•Ý ž, B CA−1 ù‡Lˆª¥n‡¦ÈÏf ^S´ØU‘¿CÄ . ¤±, éuþ¡©¬ Ý M ¦_¯K, •,• (J(¢ÚêiÝ œ¹š~ƒq, ·‚ýéØU‘¿@ ^ (1.2.21) ¥ ¦_úª, Ï•@‡úªéu©¬Ý 5`éŒUvk¿Â. −1 ,˜•¡, ·‚TXÛ)º±þ¦ M −1 (JÚ (1.2.21) ¥(Øƒm²w ƒq5Q ? c[* (1.2.44.2) ª¥Ý , ·‚ØJuy (1.2.44.1) ª Ÿþ„´ƒ uü‡•§|, 1˜‡'uC X1 , X2 , 1 ‡'uC Y1 , Y2 . ù˜:´ÚêiÝ œ¹ ˜ . ¤±, X J·‚æ^)‚5•§| *:, @o©¬Ý œ¹ÚêiÝ œ¹ƒqÝÒAOp . ·‚ • éuêiCþ ‚5•§|, ·‚Œ±^pd–e ž {. ¤±·‚¬‰XeX Ý C 1.2 Ý Žâ 61 †: 1 0 0 1 ! a−1 −ca−1 0 1 ! a 0 c b 1 0 L2 →L2 −cL1 −−−−−−−−→ ·‚• Ð {ü OŽ 0 b ! L1 →a−1 L1 1 c 0 b a−1 0 L2 →b−1 L2 1 0 0 1 a−1 −1 −1 −b ca −−−−−−−→ −−−−−−−→ 0 1 0 ! b−1 1C†Ù¢éA ´†¦Ð Ý . ¤±þ¡1C†öŠ^Ý ƒ¦ À ! ! ! ! ! 1 0 1 0 a−1 0 a 0 1 0 1 0 a−1 0 = 0 b−1 −c 1 0 1 c b 0 1 0 1 −b−1 ca−1 b−1 ÐyÒ´ yŒ• 1 0 ! 0 b−1 1 0 −c 1 ! a−1 0 ! 0 = 1 a−1 −1 −1 −b ca 0 ! b−1 @oy3k‡¯K´, XJrþ¡˜X öŠ¥ Ý ÚÐ 1C†¥Ñy a, b, c Ü†¤ Œ i1 A, B, C ¬u)ŸoQ ? ŠO h² •ÆÐ¯ Öö, @‡:s get í? ! A B g • K 1.6. M • n • , ©¬• M = , Ù¥ A , D ©O• r Ú n−r • C D (B , C KØ˜½´• ). b A Œ_. Á¦˜‡ © ªƒÓ ©¬Ý Q ¦ M Q ´ ( © ªƒÓ ) Oen . 1.2.5 SK S K 1.2.1. OŽ  −1 4   0 1 3 −1 2 −1  λ1  0 0 0 λ2 0   −1 2 1   −3 1 0 2 1  2    ! 0 1 2 i −i 1 −1 0 1 −1 1      i 0       ,    1 −1 1    2  , 1 2 + i  ,  0 2 0 1 −1  −i −1  0 1 0 −1 −1 2 −1 1 0 −2 1 1 1 −1 −1       λ1 0 0 0 0 a11 a12 a13 a14 a11 a12 a13 a14  0      0 λ2 0  0  a21 a22 a23 a24  ,  . a21 a22 a23 a24   0 0 λ3 0  λ3 a31 a32 a33 a34 a31 a32 a33 a34 0 0 0 λ4    S K 1.2.2. OŽ   a11  (x, y, z) a21 a31 a12 a22 a32   a13 x   a23  y  , a33 z    a b c 1    S K 1.2.3. A =  c b a, B = 1 1 1 1 1 T T T OŽ AB − BA, (AB) Ú A B . a11   a21 (x1 , · · · , xn )   ..  . an1  a c  b b . c a a12 a22 .. . an2 ··· ··· .. . ···  a1n    y1 a2n   .    ..    ..  . .  yn ann 1˜Ù 62 S K 1.2.4.  2  2 f (X) = X − X − 1, A = 3 1 Ý Ú‚5•§|  1  2. OŽ f (A). 0 1 1 −1 S K 1.2.5. y²: XJ A, B ∈ Mn (K) Ñ´þn þn , @o AB •´î‚þn . , @o AB •´þn  2 1 1   S K 1.2.6. •Ä¹k˜‡ëê a Ý A =  a 0 −1. ¯ a −1 0 3 B 6= 0 AB = 0. ^‡¤áž, äN‰Ñ˜‡ù Ý B. ; XJ A, B Ñ´î‚  S K 1.2.7. é?¿ 1 1 !n B¦ cos θ sin θ , − sin θ cos θ !n . A ∈ Mn (K). y²: 1. A + AT , AAT , AT A Ñ´é¡Ý A − AT ´‡é¡ , . 2. XJ A ´é¡ , @oéu?¿ P ∈ Mn (K), P T AP •´é¡ u?¿ P ∈ Mn (K), P T AP •´‡é¡ ; S K 1.2.9. †. • ê n OŽ 1 0 S K 1.2.8. ŸoŠž•3 3 A, B ∈ Mn (K) Ñ´é¡Ý . y²: AB E´é¡  0  S K 1.2.10. ¦Ñ M3 (K) ¥¤k† J = 0 0  S K 1.2.11. •Ä n y²: é?¿    J =   • † Ý .  1 0  0  1 Œ 0 .. . .. .     .  1 0 • a11   a21 A=  ..  . an1 a12 a22 .. . an2 ··· ··· .. . ··· äÛž AJ = JA.  a1n  a2n  ..  , .  ann  0 1   0 J =   .. . 0  0 1 . 0 .. . .. ··· 0 1 .. , @oé ¿©7‡^‡´ A, B Œ ê m ≥ n þk J m = 0. S K 1.2.12. ‰½ü‡ n OŽ AJ Ú JA, ¿ 0 1 0 0 ; XJ A ´‡é¡     .    . 0 1.2 Ý Žâ 63 S K 1.2.13. ‰½ü‡ n  0 ···  ..  .     A=      • 0 .. . .. . ··· .. . .. . .. . .. . a1k .. . .. . .. . a1n .. .      an−k+1, n   ,   0  ..   .  0     J =     1 0 0 .. . .. . ··· .. . .. . .. .  0 ..  .    0 .   1 0 0 OŽ JA. S K 1.2.14. é?¿ ê n OŽ  −1 1 −1 −1 1 −1   −1 −1 S K 1.2.15. ‰½ n é n −1 −1   , −1 −1 −1 1 −1 1  λ1   D=   λ2 Ñ´é .  λ  0 0 1 λ 0 n 0  1 . λ  ..   , Ù¥ λ1 , · · · , λn üüØÓ.   . λn y²: † D Œ † Ý S K 1.2.16. ‰½ A ∈ Mn (K). y²: XJ A Ú Mn (K) ¥¤kÝ þÝ , =, •3 λ ∈ K ¦ A = λIn . S K 1.2.17. n ≥ 2 •óê. y²: •3 A ∈ Mn (R) ¦ S K 1.2.18. A, B þ• n ¤á, žÞÑ‡~. • . ÑŒ †, @o A ˜½´‡X A2 + In = 0. äe¡z‡Øä´Ä˜½¤á. e¤á, ž‰Ñy². eØ 1. (A + B)2 = A2 + 2AB + B 2 . 2. e AB = B … B ´š"Ý , K A = In . 3. e A, B Ú A + B ÑŒ_, K In + BA−1 •Œ_, ¿…Ù_Ý S K 1.2.19. A ∈ Mn (R) •é¡Ý S K 1.2.20. A, B ∈ Mn (R) •é¡ K A = B = C = 0. S K 1.2.21. A ∈ M2 (K). b 1. y²•§| A xx12 = 0 0 •3 • A(A + B)−1 . . y²: XJ A2 = 0, K A = 0. , C ∈ Mn (R) •‡é¡ êm≥2¦ . y²: XJ A2 + B 2 − C 2 = 0, Am = 0. kš"). 2. y² A2 = 0. S K 1.2.22. ée z‡^‡, ©OÞ~`²•3Xê ´ ê 2 • A ÷vT^‡: 1˜Ù 64 Ý Ú‚5•§| A2 = I2 . 1. A 6= ±I2 2. A2 = −I2 . 3. A 6= I2 S K 1.2.23. A3 = I2 .  1  A = 15 4  3 2  2 0. OŽ A−1 ¿|^¤ 2 1   2x + 3y + 2z   15x + 2y    4x + 2y + z S K 1.2.24. ¦e  1  2 1 Ý 3 1 −5 −6 1  1 −8   2  1 ,  −11 1 1 1 X Ú3 • Y ¦ 0 −1 ! 2 , 8 1  Y 0 1 ··· .. . .. .  1 ..  .   ,  1 1  2 1    B=     1 0    0,   −1 S K 1.2.25. ¦ 2 • 2 −1 S K 1.2.26. ‰½ n ! 5 X= −3 • X ¦ S K 1.2.27. b1 = b2 = b3   2 1   −1   ,  −1    1  1 1 −1 −1 1 −1 1 −1 1 2 −1   −1 1   2  = 1 0 2  1 2 −1 1 1  1 2 1 2       1 2  1  0 . 1 •  1    A=   ¦n = _Ý  −1 1 1 (J)•§| 1 1  1 2 1 1 2 .. . .. . .. . 1 1 2     .    AX = B. A ∈ Mn (K) Œ_. y²: 1. XJ A é¡ (‡é¡ 2. XJ A ´þn ), K A−1 •´é¡ (en (‡é¡ ), K A−1 •´þn ). (en S K 1.2.28. A ∈ Mn (K) ÷v Am = 0, Ù¥ m ∈ N∗ . y²: In − A Œ_, ¿…•3gê u m Ä˜õ‘ª f ¦ ). (In − A)−1 = f (A). S K 1.2.29. b g(X) ∈ K[X] ´ m gõ‘ª, m ≥ 1, … g(0) 6= 0. A ∈ Mn (K) ÷v g(A) = 0. −1 y²: A Œ_, ¿…•3gê u m õ‘ª f ¦ A = f (A). S K 1.2.30. b A, B ∈ Mn (K) ÷v A + B = AB. y²: 1.2 Ý 1. Ý Žâ 65 A − In Œ_. 2. A Ú B Œ S K 1.2.31. †. A•n • . y²: e A2 = In … A 6= I, K A + I ØŒ_. S K 1.2.32. B ∈ Mn (K) Œ_, u, v ∈ K n×1 , A = B + uv T . b c := 1 + v T B −1 u Ø u 0. y²: A Œ_, … A−1 = B −1 − c−1 B −1 uv T B −1 . (dúª¡• Sherman–Morrison úª.) S K 1.2.33. A ∈ Mn (R) ÷v A2 + 2A + In = 0. 1. y²: é?¿ a ∈ R, Ý 2. ¦ A + 4In A + aIn Œ_. _. S K 1.2.34. éu?¿ M ∈ Mn (K), In + M Œ_ž·‚½ÂÙ Cayley C† (Cayley transformation) • f (M ) := (In − M )(In + M )−1 . b A ∈ Mn (K), In + A Œ_. y²: 1. (In + f (A))(In + A) = 2In . (Ïd In + f (A) Œ_.) 2. f (f (A)) = A. S K 1.2.35. n ≥ 2, A ∈ Mn (K). b A z˜1 ƒƒÚÑ u a. 1. y²: e a = 0, K A ØŒ_. 2. b A Œ_, y² A−1 S K 1.2.36. z˜1 A, B ∈ Mn (K). b u a−1 . ƒƒÚÑ In − AB Œ_. y²: In − BA •Œ_, … (In − BA)−1 = In + B(In − AB)−1 A . S K 1.2.37. b A, B ∈ Mn (K) þŒ_. 1. Þ~`²: A + B kŒUØŒ_. 2. b A + B •Œ_. y²: A−1 + B −1 Œ_, … (A−1 + B −1 )−1 = A − A(A + B)−1 A = A(A + B)−1 B = B(A + B)−1 A . S K 1.2.38. 1. éÑk•õ‡1naÐ 2. Ý ¦§‚ λ •š"~ê. éÑk•õ‡1naÐ 3. y²: ?Û1 ª u1 • ÑŒ± ¦È Ý u 0 −1 ¦§‚ ! 1 . 0 ¦È u ¤k•õ‡1naÐ Ý λ 0 ! 0 . λ−1 ¦È. 1˜Ù 66 S K 1.2.39. a1 , · · · , an ∈ K þ•š"~ê, ¦±e n  0 a1 .  .. 0 a2  . .. . . ..  A =  .. . . .  . ..  .. . 0 • _  1   n  n − 1  A= .  ..    3 2 S K 1.2.41. ¦±e n • ··· 2 3 1 2 n 1 .. .. . . 4 5 3 4 ··· ··· ··· .. . n−1 n−2 n−3 .. . 1 n  n  n − 1  n − 2  ..  .    2  1 _  1  1  1  A= 1 . . . 1 2π Ù¥ ω = cos 2π n + i sin n . S K 1.2.42. ¦±eÝ _         an−1  0 ··· ··· ··· 1 ω ω2 ω3 .. . ω n−1 1 ω2 ω4 ω6 .. . 1 ω3 ω6 ω9 .. . ··· ··· ··· ··· .. . ω 2(n−1) ω 3(n−1) ···  1  2  1  0   0 0 1 1 −1 0 0 0 1   ω n−1   ω 2(n−1)   ω 3(n−1)   ..   .  2 ω (n−1) _Ý −1 0 0 0 0 0 0 0 0 2 1 −1  0 0  0 0  0 0  2 3   −1 0 2 1 S K 1.2.43. y²: ?‰ m + 1 ‡š"Ý þƒ-. A1 , · · · , Am+1 ∈ Mm×n (K), Ù¥– S K 1.2.44. ^Ð Ý 1C†Ú C†òe  −1  3 0 S K 1.2.45. Ú‚5•§|  0 an S K 1.2.40. ¦±e n • Ý z•ƒ-IO/:    1 2 3 0 1 2 −1 0 1    1 0 −1 ,    2 1 4 2 1 4 0 2 3 äe¡ü‡Ý A, B ´Ä1 d ? ´Ä d?    1 −1 0 1 2 −2 −2    A= 0 0 1 0  ; B = −1 1 3 −1 1 1 −1 0 0 0  2  −1 0 kü‡Ý 3K 1.2 Ý Žâ S K 1.2.46. 67 A, B ∈ Mn (K). 1. y²: XJ A Œ_, @o±e^‡ (a) A † B 1 d: d. (b) B Œ_. (c) A † B d. 2. Øb½ A, B Œ_. b A † B 1 ž‰Ñ‡~.   1 0 0   S K 1.2.47. B = 0 −1 1 . 0 1 −1 1. Ñ˜‡ 2 × 3 Ý A ¦ AB 1˜m† B 0 A B 1˜m† B 1˜mØƒ . 2. b C ´ 3 Œ_• Ä, ž‰Ñ‡~. 3. d. ´Ä A † B ˜½ Ñ˜‡ØŒ_ 3 • D0 , ¦ BD0 . ´Ä BC • ˜m† B D, ¦ ˜m† B b ;2 Ñ˜‡ 2 × 3 Ý ˜m˜½ƒ BD ˜m† B ˜mØƒ . S K 1.2.48. A • n • , B • n×m Ý . b C†ƒ C¤ (In X) /ª. y² X = A−1 B. S K 1.2.49. 1˜mƒ d ? e´, ž‰Ñy²; eÄ, A ∈ Mm (K) Ú B ∈ Mn (K) þ•Œ_Ý ( ? e´, ž‰Ñy²; e ˜mƒ n × (n + m) Ý Œ ; Ø˜½ƒÓ). - C = S K 1.2.50. •Ä©¬Ý b A12 A22 A32  A13  A23  , A33 Ù¥z‡ Aij ∈ Mn (K) . A11 ®•Œ_. Á¦ 3n S K 1.2.51. b Œ_Ý P, Q ¦ P AQ äkXe/ª:   A11 0 0   P AQ =  0 Ù¥z‡ Bij ∈ Mn (K) . B22 B23  , 0 B32 B33 A ∈ Mn (K) Œ‰Xe©¬  A11  A21 A=  ..  . Ak1 A12 A22 .. . Ak2 ··· ··· .. . ···  A1k  A2k  ..   .  Akk Ñ˜‡ØŒ_ (A B) ²L˜X A−1 , B −1 ®•. ¦ C −1 .  A11  A = A21 A31 A0 ¦ Ù¥z‡ Aii þ•• . 3 Ð 1 0 B ! A . 0 1˜Ù 68 b A11 ®•Œ_. Á¦ n Œ_Ý  P ¦ S K 1.2.52. •ÄOé  A1k ..  .    Ai−1, k   Bik  .  Ai+1, k  ..   .  Akk ··· .. . ··· ··· ··· .. . ··· A12 .. . Ai−1, 2 Bi2 Ai+1, 2 .. . Ak2  J1  J = Ú‚5•§| P A äkXe/ª: A11  .  ..    Ai−1, i  PA =   0  Ai+1, 1  .  .  . Ak1  Ý  , Ù¥ J2 J3 J1 = −1 0 ! 1 −1 ,  −1  J2 =  0 0  0  1 , −1 1 −1 0 J3 = 2 0 1 2 ! . Á¦˜‡ 5 gÄ˜õ‘ª f (X) ¦ S K 1.2.53. •Ä n f (J) = 0.  J1  J2  J =   Oé  ..   , Ù¥éuz‡ i ∈ [[1, s]], Ji •Xe ni   . Js •  λi     Ji =     1 λi  1 .. . .. . .. .     .   1 λi Á¦˜‡gêØ‡L n S K 1.2.54. b Ä˜õ‘ª f (X) ¦ ˜‡ n ¢XêÝ y²: XJ A Ú AT Œ S K 1.2.55. b S K 1.2.56. S K 1.2.57. A, B ´Œ A•n • …= A Œ±©¬• A = A1 0 ! A2 , Ù¥ A1 , A3 þ•• A3 . †, K A2 = 0. ƒÓ • , ¿… AB = BA. é?¿ … A3 = In . OŽ !2000 0 −In ; A 0 A, B ∈ Mn (K), M = 1. y²: M Œ_ f (J) = 0. A B 1 A √2 3 2 A ! B . A A + B Ú A − B þŒ_. √ − 23 A 1 2A !2000 ê n, ¦ A 0 B A !n . 1.2 Ý 2. Žâ 69 M Œ_ž¦Ñ M −1 . S K 1.2.58. A, B, C ∈ Mn (K), M = 1. y²: M Œ_ 2. …= A C −B ! A . C A Ú B þŒ_. M Œ_ž¦Ñ M −1 . S K 1.2.59. •Ä˜‡©¬Ý M= y²: XJ M Œ_K B, C þŒ_. A C ! B , Ù¥ B, C þ•• D ( Œ ™7ƒÓ). 1 Ù •þ˜m9Ùf˜m ÏLc˜Ù'u‚5•§|ÚÝ ?Ø, ·‚F"Öö®²é“‚5“ê”¥ “‚5”˜c k ˜½ N¬. Ùm©·‚ò¬Ú\•\°( êÆŠó5£ã“‚5” é–Ú§‚ƒm“‚ 5” éX. Ù‡?Ø ¥%Vg´‚5˜m, •¡•þ˜m. ·‚o K •Eê• C ˜‡f•. 1½ 2.1 2.1.1 •þ˜m •þ˜m K n 9Ùf˜m (2.1.1) X (1.2.1) ã¤ã, é?¿ ê n, ·‚± K 1×n Ú K n×1 ©OL« K þ n ‘1•þÚ n ‘ •þ N ¤ 8Ü. ·‚Ø I‡«©1•þ½ö •þž, ½ö ·‚Ø¬·Ü¦^1 •þÚ •þž, •Œ±ò öÑ<ÚP• K n . Š•Ý \{ A~, K n ¥?¿ü‡ ƒƒmŒ±‰\{, Ù•ªÚì‹I©þ©OéAƒ \. =, e α = (a1 , . . . , an ) ∈ K n , β = (b1 , . . . , bn ) ∈ K n K α + β := (a1 + b1 , . . . , an + bn ) . d , ·‚„• Ý Ú~êƒmkê¦$Ž. ÏdŠ•A~, K n ¥ ƒÚ K ¥~ê•kê¦$ Ž, Ù•ª•Úì‹I©þ5†~ê‰ƒ¦. =, e c ∈ K, α = (a1 , . . . , an ) ∈ K n , K c.α := (ca1 , . . . , can ) . ÏL·ÜöŠ\{Úê¦, Œ± ¤¢‚5|Ü Vg. äN5`, e α1 , . . . , αr ∈ K n , Kù| ƒ ˜‡ K-‚5|ÜÒ´•˜‡UìXe•ª?1$Ž K n ¥ ƒ: c1 α1 + · · · + cr αr Ù¥z‡ ci ∈ K . Ï• K ¥•3~ê −1, ¤±é?¿ü‡•þ α, β ∈ K n , Œ±ÏL 1 · α + (−1) · β ù«‚5|Ü ˜‡# •þ, ù‡•þ ¡• α ~ β . ¤±, ~{ù«$ŽŒ±@•´d\{Úê¦ ) Ñ5 , Ÿþ·‚Œ±Ør§ Š˜«# $Ž. e¡·K¥ ˆ‡(ØŒ±† Uì½Â5Å˜ ·K 2.1.2: - V = K n . Uìƒc¤½Â ~êƒm ) ê¦$Ž, e 5Ÿ¤á: y. (V ¥?¿ü‡ ƒƒm 1. \{(ÜÆ: é?¿ α, β, γ ∈ V , (α + β) + γ = α + (β + γ). 71 ) \{Ú (V ¥ ƒÚ K ¥ 1 72 2. \{ Ù •þ˜m9Ùf˜m †Æ: é?¿ α, β ∈ V , α + β = β + α. 3. \{kð : •3 ƒ 0 ∈ V (¡• V ¥\{ α ∈ V þk α + 0 = α = 0 + α. (8 ÖöÙGù‡Vgƒ , ·‚•¬²~† Ù ¤ 0 ù«çN f.) ˜‡ð (identity element)) ¦ ± 0 5L« V ¥ \{ð , é¤k Ø72ò 4. \{_o•3: é?¿ α ∈ V , •3˜‡ ƒ α0 ∈ V ¦ α + α0 Ú α0 + α Ñ´˜‡\{ð . ?Û÷vd^‡ α0 Œ±¡• α ˜‡\{_ (additive inverse) ½öK (negative). ±þ 4 ‡5Ÿ•Ú V ¥ \{$Žk'. Uì8 Öö¬Æ “êÆŠó, Ï~±þ 4 ^ Œ±o(•ù ˜é{: V 'u\{$Ž/¤˜‡ †+ (commutative group) ½C + ∗ (Abelian group) . 5. ê¦kð : K ¥~ê 1 éuê¦$Ž÷v: 1 · α = α é¤k α ∈ V ¤á. 6. ê¦†~ê¦{÷v(ÜÆ: é?¿ c, k ∈ K Ú?¿ α ∈ V , (ck)α = c(kα). 7. ê¦Ú~ê\{÷v© Æ: é?¿ c, k ∈ K Ú?¿ α ∈ V , (c + k)α = cα + kα. 8. ê¦Ú•þ\{÷v© Æ: é?¿ c ∈ K Ú?¿ α, β ∈ V , c(α + β) = cα + cβ. Ø• Öö M¥dž´Ä®²,å˜‡ŒŒ ¯Ò. ±þù (Øéu K n óØ´š~w , ¯œí ? ·‚Zí‡ùo¤• rù w, ¯¢©‘Å^ QãÑ5, „‡x-Ù¯/r§ Š˜‡·KQ ? Ï•é{ü: Ï•±þù ¯¢•,{ü, ´¿Â›©-Œ. Ù¿Âƒ¤±-Œ, ´Ï• ù«y–Œ±‰š~2• í2. †é{`, ·K 2.1.2 •,´éu V = K n ù‡8Ü5•ã, Ù ¢Œ±kNõÙ¦ 8Ü V •U÷vÓ ˜|5Ÿ. ~ 2.1.3. ?¿ ½˜‡•þ α ∈ K n . ·‚P Kα := {cα | c ∈ K} . ÖöŒ± y, Š• K n ˜‡f8, Kα 'u\{Úê¦´µ4 . •Ò´`, XJü‡•þÑ5 g Kα ù‡f8, @o§‚‰\{ (JE,áu Kα (ù‡(Ø Ÿþ u·K 2.1.2 ¥ 5Ÿ 7); XJ˜‡•þ5gu Kα, @o§Ú?Û K ¥~ê‰ê¦ (JE,áu Kα (ù‡(Ø Ÿþ u·K 2.1.2 ¥ 5Ÿ 6). ¤±, ·‚Œ±@• Kα ù‡8Üg •‘k\{Úê¦$Ž. ÖöéN´ y, e V = Kα, K V 'u§ \{Úê¦•÷v·K 2.1.2 ¥ ¤kl^5 Ÿ. (éuÙ¥ , 5Ÿ, X5Ÿ1, 2, 5–8, Q,ù 5Ÿéu¤k K n ¥ ƒÑ¤á, @o , §‚éuf8 Kα ¥ ¤k ƒ•Ñ¤á. ¤±, ¢Sþ•I‡ y\{ð •áu Kα ù‡f 8, …, éu Kα ¥ ƒ5`, 3 Kα ù‡f8¥Ò®²•3§ \{_ .) ~ 2.1.4. - V • K n ¥1˜‡‹I©þ u0 ¤k•þ ¤ f8, =, V := {(a1 , . . . , an ) ∈ K n | a1 = 0} . ØJ y, V 'u\{Úê¦Ñ´µ4 •Ñ¤á. ∗C (Abel, 1802–1829) ´é%Í¶ . †~ 2.1.3 aq, ·K 2.1.2 ¥ UâêÆ[. ¤kl^5Ÿéù‡ V 2.1 1½ •þ˜m 73 ±þ·‚w , U ÷v·K 2.1.2 ¥l^5Ÿ 8Ü V ýØŽ K n ù˜«a. 8Ü. XJ· ‚rÙ¥¤k Ÿ únz 5ŸÄ–Ñ5, Œ± ˜„ ‚5˜m½•þ˜m Vg. ØL, · ‚Žrù‡˜„ Ä–½Âí´ ¡ !. !¥·‚k5?Ø˜ •äN ~f, ù ~f Ä þÑ5gu K n ˜ AÏf8. ½ Â 2.1.5. V • K n ˜‡f8. XJ±e¤k^‡¤á, ·‚Ò¡ V ´ K n ˜‡‚5f˜ m (linear subspace)!•þf˜m (vector subspace) ½ö (•{Ñ/`¤) f˜m (subspace): 1. K n ¥ \{ð 2. V 'u K n ¥ , ="•þ 0 = (0, . . . , 0) áu V . \{µ4, =, éu?¿ u, v ∈ V þk u + v ∈ V . Šâù‡^‡, ·‚Œ±r8Ü K n þ®k \{$Ž•› V ¥ ƒ5•Ä. ¤±·‚Œ± ` V l K n @pU« (inherit) ˜‡\{$Ž, ½ö` K n þ \{p (induce) V þ ˜‡\{. 3. V 'u K n Ú K ¥~ê ê¦$Žµ4, =, éu?¿ v ∈ V Ú?¿ λ ∈ K, þk λv ∈ V . Šâù‡^‡, ·‚Œ±r8Ü K n † K ¥~êƒm ê¦$Ž•› V ¥ ƒ5•Ä. ¤ ±·‚Œ±` V l K n @pU« ˜‡ê¦$Ž, ½ö` K n þ ê¦p V þ ˜‡ê ¦. ØJwÑ, e V ´ K n f˜m, K V ¥?¿˜| ƒ ?¿ K-‚5|ÜE,áu V . ~ 2.1.6. e V ´ K n ¥"•þ 0 |¤ •k˜‡ ƒ f8, K V ´ K n f˜m, ù‡f˜m ¡•"•þ˜m (zero vector space, zero space). , , K n gC ,•´ K n f˜m. ùü‡f˜m¡•²… (trivial) f˜m. ·K 2.1.7: b V ´ K n ¥ l^5Ÿ Ñ¤á. ˜‡f˜m. K, é V l K n U« \{$ŽÚê¦$Ž, ·K 2.1.2 y². ·‚3~ 2.1.3 ¥Q`L, ·K 2.1.2 ¥ 5Ÿ 1, 2, 5–8 3 V 'u\{Úê¦µ4 cJeg ,¤á. 5Ÿ 3 d½Â 2.1.5 ¥ 1˜‡^‡ y. Šâ\{Úê¦ $Žµ45, ·‚• V ¥ ?¿ü‡ ƒŒ±‰~{. éu?Û α ∈ V , Ï•®• 0 ∈ V , ¤± V 'u~{ µ45wŠ·‚ α0 := 0 − α ´ V ¥ ƒ. † U½Â yŒ• α0 ´ α ˜‡\{_. ù Ò y ·K 2.1.2 ¥ 5Ÿ 4. g • K 2.1. V • Kn f8. y²: V ´ K n f˜m …= ±eü‡^‡Óž¤á: 1. V ´š˜8Ü. 2. éu?¿ u, v ∈ V Ú?¿ λ ∈ K, þk u + λv ∈ V . 2.1.2 ‚5|Ü†‚5ƒ' y3·‚5?Ø K n ¥f˜m ã-‡~f. XJØ´kƒ‡ (², Ö¥J •þ| (vector system) o´•=kk•õ‡ (Œ±´" ‡) •þ •þ|. 3˜‡•þ|¥, #N˜ •þ-EÑyõg. ·K 2.1.8: r ∈ N∗ , α1 , . . . , αr ∈ K n . ·‚P spanK (α1 , . . . , αr ) = {c1 α1 + · · · + cr αr | ci ∈ K} . 1 74 Ù •þ˜m9Ùf˜m =, spanK (α1 , . . . , αr ) ••þ| α1 , . . . , αr ¤k K-‚5|Ü ¤ 8Ü. K spanK (α1 , . . . , αr ) ´ K n f ˜ m, ¡ • • þ | α1 , . . . , αr ) ¤ (subspace generated (or spanned) by the vector system α1 , . . . , αr ). y². U½Â (½ Ü ¤ ) f˜m y=Œ. [!3‰Öö. •‡Ø–uÚåÜÂ, ·‚~~rPÒ spanK ¥ eI K Ž ,† { • span. 5P 2.1.9. 8 Öö¬N¬ , kžÿ#N¦^“˜•þ|”ù‡Vg¬‰·‚‘5B|. ,, ¶ gÂ, ˜•þ| (empty system) Œ±@•´"‡•þ ¤ •þ|. ·‚ ½, ˜•þ|)¤ f˜ m•"•þ˜m. •Ò´`, ·‚ ½: d˜•þ| ‚5|Ük…=k"•þ. (2.1.10) A ∈ Mm×n (K). Šâ (1.2.3) ˜ã¥ ½Â, Ý A m×1 3K ¥)¤ f˜m, =, C (A) = span(α1 , . . . , αn ) ˜m C (A) Ò´ A •þ| Ù¥    a1n a11  .   .   .  .  α1 =   .  , · · · , αn =  .  amn am1  •A •þ|. (aq/, A 1˜mÒ´ A 1•þ|3 K 1×n ¥)¤ f˜m.) éu?¿ •þ b = (b1 , . . . , bm )T ∈ K m×1 , 'uC X = (x1 , . . . , xn )T ‚5•§| AX = b Œ± ¤   x1  .   (2.1.10.1) (α1 , . . . , αn )  ..   = x1 α1 + · · · + xn αn = b . xn ùL², ¦)‚5•§| AX = b ¢SþÒ´¦Xê x1 , . . . , xn ¦ A •þ|±ù Xê‰‚ 5|ÜžU •þ b. Ïd, AX = b k) …= •þ b áu A ˜m. u´·‚w , •§| AX = b ) •35 ûu •þ b Ú A ˜m´Äkäá'X. ùÒ^•þ˜m Šó•x ‚5•§|) •35. e52?Ø‚5•§|) ½Â´š~k^ . ½ Â 2.1.11. ¦ α1 , . . . , αn • •˜5¯K. XJ·‚k•Äàg •þ˜m K m×1 ¥ ƒ. XJ•3Ø ‚5•§|, ØJuye¡ •0 ~ê c1 , . . . , cn ∈ K c1 α1 + · · · + cn αn = 0 K ¡ • þ | α1 , . . . , αn 3 K þ ‚ 5 ƒ ' (linearly dependent). Ä K ¡ α1 , . . . , αn 3 K þ ‚ 5 Ã ' (linearly independent). XJP A ∈ Mm×n (K) •± α1 , . . . , αn Š• •þ| Ý , @o(Ü (2.1.10.1) =•: α1 , . . . , αn ‚5Ã' …= àg‚5•§| AX = 0 vkš²…). ·‚Ö¿5½: ˜•þ|´‚5Ã' . ØJwÑ, •¹"•þ •þ|o´‚5ƒ' ! (2.1.12) XJ• ˜‡•þ α ∈ K m ¤˜‡•þ|, KT•þ|‚5ƒ' ¿©7‡^‡´•3 š"~ê c ∈ K ¦ cα = 0. ù , du α = 0. ¤±, ˜‡•þ α ¤ •þ|‚5Ã' … = α ´š"•þ. XJ ü‡•þ α, β ∈ K m ¤˜‡•þ|, Ke •ã d (äN yL§3‰Öö): 2.1 1½ •þ˜m 75 • •þ| α, β ‚5ƒ'. • α Ú β öƒ¥– k˜‡Œ± ¤,˜‡ „~ 2.1.3 ¥PÒ), ‡o β ∈ span(α) = Kα. • ‡o α Ú β ~ê . •Ò´`, ‡o α ∈ span(β) = Kβ (ë öƒ¥•¹"•þ, ‡o α Ú β Ñ´é• š"~ê . XJl‹IAÛ Ý5•Ä, b α Ú β ©O´ R2 (½ R3 ) ¥l : O ••ü‡: P Ú Q − − → −−→ −−→ −−→ AÛ•þ† : α = OP , β = OQ, K α, β ‚5ƒ' AÛ)ºÒ´ OP † OQ ü‡•þ ‚, ½ö ` O, P, Q n‡: ‚. • Qã•B, XJ˜‡•þ β Œ± ¤•þ| α1 , . . . , αn ‚5|Ü, =, β ∈ span(α1 , . . . , αn ), ·‚•¡ β Œd α1 , . . . , αn ‚5LÑ (linearly represented). U·‚ƒc ½, ˜•þ| ‚5| Ük…=k"•þ. ¤±, •k"•þŒ±@• ˜•þ|‚5LÑ. e¡ù^{ü (Ø·‚3‰Öög1y². Ún 2.1.13: α1 , . . . , αn , αn+1 , . . . , αn+r • K m ¥ •þ|. K span(α1 , . . . , αn ) ⊆ span(α1 , . . . , αn , αn+1 , . . . , αn+r ). ùL², •þ|¥•¹ •þ|U ‚5LÑ •þ• õ. ·K 2.1.14: K m ¥ •þ| α1 , . . . , αn (Ù¥ n ≥ 1) ‚5ƒ' ± Ù{•þ‚5LÑ. •þ õ, T ¿©7‡^‡´Ù¥˜‡•þŒ y². ky7‡5. =‡y: XJ α1 , . . . , αn ‚5ƒ', KÙ¥˜‡•þŒ± Šâ‚5ƒ' ½Â, •3Ø • 0 ~ê c1 , . . . , cn ∈ K ¦ Ù{•þ‚5LÑ. c1 α1 + · · · + cn αn = 0 . Ø” cn 6= 0. KþªŒ±U • αn = c1 − cn cn−1 α1 + · · · + − αn−1 . cn ùÒ`² αn Œ± α1 , . . . , αn−1 ‚5LÑ. 2y¿©5. =‡y: e α1 , . . . , αn Ù¥˜‡Œ± Ù{ •þ‚5LÑ, K•þ| α1 , . . . , αn ‚5ƒ'. Ø” αn Œ± Ù{•þ‚5LÑ. =, •3~ê λ1 , . . . , λn−1 ∈ K ¦ αn = λ1 α1 + · · · + λn−1 αn−1 . cn = −1, c1 = λ1 , . . . , cn−1 = λn−1 =• c1 α1 + · · · + cn−1 αn−1 + cn αn = 0 . Ï• cn = −1 6= 0, ¤±þª`² α1 , . . . , αn ‚5ƒ'. ·‚Q²J , •¹"•þ •þ|˜½‚5ƒ'. ù‡(Ø ,Œ±† ½Â5 y. ,˜• ¡, ù•´·K 2.1.14 ˜«AÏœ¹, Ï•"•þŒ± ?¿•þ| (•)˜•þ|) ‚5LÑ. e¡ (ØŒ±† † X·‚3 35 • VgL •~ l½Â5 y, •Œ±Š•·K 2.1.14 íØ : ˜‡ 5¥¤`, Ö^“AÛ•þ”ù‡c5LãÖö3AÛ†*½ÔnÆ¥n) ¡•“¥þ”, ¿g´QkŒ qk•• þ. •þVg. ù‡ 1 76 Ù •þ˜m9Ùf˜m íØ 2.1.15: α1 , . . . , αn , αn+1 , . . . , αn+r • K m ¥•þ|. e α1 , . . . , αn ‚5ƒ', K α1 , . . . , αn , αn+1 , . . . , αn+r •‚5ƒ'. †óƒ, e α1 , . . . , αn , αn+1 , . . . , αn+r ‚5Ã', K α1 , . . . , αn •‚5Ã'. ~ 2.1.16. α ∈ K m , c ∈ K. K•þ| cα, α ‚5ƒ', Ï• cα Œ±dü‡•þ α ¤ •þ| ‚5LÑ. ŠâíØ 2.1.15, éu?¿ α1 , . . . , αn ∈ K m , •þ| cα, α, α1 , . . . , αn 7,•‚5ƒ'. AO/, ù‡~f`²˜‡‚5Ã' •þ|¥ØŒUk-EÑy •þ. íØ 2.1.15 wŠ·‚, 3˜‡‚5Ã' •þ|¥ K˜ •þ, •{ •þ|˜½E,‚5 Ã'. ´, XJ3˜‡‚5Ã' •þ|¥V\˜ •þ, @o*Œ •þ|ÒØ˜½E‚5Ã ' . ·K 2.1.17: α1 , . . . , αn • K m ¥‚5Ã' •þ|. Kéu?¿ β ∈ K m , •þ| α1 , . . . , αn , β E,‚5Ã' ¿©7‡^‡´ β ØU ‚5LÑ, ½=, β ∈ / span(α1 , . . . , αn ). †óƒ, α1 , . . . , αn , β ‚5ƒ' ¿©7‡^‡´ β Œ± α1 , . . . , αn ‚5LÑ. α1 , . . . , αn y². e β Œ α1 , . . . , αn ‚5LÑ, KŠâ·K 2.1.14 =• α1 , . . . , αn , β ‚5ƒ'. ‡L5, b ®• α1 , . . . , αn , β ‚5ƒ', 5y² β Œ± α1 , . . . , αn ‚5LÑ. ¯¢þ, ‚5 ƒ' ½ÂL², •3Ø • 0 ~ê c1 , . . . , cn , b ∈ K ¦ c1 α1 + · · · + cn αn + bβ = 0 . e b = 0, K c1 , . . . , cn Ø • 0. džþªL² c1 α1 + · · · + cn αn = 0. ù ù‡K ^‡gñ. ¤±7, b 6= 0. u´ c c 1 n β= − α1 + · · · + − αn . b b † α1 , . . . , αn ‚5Ã' ¤±, β Œd α1 , . . . , αn ‚5LÑ. (2.1.18) 3 (2.1.12) ˜ã¥·‚?ØLü‡•þ‚5ƒ' AÛ¿Â. y35wn‡•þ œ¹. −→ −−→ b α, β , γ © O ´ R3 ¥ l : O • • n ‡ : A, B, C A Û • þ: α = OA, β = OB, −−→ γ = OC. ·‚5`², •þ| α, β , γ ‚5ƒ' AÛ¹ÂTÐÒ´ O, A, B, C o‡: ¡, =, U •¹3Ó˜‡²¡S. (5¿: ·‚ùp` “ ¡”#N“òz” œ¹. =, XJ O, A, B, C o‡: Ó3˜^†‚þ, @ow,•3˜‡²¡Óž²Lùo‡:. •,džù ²¡¿Ø•˜, ·‚ E,@•dž O, A, B, C o: ¡.) XJ α, β ‚5ƒ', KŠâ (2.1.12) ¥ ?Ø, O, A, B n:´ ‚ . ¤±dž O, A, B, C ˜ ½´?u,‡ Ó ²¡S. íØ 2.1.15 wŠ·‚, dž α, β, γ n‡•þ ¤ •þ|•‚5ƒ '. ¤±, ±e·‚•‡2?Ø α, β ‚5Ã' œ¹. dž O, A, B n‡:Ø ‚. Ïd•3•˜ ˜‡²¡²L O, A, B n‡:. ù‡²¡¡ •²¡ OAB. 5¿ α ~ê ÒéAul O ••†‚ OA þ,: AÛ•þ, β ~ê KéAul O ••†‚ OB þ,: AÛ•þ. ŠâAÛ•þ\{ ²1o>/{K, α ˜‡~ê \þ β ˜‡~ê ¤ AÛ•þƒª:˜½´²¡ OAB S ˜:. ‡ƒ, XJ P ´²¡ OAB S ˜ −−→ −→ −−→ :, K7,•3~ê c1 , c2 ∈ R ¦ OP = c1 OA + c2 OB. Ïd, O, A, B, C o: ¡ (=, : C −−→ −→ −−→ u²¡ OAB ƒS) …= OC ´ OA, OB ‚5|Ü, =, γ Œd α, β ‚5LÑ. Ï•·‚®² b½ α, β ‚5Ã', ¤±Šâ·K 2.1.17, ù du` α, β , γ ‚5ƒ'. –d·‚ ² ƒc Øä. 2.1 1½ 2.1.3 •þ˜m •þ| 77 •Úf˜m‘ê þ˜!·‚QJ , ˜‡•þ|¥ •þ õ, §U‚5LÑ •þ• õ (Ún 2.1.13). ¤±, XJ‰½ ˜‡f˜m V ⊆ K n , Ž‡ÏL‚5|Ü •ª V ¥¦ŒUõ •þ ($– V ¥¤k •þ), ·‚F"3À •þ|¥•¹¦ŒUõ •þ. ´, •þ|¥ •þõ N´ E¤‚5ƒ' y– (·K 2.1.17), ù¿›X•þ|¥k •þ¢Sþ´“P{ ”, §‚ŒUŒ± Ù{ •þ‚5LÑ (·K 2.1.14). !¥·‚ò‡?Ø, éu‰½ ˜‡f˜m ó, XÛÀ ˜ ‡•þ|âw “ØõØ ffÐ”. 3±e?Ø¥·‚~~ØI‡ Þ˜‡•þ|¥ ¤k•þ, ¤±·‚#NgC† æ^ ˜‡i1“LeZ‡•þ ¤ •þ|. e S • K n ¥ •þ|, ± span(S) L«T•þ|Ü¤ f˜m (ë„·K 2.1.8 ¥ PÒ). k5y²'uÜ¤f˜m ˜‡{ü(Ø. Ún 2.1.19: T • K n ¥ •þ|. K span(T ) ´ K n ¥•¹ T • f˜m. =, span(T ) ´ n n K ¥ ˜‡f˜m, ¿…éu K ¥?Û•¹ T f˜m V þk span(T ) ⊆ V . y². 3·K 2.1.8 ¥·‚®²`² span(T ) ´‡f˜m. Ï•?¿f˜m'u‚5|ÜöŠ´µ 4 , ¤±, XJ˜‡f˜m V •¹ T ¥¤k•þ, @o§•7,•¹•þ| T ¤k‚5|Ü, = V •¹ span(T ). 5¿, T w,•¹u span(T ). ¤±, l span(T ) ⊆ V •Œ±íÑ T ⊆ V . †óƒ, éu K n ¥ ?¿f˜m V 5`, T ⊆ V du span(T ) ⊆ V . −→ lAÛ†*þ`, bX α éAu˜m R3 ¥l :Ñu AÛ•þ OA, @o•¹•þ α ? −−→ ¿f˜m V ⊆ R3 •˜½ •¹†‚ OA. XJ β éAu˜m¥ AÛ•þ OB, ¿… β † α Ø ‚, @o•¹•þ| α , β f˜m˜½ •¹²¡ OAB. ±e(ØÙ¢3g•K 1.2 ¥ÒJ L . íØ 2.1.20: S Ú T • K n ¥ •þ|. XJ T ¥z‡•þÑŒ± •þ| S ‚5LÑ, K? ÛU T ‚5LÑ •þ•U S ‚5LÑ. •Ò´`, XJ T ⊆ span(S), K span(T ) ⊆ span(S). y². 3Ún 2.1.19 ¥ V = span(S) =Œ. ½ Â 2.1.21. S Ú T • K n ¥ •þ|. XJ S Ú T Œ±pƒ‚5LÑ (½ö`p•‚5|Ü), = S ¥z‡•þÑŒ± •þ| T ‚5LÑ, T ¥z‡•þ•ÑŒ± •þ| S ‚5LÑ, @o¡ S Ú T ùü‡•þ|‚5 d (linearly equivalent). ŠâíØ 2.1.20, S Ú T ‚5 d …= span(S) = span(T ). XJr K n ¥ z‡•þ| Nþ Š˜‡ÔN, ±¤kù ÔN ¤˜‡8Ü S , K‚5 d´8Ü S þ ˜‡ d'X (žÖösžm(@gCn)ùé{ ¹Â). g • K 2.2. Þ~`²: ¤¹•þê8ØÓ, ¿…Ø¹ú «: ù«~f3‘êé$ œ¹Òkí.) ½ Â 2.1.22. S • Kn ¥ •þ|, V ´ K n •þ ü‡•þ|kŒU‚5 d. (J ˜‡f˜m. 1. b T ´ S ˜‡f•þ| (= T ´l S ¥À ˜ •þ •þ|). XJ•þ| T ´ ‚5Ã' , ¿…ò S ¥?˜•þV\– T ¥ƒ *¿•þ|Ñ‚5ƒ', ·‚Ò¡ T ´ S ¥ ˜‡4Œ‚5Ã'| (maximal linearly independent system). 1 78 Ù •þ˜m9Ùf˜m • ˜ „ /, · ‚ F • K n ¥ ? ¿ õ ‡ (Œ ± ´ Ã • õ ‡) • þ ¤ • þ x (family of vectors) (Œ±#N˜ •þ3 F ¥-EÑy). XJ T ´ g F ¥ ˜ •þ ¤ •þ |, ·‚•Œ±aq/?Ø T 3 F ¥´Ä´4Œ‚5Ã'|. •Ò´`, XJ T ¥•þÑ5 gu F , T ´‚5Ã'|, ¿…éu F ¥ ?¿ ƒ γ, ò γ V\ T ƒ Ñ¬¦*¿ • þ|C¤‚5ƒ' , @o·‚¡ T ´ F ¥ ˜‡4Œ‚5Ã'|. AO/, ·‚Œ± F •f˜m V , l V ¥4Œ‚5Ã'| Vg. 2. XJ V = span(S), ·‚¡ S ´f˜m V ˜‡ (‚5) Ü¤| ((linearly) spanning system) ½ (‚5) )¤| ((linearly) generating system). XJ S Q´ V )¤|, q´‚5Ã' •þ|, ·‚¡ S ´ V ˜|Ä (basis). 3. XJ S ´ V ˜‡)¤|, ´l S ¥?¿ K˜‡•þƒ BØ2´ V ´V ˜‡4 (‚5) )¤| (minimal (linearly) generating system) ½4 (minimal (linearly) spanning system). ~ 2.1.23. V = K n . KXe•þ| ¤V )¤|, K¡ S (‚5) Ü¤| ˜|Ä: e1 = (1, 0, . . . , 0) , e2 = (0, 1, 0, . . . , 0) , . . . , en = (0, . . . , 0, 1) . (= ei • 1 × n Ä "˜Ý , Ù¥ ƒ 1 u1 i .) ±þù|Ä¡• K n basis)!g,Ä (natural basis) ½ ;‰Ä (canonical basis). IOÄ (standard ~ 2.1.24. •Ä K 3 ¥Xe•þ α1 = (1, 0, 0) , α2 = (0, 1, 0) , α3 = (0, 0, 1) , α4 = (1, 1, 1) , α5 = (1, 1, 0) . 1. •þ| α1 , α2 , α3 ´ K 3 ¥ ˜‡4Œ‚5Ã'|. •þ| α1 , α3 , α4 ´ K 3 ¥,˜‡4Œ 3 ‚5Ã'|. ùü‡•þ|•Ñ´ K ¥ 4 )¤|. AO/, ùü‡•þ| (ˆg) Ñ´ K 3 Ä. 2. ò•þ| α3 , α4 , α5 P• S, - V = span(S). K S ‚5ƒ', l S ¥? ü‡•þ •þ|Ñ´ S ¥ 4Œ‚5Ã'|. , , S ´ V )¤|, Ø´ V 4 )¤|. l S ¥ ?¿ K˜‡•þþŒ V ˜‡4 )¤|. ÖöŒU®²5¿ , 3~ 2.1.24 ¥·‚‘ œ/ÑÎÜ±eA : éuÓ˜‡f˜m V , ˜ ‡•þ|´§ 4Œ‚5Ã'|ž•˜½´§ 4 )¤| (ÏdAO/•´˜|Ä), …‡ƒ½ ,. d , V ?Û4Œ‚5Ã'|¤¹ •þê8ÑƒÓ. ·‚ e5 Ì‡8IÒý²ù 5 ŸØ´=3˜ AÏ ~f¥¤á, ´3•˜„ œ¹eÑ¤á. Äk, e¡ Ún Ÿþć¡˜ (Ø -#Qã. Ún 2.1.25: 1. S ´ V ¥ V ´ Kn ¥ f˜m, S ´ V ¥•þ ¤ •þ|. Ke 4Œ‚5Ã'|. 2. S ´ V ˜|Ä, = S ‚5Ã'¿…´ V 3. S ´ V ˜‡4 )¤|. ˜‡)¤|. •ã d: 2.1 1½ •þ˜m 79 y². (1)⇒(2). éu?¿ v ∈ V , Šâ4Œ‚5Ã'| “4Œ”5Ÿ, v V\ S ¥ƒ ¤ •þ| ‚5ƒ'. Šâ·K 2.1.17 Œ• v ∈ span(S). ù`² V ¥¤k•þáu span(S), = S ´ V )¤ |. (2)⇒(1). Ï• S ´ V )¤|, ¤± V = span(S). ùL²?Û v ∈ V Ñáu span(S), •Ò ´Œ± ¤ S ¥•þ ‚5|Ü, Ïd v V\ S ¥ƒ *¿ •þ|‚5ƒ'. ùÒ y S “4Œ”5. (2)⇒(3). b S0 ´ l S ¥ K ˜ ‡ • þ β • þ |. X J S 0 ‚ 5 Ã ', @ o Š â · 0 K 2.1.17, β ∈ / span(S ). ù V = span(S) ¥•¹ •þ β Ø2áu span(S 0 ), Ïd S 0 Ø2´ V )¤|. ùÒy² S ´4 )¤|. (3)⇒(2). •‡y²^‡ (3) ¤áž S 7½‚5Ã'. ¯¢þ, XJ S ‚5ƒ', K•3 β ∈ S, ¦ β Œ± •{ •þ| S 0 ‚5LÑ (·K 2.1.14). y3 span(S 0 ) •¹ S 0 Ú β, •Ò´•¹ S. ŠâíØ 2.1.20, span(S) = V •¹u span(S 0 ). ù`² span(S 0 ) = V , = S 0 •´ V )¤|. ù† S “4 ”5gñ. íØ 2.1.26: |). K: 1. T ´ S ¥ S • Kn ¥ ?¿•þ|, T • S 4Œ‚5Ã'| …= 2. S ¥˜½•3,‡f•þ|´ S ¥ f•þ| (=l S ¥ÀÑ˜ T ‚5Ã'¿…† S ‚5 •þ ¤ •þ d. 4Œ‚5Ã'|. y². V = span(S). (1) éu S ?Ûf•þ| T , ` T † S ‚5 dÒ´` T •´ V )¤|. XJq• T ‚5Ã', KŠâÚn 2.1.25, T ´ V ¥´4Œ‚5Ã'|. Ï• S ⊆ V , , T Š• S f•þ |•´ S ¥ 4Œ‚5Ã'|. ‡L5, bX®• T ´ S ¥ 4Œ‚5Ã'|. @o S ¥?Û• þ α V\? T ƒ #•þ|‚5ƒ'. Šâ·K 2.1.17, α ∈ span(T ). ù`² S ⊆ span(T ). u´, dÚn 2.1.19 Œ• V = span(S) ⊆ span(T ). du T ⊆ S, span(T ) ⊆ span(S) = V . ¤± span(T ) = V = span(S), = T † S ‚5 d. ¤±ùÒy² Øä 1. (2) Š â c ˜ Ø ä, · ‚ • ‡ y ² S ¥ Œ ± À Ñ ˜ • þ ¦ § ‚ ¤ f • þ | • ´ V = span(S) )¤|, …‚5Ã'. b S d•þ α1 , . . . , αr ¤, P Si , 1 ≤ i ≤ r • S K αi f•þ|. XJ?Û Si Ñ Ø÷v span(Si ) = span(S) = V , KŠâ½ÂŒ• S ´ V 4 )¤|. u´, dÚn 2.1.25 Œ•, S ´‚5Ã' . Ïdù«œ¹e·‚Ž‡ (Ø¤á. bX,‡ Si ÷v span(Si ) = span(S) = V , ~X span(Sr ) = V , @o·‚Œ±^ Sr “Ofâ S ?1Ó ?Ø. =, ·‚Œ±UY•Äl Sr 0 0 ¥ K˜‡•þŒ± r − 1 ‡f•þ| S1 , . . . , Sr−1 , XJ§‚¥z˜‡ÑØ´ V = span(Sr ) )¤|, K Sr Ò´ V 4 )¤|. ¤±ÚƒcÓn, Sr ‚5Ã'. džf•þ| Sr =Œ÷v ¤I ^‡. XJ,‡ Sj0 ´ V = span(S) = span(Sr ) )¤|, KUY•Ä Sj0 ¥íØ˜‡•þ @ f•þ|. oƒ, ÏL±þöŠŒ±l S ƒ¥çÀÑ V = span(S) ˜‡4 )¤| T . ŠâÚn 2.1.25, T ‚5Ã'. –dØä 2 y. 5¿, 3íØ 2.1.26 ¥, XJ S •˜•þ|, ½ö S =•¹"•þ, K S •þ|. 4Œ‚5Ã'|´˜ ·K 2.1.27: V ´ K n ¥ f˜m. Kéu?¿•¹u V ¥ ‚5Ã'•þ| T Ú V ?¿‚ 5)¤| S, 7, T ¥¤¹•þ ê8 u u S ¥¤¹•þ ê8. •Ò´`, XJ V ¥ k•õ‡•þ v1 , . . . , vr ∈ V ¤˜‡‚5)¤|, u1 , . . . , um ∈ V ´ l V ¥ÀÑ ?¿‚5Ã'•þ|, K7k m ≤ r. †óƒ, XJ V Œ±d r ‡•þ‚5Ü¤, @o V ¥?¿ÀÑ •þ|•‡¤¹•þê8î‚Œu r Ò7,‚5ƒ'. 1 80 Ù •þ˜m9Ùf˜m y². b v1 , . . . , vr ∈ V ´ V ?˜‚5)¤|. ·‚•‡y², éu?¿•þ| u1 , . . . , um ∈ V , XJ m > r, K u1 , . . . , um 7,‚5ƒ'. éz‡ j ∈ [[1, m]], Ï• uj ∈ V = span(v1 , . . . , vr ), ˜½•3~ê a1j , . . . , arj ∈ K ¦ uj = a1j v1 + a2j v2 + · · · + arj vr . y3•Ä r × m Ý A = (aij ). þª ¤Ý /ª=• (u1 , . . . , um ) = (v1 , . . . , vr )A . du A ê8õu1 ê8 (= m > r), àg‚5•§| AX = 0 9 Cþê8Œu•§ ê8, ¤±T•§|˜½k,‡š") (íØ 1.1.20), • X = (c1 , . . . , cm )T . u´     c1 c1  .   .   .  .  c1 u1 + · · · + cm um = (u1 , . . . , um )   .  = (v1 , . . . , vr )A  .  = (v1 , . . . , vr ) · 0 = 0 . cm cm Šâc¡¤ã, ùp íØ 2.1.28: ci Ø • 0. ¤±þªL²•þ| u1 , . . . , um ‚5ƒ'. ·K V • Kn f˜m. K V ?¿ü|Ä (XJ•3 y². S, S 0 ´ V ü|Ä. ˜•¡ S ´ V )¤|, S 0 ´ V ¥ Œ± |S 0 | ≤ |S|. Šâé¡5, |S| ≤ |S 0 |. ¤± |S| = |S 0 |. íØ 2.1.29: S • Kn ¥ •þ|. K S ¥ {) ¤¹ y. •þê87,ƒÓ. ‚5Ã'|, ¤±d·K 2.1.27 ?¿ü‡4ŒÃ'|¤¹ •þê87,ƒÓ. y². V = span(S). XJ T, T 0 ´ S ¥ ü‡4Œ‚5Ã'|. @oŠâíØ 2.1.26, §‚´ V = span(S) ü|Ä. ¤±(ØdíØ 2.1.28 Œ . ½ Â 2.1.30. S • K n ¥ •þ|. ? S ¥ 4Œ‚5Ã'| T (Ù•35díØ 2.1.26 y), ·‚ò T ¥¤¹•þ êþ½Â• S • (rank), P• rank(S). ŠâíØ 2.1.29, S •Ø•6u4Œ‚5Ã'| À , =, XJ T 0 ´ S ¥, ˜‡4Œ‚ 5Ã'|, K |T | = |T 0 |. y3·‚y² K n ¥ ?Ûf˜mÑkÄ. ·K 2.1.31: F ´ K n ¥?¿•þx, S0 ´l F ¥ Ñ ˜ (k•õ‡) •þ ¤ •þ |. XJ S0 ‚5Ã', @o˜½Œ±l F ¥2À eZ (k•õ‡!ŒU´"‡) •þV\ S0 ¥, ¦ *¿ •þ| S ´ F ¥ 4Œ‚5Ã'|. y². Ï• K n k˜|IOÄ (~ 2.1.23) Tk n ‡•þ, ¤± K n k‡)¤|d n ‡•þ ¤. Š â·K 2.1.27, |S0 | ≤ n. XJ F ⊆ span(S0 ), @o F ¥?Û•þÑ´ S0 ‚5|Ü, ¤± S0 ´ F ¥ 4Œ‚5Ã'|. eØ,, KŒ±é •þ β1 ∈ F , ¦ β1 ∈ / span(S0 ). Šâ·K 2.1.17, S1 := S0 ∪ {β1 } E,‚5Ã'. dž |S1 | = |S0 | + 1. XJ F ⊆ span(S1 ), K S1 ´ F ¥ 4Œ‚5 Ã'|. eØ,, Œ±UYÀÑ•þ β2 ∈ F , β2 ∈ / span(S1 ), ¦ S2 := S0 ∪ {β1 , β2 } E‚5Ã'. Ø ä-Eù‡L§, ò¬Øä # ‚5Ã'•þ| Sk , k ≥ 1, … |Sk | = |S0 | + k. Ï• K n ¥ ‚5Ã'•þ|•õ¹k n ‡•þ (·K 2.1.27), ¤±±þL§7,3k•õÚÊŽ. ùÒ´`, S0 ƒ¥· V\ F ¥ k•õ‡•þƒ , ˜½Œ± ˜‡4Œ‚5Ã'|. 2.1 1½ •þ˜m 81 íØ 2.1.32: K n ?Ûf˜m V ÑkÄ. Š â í Ø 2.1.28, V ¤ k Ä ¤ ¹ • þ ê 8 ƒ Ó, ù ‡ ê ¡ • V ‘ ê (dimension), P • dim V ‡ . (5¿, dim V = 0 …= V ´"•þ˜m. ÄK dim V ´ ê.) y². e V =¹"•þ, K˜•þ|´ V Ä. eØ,, S0 • V ¥,š"•þüÕ ¤ •þ |. ò·K 2.1.31 A^ F = V œ¹Œ•, •3 S0 *¿ V ¥4Œ‚5Ã'| S. ŠâÚ n 2.1.25, S ´ V ˜|Ä. y3ò !¥ ˜ ‡:o(Xe: ‚5Ã' •1, ƒ'N´|Ü". Ã'o')¤ , Ø ØõÄÀ¤. ‘ê VgÙ¢Ú•þ| •´—ƒƒ' . ·K 2.1.33: S • K n ¥ •þ|, V = span(S). K rank(S) = dim V . •Ò´`, ˜‡•þ| • uT•þ|Ü¤ f˜m‘ê. y². ŠâíØ 2.1.26, e T ´ S ¥ 4Œ‚5Ã'|, K T ‚5Ã'… span(T ) = span(S) = V . ùƒ u` T ´ V ˜|Ä. ¤± dim V = |T | = rank(S). ·‚±e¡˜^{ü ·K 2.1.34: íØ(å V, U Ñ´ K n f˜m. XJ U ⊆ V , K dim U ≤ dim V . y². U ?¿˜|ÄÑ´ V ¥ ·K 2.1.27 Œ . g • K 2.3. V, U Ñ´ K n g • K 2.4. V ´ Kn 1. α1 , . . . , αr ´ V !. ‚5Ã'|, V ?¿˜|ÄÑ´ V f˜m. y²: XJ U ⊆ V f˜m, α1 , . . . , αr • V ¥•þ )¤|. ¤±(Ød U 6= V , K dim U < dim V . ¤ •þ|. y²e ^‡ d: ˜|Ä; 2. r = dim V ¿… α1 , . . . , αr ‚5Ã'; 3. r = dim V ¿… α1 , . . . , αr ´ V 2.1.4 Ü¤|. SK S K 2.1.1. •Ä K 4 ¥ ±e•þ α1 = (0, −1, 1, 1) , α2 = (1, −1, 1, 2) , α3 = (0, 1, 1, 2) , α4 = (2, 2, 1, 3) , α5 = (0, 1, −1, −1) . β = (3, 1, 4, 8). S K 2.1.2. äe ä β ´Äáu span(α1 , · · · , α5 ) ¿`²nd. •þ|´Ä‚5ƒ': ‡ î‚5`, Ï•‚5|Üù‡Vg•6uXê À ‰Œ, ¤±‘êù‡Vg•6u• K À . ¤±, 3k7‡ ž ÿ·‚ATrùpJ dim V P• dimK V , ¡ƒ• V 3 K þ ‘ê½ö V Š• K- •þ˜m ‘ê. 3 ©¥·‚ ¬‰Ñù˜: ?˜Ú`². 1 82 Ù •þ˜m9Ùf˜m 1. α1 = (1, 3, 5, −4, 0), α2 = (1, 3, 2, −2, 1), α3 = (1, −2, 1, −1, 1), α4 = (1, −4, 1, 1, −1). 2. α1 = (1, 2, 3, −1), α2 = (3, 2, 1, −1), α3 = (2, 3, 1, −1), α4 = (2, 2, 2, 1), α5 = (0, 2, 1, −1). 3. α1 = (1, −1, 0, 0, 0), α2 = (0, 1, −1, 0, 0), α3 = (0, 0, 1, −1, 0), α4 = (0, 0, 0, 1, −1), α5 = (−1, 0, 0, 0, 1). S K 2.1.3. y²: é?¿ α1 , α2 , α3 ∈ K n , span(α1 , α2 , α3 ) = span(α1 + α2 , α2 + α3 , α1 + α3 ) . S K 2.1.4. {i1 , i2 , · · · , is } ´ [[1, n]] ýf8. •Ä K n ¥‰½ •þ| α1 = (a11 , · · · , a1n ) , α2 = (a21 , · · · , a2n ) , ············ αm = (am1 , · · · , amn ) . y3l α1 , · · · , αm ù 0 | α10 , · · · , αm . •þ¥ z˜‡ K1 i1 , i2 , · · · , is ‡©þ, # ˜| K n−s ¥ 0 ‚5Ã', K α1 , · · · , αm •‚5Ã'. 1. y²: XJ α10 , · · · , αm 2. b 0 ‚5ƒ', UÄä½ α1 , · · · , αm •‚5ƒ' ? α10 , · · · , αm S K 2.1.5. ée z‡•þ|, éÑÙ¥ ˜‡4Œ‚5Ã'|: 1. α1 = (−1, 0, 1, 0, 0), α2 = (1, 1, 1, 1, 0), α3 = (0, 1, 2, 1, 0). 2. α1 = (1, 0, 0), α2 = (1, 1, 0), α3 = (1, 1, 1), α4 = (0, 1, 1), α5 = (0, 0, 1). 3. α1 = (2, 3, 1, 1), α2 = (4, 6, 2, 2), α3 = (0, 1, 2, 1), α4 = (0, −1, −2, −1). S K 2.1.6. α1 , α2 , α3 ´˜‡‚5Ã' •þ|, β1 = α1 − α2 , β2 = α2 − α3 , β3 = α3 − α1 . ¦•þ| β1 , β2 , β3 ¥ ˜‡4Œ‚5Ã'|. S K 2.1.7. ¦±eˆ•þ| •: 1. α1 = (4, 1, −1, 2, 6), α2 = (0, 2, 3, −4, 1), α3 = (4, −9, −16, 22, 1), α4 = (1, 0, −1, 3, 7). 2. α1 = (3, 0, 7, 14), α2 = (0, 3, 1, 2), α3 = (1, −1, 2, 4), α4 = (2, 1, 5, 6), α5 = (1, −1, 2, 0). S K 2.1.8. ü‡•þ| α1 , · · · , αn Ú β1 , · · · , βn ƒm÷vXe'X: β1 = α2 + α3 + · · · + αn−1 + αn = β2 = α1 + α3 + · · · + αn−1 + αn = n X i=1 n X αi − α1 , αi − α2 , i=1 ······································· βn−1 = α1 + α2 + · · · + αn−2 + αn = n X αi − αn−1 , i=1 βn = α1 + α2 + · · · + αn−2 + αn−1 = n X i=1 y²: α1 , · · · , αn Ú β1 , · · · , βn ùü‡•þ| •ƒÓ. αi − αn . •þ 2.2 ˜ A^ 83 S K 2.1.9. n > r ≥ 1. α1 , · · · , αr ´•þ| α1 , · · · , αr , · · · , αn ¥ α = α1 + · · · + αn . 2b •3~ê c1 , · · · , cr ÷v α = c1 α1 + · · · + cr αr , Á¦•þ| α − α1 , · · · , α − αn ¥ ˜‡4Œ‚5Ã'|. - c1 + · · · + cr 6= 1 . ˜‡4Œ‚5Ã'|. S K 2.1.10. α1 , · · · , αn ´ K n ¥ •þ|. y²: α1 , · · · , αn ‚5Ã' ?¿•þÑŒ± α1 , · · · , αn ‚5LÑ. S K 2.1.11. b •þ| S •´ r. 1. y²: e S0 ´d S ¥, r ‡•þ ¤ ‚5Ã'|, K S0 ´ S ¥ 2. y²: e•þ| S 0 ¥z‡•þÑŒ±^ S ‚5LÑ, K S 0 3. S1 ´ S ¥, r ‡•þ ¤ f•þ|. y²: XJ S ¥ K S1 ´ S ¥ 4Œ‚5Ã'|. S K 2.1.12. Ý ¿©7‡^‡´ K n ¥ ˆÝ A = (aij ) ∈ Mn (K) n X |aii | > |aij | 4Œ‚5Ã'|. • ≤ r. z‡•þÑŒ±^ S1 ‚5LÑ, Xê÷v i = 1, 2, · · · , n . j=1, j6=i y²: A •þ| α1 , · · · , αn ‚5Ã'. S K 2.1.13. þ a1 , · · · , an ∈ K þ•š"~ê, … Pn 1 i=1 ai 6= −1. •Ä•þ˜m K 1×n ¥ e • η1 := (1 + a1 , 1, 1, · · · , 1) , η2 := (1, 1 + a2 , 1, · · · , 1) , ····················· ηn := (1, · · · , 1, 1 + an ) . y²: •þ| η1 , · · · , ηn ‚5Ã'. 2.2 2.2.1 Ý ˜ A^ • ½ Â 2.2.1. A ∈ Mm×n (K). ·‚ò A 1•þ| •¡• A 1• (row rank), P• rank(A). Šâ·K 2.1.33, A 1• uÙ1˜m ‘ê. aq/, A •þ| •¡•§ • (column rank), 6žP• rankc (A). • u ˜m ‘ê. ÏL1˜mÚ ˜mƒm éA'XN´wÑ, A • uÙ=˜Ý AT 1•. žÖöUì½Â Ún 2.2.2: y: XJ A ´1 F/Ý , K rank(A) A ∈ Mm×n (K). 1. é?¿Œ 2. e P • m Ü· Ý Œ_Ý P , þk rank(P A) ≤ rank(A). , K rank(P A) = rank(A). uA š"1ê8. 1 84 3. XJ A0 ´d A ²Lk•õgÐ 1C† Ý Ù •þ˜m9Ùf˜m , K rank(A0 ) = rank(A). y². (1) Ý 1• uÙ1˜m‘ê. ŠâíØ 1.2.9, P A 1˜m•¹u A 1˜m. ¤±( ØdíØ 2.1.34 Œ . (2) Ï• A Œ± ¤ P −1 (P A), ¤±Šâ (1) Œ• rank(A) = rank(P −1 (P A)) ≤ rank(P A) ≤ rank(A). (3) ‰Ð 1C†ƒ u†>¦±Ð Ý (·K 1.2.26), k•õ‡Ð Ý ¦È´Œ_Ý (íØ 1.2.29). ¤± A0 u A †>¦±,‡Œ_Ý , Ïd(Ød (2) = . ±þÚnwŠ·‚, Ð 1C†ØUCÝ •. •d·‚kw˜‡{ü Ún. 1•. ·‚ e5y²: Ð 1C†•ØUCÝ Ún 2.2.3: α1 , . . . , αr • K m×1 ¥ •þ|, P ∈ Ms×m (K). XJ•þ| P α1 , . . . , P αr ‚5Ã', K•þ| α1 , . . . , αr ˜½•‚5Ã'. †óƒ, e•þ| α1 , . . . , αr ‚5ƒ', K•þ| P α1 , . . . , P αr ˜½•‚5ƒ'. y². e•þ| α1 , . . . , αr ‚5ƒ', K•3 c1 , . . . , cr ∈ K Ø •", ¦ c1 α1 + · · · + cr αr = 0. u ´ c1 P α1 + · · · + cr P αr = P (c1 α1 + · · · + cr αr ) = P · 0 = 0 . ù`², •þ| P α1 , . . . , P αr ‚5ƒ'. Ún 2.2.4: A ∈ Mm×n (K). 1. é?¿Œ 1j . Ü· Ý P , XJ α ∈ K m×1 ´ A 2. é?¿Œ Ü· Ý P , þk rankc (P A) ≤ rankc (A). 3. e P • m Œ_Ý 1j (Ù¥ j ∈ [[1, n]]), K P α ´ P A , K rankc (P A) = rankc (A). 4. XJ A0 ´d A ²Lk•õgÐ 1C† Ý , K rankc (A0 ) = rankc (A). y². (1) Š â Ý ¦ { ½ Â = . , ˜ « w { ´, X J P e1 , . . . , en • K n×1 I O Ä (ë „ ~ 2.1.23), K A 1 j Ù¢Ò´ Aej ù‡¦ÈÝ . Ón, P A 1 j Ò´¦ÈÝ (P A)ej . ùp¤` (ØÃšÒ´ (P A)ej = P (Aej ) ù‡ ª. (2) α1 , . . . , αn • A • þ |. K Š â (1), P α1 , . . . , P αn ´ P A • þ |. b P αj1 , . . . , P αjr ´ Ù ¥ ˜ ‡ 4 Œ ‚ 5 Ã ' | (A O /, r = rankc (P A)). Š â Ú n 2.2.3, • þ | αj1 , . . . , αjr ‚5Ã'. 5¿ù Ñ´ A •þ, ¤±T•þ| •þê8 u½ u A •þ |ƒ¥ 4Œ‚5Ã'|¤¹•þê8, =, r ≤ rankc (A). ù ´–y Ø ª. (3) Ú (4) y²3‰Öö. íØ 2.2.5: eÝ A, A0 ∈ Mm×n (K) ƒ-, K rank(A) = rank(A0 ), rankc (A) = rankc (A0 ). y². ·‚•y1˜‡ ª. 1 ‡ ª y²3‰Öö. Šâƒ- ½Â, •3ü‡Œ_Ý P ∈ Mm (K) Ú Q ∈ Mn (K) ¦ A0 = P AQ. P B = AQ. K rank(B) = rankc (B T ) = rankc (QT AT ) = rankc (AT ) = rank(A) . ùp1n‡ Ò •â´Ún 2.2.4 (3). Ï• A0 = P B, dÚn 2.2.2 (2) Œ• rank(A0 ) = rank(B). ¤± rank(A0 ) = rank(A). 2.2 ˜ A^ 85 ½n 2.2.6: é?¿Ý rank(A). A ∈ Mm×n (K), Ù1• uÙ •. ·‚8 Ú¡• A • (rank), P• ! Ir 0 y². Š â ½ n 1.2.35, A ƒ - u ˜ ‡ © ¬ Ý A = . é u A0 Œ ± † 0 0 rank(A) = r = rankc (A). A^íØ 2.2.5 = rank(A) = rankc (A). 0 íØ 2.2.7: 1. •3•˜ A ∈ Mm×n (K). šK ê r ≤ min{m, n} ¦ ´d A •˜(½ 2. é?ÛŒ d½ÂwÑ Ü· Ir 0 A ƒ-uÝ ! 0 . •Ò´`, A 0 ƒ-IO/ . Ý B, þk rank(AB) ≤ min{rank(A), rank(B)}. y². (1) •35®3½n 1.2.35 ¥?Ø. Ï•ƒ- Ý äkƒÓ •, ¤±ùp r ˜½ u A •, ù‡ê A •˜(½. (2) Ø ª rank(AB) ≤ rank(B) ®²3Ún 2.2.2 (1) ¥Ñy. íØ 1.2.9 (Ü·K 2.1.34 wŠ· ‚ rankc (AB) ≤ rankc (A). Ï•1• u • (½n 2.2.6), ¤± rank(AB) ≤ rank(A). (Øy.. g • K 2.5. A ∈ Mm×n (K). b A ²Lk•õgÐ 0 •þ|, α1 , . . . , αn0 • A0 •þ|. 1C†C¤Ý A0 . P α1 , . . . , αn • A 1. y²: éu?¿l [[1, n]] ¥ÀÑ þ| αj0 1 , . . . , αj0 s ‚5Ã'. •I8 j1 , . . . , js , •þ| αj1 , . . . , αjs ‚5Ã' 2. Šâþ˜ •{. •{B ! l‰½ g • K 2.6. K (Ø, O˜« •þ|¥ÀÑ4Œ‚5Ã'| • OŽ A ∈ Mm×n (K). 1. y²: rank(A) ≤ min{m, n}. (e rank(A) = min{m, n}, k<¡ A ´÷• 2. XJ rank(A) = m, ·‚¡ A ´1÷• ÷• (of full column rank). y²: (a) A 1÷• …= (b) XJ A ´• 2.2.2 …= A km_; A , K A 1÷• (of full rank).) (of full row rank); XJ rank(A) = n, ·‚¡ A ´ ÷• …= …= A A k†_. ÷•. 2Ø‚5•§| !¥, ·‚ò±•þ˜m *:25• ‚5•§| )8. ½ Â 2.2.8. A ∈ Mm×n (K), b ∈ K m×1 . ò?Û÷v Av = b •þ v ∈ K n×1 ¡•‚5•§| AX = b (3 K þ) ˜‡)•þ (solution vector over K). P Sol(A; b) (½•°(/ SolK (A; b)) •• §| AX = b )8, =¤k)•þ ¤ 8Ü. b •"•þž, ·‚~ò Sol(A; 0) UP• N (A) (½ö•°(/ NK (A)). e¡·‚klàg•§|m©?Ø. Ún 2.2.9: é?¿ A ∈ Mm×n (K), àg‚5•§| AX = 0 )8 N (A) ´ K n×1 ˜‡f˜m. 1 86 Ù •þ˜m9Ùf˜m y². 3‰ÖööS. (2.2.10) Äu±þ(J, ·‚•¡ N (A) • AX = 0 )˜m (solution space) ½öÝ A " z˜m (null space)§ . )˜m N (A) ?¿˜|Ä¡•àg‚5•§| AX = 0 ˜‡Ä:)X (fundamental system of solutions). Ó˜‡àg‚5•§| AX = 0 ?¿ü‡Ä:)X¤¹•þê 87,ƒÓ, Ï•ù‡ê87, u N (A) ‘ê. ·‚•¡ dim N (A) • A "zÝ (nullity). XJ N (A) ´"•þ˜m, `²àg•§| AX = 0 =k"). dž·‚@•Ä:)X•˜• þ| (=, ¹k"‡•þ). k Ö (~X [18]) r=˜Ý AT "z˜m¡• A †"z˜m (left null space). XJØ •Ä1•þÚ •þƒmÖ •ª O, A †"z˜mŒ±@•´÷v uA = 0 ¤k•þ u ∈ K 1×m ¤ ˜m. y3·‚ÏL˜‡~f5wXÛéÑàg‚5•§|   1 −2 0 −1 0 0 0 0 1 0 0 1    ~ 2.2.11. A= . K 0 0 0 0 1 −4 0 0 0 0 0 0 ˜‡Ä:)X. N (A) = (2w + t , w , −s , t , 4s , s)T w , s , t ∈ K = w.(2 , 1 , 0 , 0 , 0 , 0)T + s.(0, 0, −1, 0, 4, 1)T + t.(1, 0, 0, 1, 0, 0)T w , s , t ∈ K . ØJ y, •þ| η1 = (2 , 1 , 0 , 0 , 0 , 0)T , η2 = (0, 0, −1, 0, 4, 1)T , η3 = (1, 0, 0, 1, 0, 0)T ´ AX = 0 ˜‡Ä:)X. [% ÖöŒU®²5¿ , ±þ η1 , η2 , η3 n‡•þ¢SþŒ±l ±þ'u N (A) Lˆª1˜1¥“©l”n‡ëCþ w, s, t . ù«•{Œ±¡ƒ•©lCþ { (method of isolating variables). ©lCþ{¤ (JÙ¢Ú±eöŠ J˜ : 1˜g w = 1, s = t = 0, @oLˆª (2w + t , w , −s , t , 4s , s)T ‰Ñ•þ η1 ; 1 g s = 1, w = t = 0 K η2 , 1ng w = s = 0, t = 1 Ò η3 . ØJŽ , ±þ~f¥ÏéÄ:)X ½n 2.2.12: ˜„ œ/. A ∈ Mm×n (K), r = rank(A). 1. àg‚5•§| AX = 0 n − r. 2. ?˜Úb K: •{Œ±í2 ?¿Ä:)X¤¹ A ´1•{/Ý . b Ø¹ A •þê8• n − r, ½=, )˜m N (A) ‘ê´ š"1Ì @ Á 1 j1 , . . . , jn−r . (a) •3•˜˜|•þ η1 , . . . , ηn−r ∈ N (A) ÷veã^‡: é?¿ s, t ∈ [[1, n − r]], •þ ηs δst = 1, ÄK δst = 0). (5¿, éu j1 , . . . , jn−r ƒ (b) ±þÀÑ § éõÖòùp 1 jt ‡‹IŠ• δst (Kronecker delta ÎÒ, =, s = t ž •I j, •þ ηs •þ| η1 , . . . , ηn−r ´ AX = 0 null space €È•“"˜m”. ¦þ;•¦^“"˜m”ù‡c. • 1 j ‡‹IŠŒUQØ´ 1 •Ø´ 0.) ˜‡Ä:)X. ;•Ú“"•þ˜m (zero space)” )· , 3¥©Lãž·‚¬ 2.2 ˜ A^ 87 y². (1) Š â · K 1.2.34, • 3 Œ _ Ý P ∈ Mm (K) ¦ A0 := P A • 1 • { / Ý . d u P Œ _, ¤ ± AX = 0 d u A0 X = 0. Ï d N (A) = N (A0 ). , , Ú n 2.2.2 (2) w Š · ‚ 0 rank(A ) = rank(A) = r. ¤±, •‡é1 F/ Ý A0 y² dim N (A0 ) = n − rank(A0 ) =Œ. ù˜ :ÏLØä (2) y²=Œ² . (2) • PÒ{B, Ø” j1 = r + 1, . . . , jn−r = n. •Ò´`, 31•{/ àg•§| AX = 0 ¥, ·‚b A š"1Ì uc r . Šâ·K 1.1.19 (3), dž•§| C xr+1 , . . . , xn • gdCþ, §‚Œ±k?¿ Š, , Ù¦Cþ x1 , . . . , xr Šd§‚ÏL•§|•˜/û½. AO /, zg3 xr+1 , . . . , xn ù n − r ‡Cþ¥ Ù¥˜‡• 1 Ù¦•", Œ± •˜(½ ˜|) •þ. Šâ (2.a) ¥ £ã, ù )•þ ´¤¦ η1 , . . . , ηn−r . y3·‚•I‡ y η1 , . . . , ηn−r ´)˜m N (A) ˜|Ä, =y²Øä (2.b). Äk, Šâ±þ£ã, η1 , . . . , ηn−r ù •þ ‹I/ªXe: η1 = (∗ , . . . , ∗ , 1, 0, . . . , 0) , η2 = (∗ , . . . , ∗ , 0, 1, . . . , 0) , ··························· ηn−r = (∗ , . . . , ∗ , 0, 0, . . . , 1) . ùp·‚^(Ò ∗ L«êŠØ‡äN•² @ ‹I ƒ. é?¿Xê k1 , . . . , kn−r ∈ K, k1 η1 + · · · + kn−r ηn−r = (∗ , · · · , ∗ , k1 , k2 , . . . , kn−r ) . (2.2.12.1) ¤± k1 η1 + · · · + kn−r ηn−r = 0 •k3 k1 = · · · = kn−r = 0 žâU¤á. ùÒ`² η1 , . . . , ηn−r ‚5Ã'5. • I ‡ ` ² η1 , . . . , ηn−r ´ ) ˜ m N (A) Ü ¤ |. =, é u ? ¿ η ∈ N (A), þ • 3 ~ ê k1 , . . . , kn−r ∈ K ¦ η = k1 η1 + · · · + kn−r ηn−r . ¯¢þ, XJ- k1 , . . . , kn−r • η • n − r ‡‹I, 2- η 0 := k1 η1 + · · · + kn−r ηn−r , @o˜ •¡·‚• η 0 ∈ N (A), ù´Ï•f˜m N (A) é‚5|ÜöŠ´µ4 , η1 , . . . , ηn−r ù • þŠâÀ •ª• §‚Ñáu N (A). ,˜•¡, (2.2.12.1) ª`² η 0 • n − r ‡‹IÚ η ƒ Ó. Š••{/‚5•§| AX = 0 ), gdC xr+1 , . . . , xn Š(½ž•U •˜˜‡ 0 ). ¤± η = η = k1 η1 + · · · + kn−r ηn−r . ù ´·‚I‡y² (Ø. ½n–dy..   0 1 −1 1 −1 1 0 1 2 −1   ~ 2.2.13. A= . ¦ AX = 0 ˜‡Ä:)X. 1 1 0 3 −2 2 ˜‡ë•‰Y•: 2 0 6 −3 η1 = (−1 , 1 , 1 , 0 , 0)T , η2 = (−2 , −1, 0, 1, 0)T . äNOŽL§lÑ. y3·‚5?Ø˜„ ½n 2.2.14: (™7àg A ∈ Mm×n (K), b ∈ K m×1 . P Ã = (A; b) •‚5•§| AX = b 1. ‚5•§| AX = b k) 2. b ) ‚5•§|. ¿©7‡^‡´ rank(A) = rank(Ã). rank(A) = rank(Ã) ù‡^‡®¤á. (a) XJ A ÷•, =, rank(A) = n, K•§| AX = b )•˜. O2Ý . 1 88 (b) XJ r := rank(A) < n, K•§| AX = b Ù •þ˜m9Ùf˜m )Ø•˜. ¯¢þ, XJ γ0 ∈ Sol(A; b) ´˜‡)•þ, η1 , . . . , ηn−r ´ N (A) ‚5•§| AX = 0 ˜‡Ä:)X), @odLˆª (2.2.14.1) γ0 + k1 η1 + · · · + kn−r ηn−r , Œ±‰Ñ• | AX = b (2.2.14.2) ˜|Ä (•Ò´àg Ù¥ k1 , . . . , kn−r Œ3 K ¥?¿ Š Ü). †óƒ, Sol(A; b) = γ0 + k1 η1 + · · · + kn−r ηn−r k1 , . . . , kn−r ∈ K = {γ0 + η | η ∈ span(η1 , . . . , ηn−r )} = {γ0 + η | η ∈ N (A)} . (Ï~·‚Œ±ò Sol(A; b) ¥?¿ ½ )•þ γ0 ¡••§| AX = b ˜‡A) (particular solution, special solution), òLˆª (2.2.14.1) ‰Ñ )•þ˜„/ª¡• AX = b Ï) (general solution).) y². (1) `‚5•§| AX = b k)ƒ u` •þ b áuXêÝ A ˜m C (A). O2Ý Ã •þ|´d A •þ|V\ •þ b . ¤± C (A) ⊆ C (Ã). 0 XJ b ∈ C (A), @o A z‡ Ñáu C (A), l C (Ã) ⊆ C (A) (íØ 2.1.20). ù`², XJ AX = b k), @o C (A) = C (Ã). ¤± rank(A) = dim C (A) = dim C (Ã) = rank(Ã). ‡ ƒ, X J rank(A) = rank(Ã), @ o C (A) Ú C (Ã) ‘ ê ƒ Ó. Š â g • K 2.3, d ž 7 , k C (A) = C (Ã). Ï• b ´ A0 ˜‡ , ¤± b ∈ C (Ã). ù b ∈ C (A), Ïd AX = b k). (2) ·‚• , ÏLÐ 1C†Œ±ò A z¤˜‡1•{/ Ý A0 . ùƒ u`•3,‡Œ _Ý P ∈ Mm (K) ¦ A0 = P A •1•{/Ý . - b0 = P b. K•§| AX = b Ú A0 X = b0 )8ƒÓ. ¤±, AX = b )•˜ …= A0 X = b0 )•˜. Šâ·K 1.1.19 (3), ö 0 0 d^‡• rank(A ) = n. Ï• A Ú A •ƒ (Ún 2.2.2 (2)), ¤± AX = b )•˜ …= rank(A) = n. b r = rank(A) = rank(A0 ) < n. ·‚Žy²8Ü Sol(A; b) u {γ0 + η | η ∈ N (A)}, Ù¥ γ0 ´ AX = b ˜ ‡ A ). Ä k, é u ? ¿ η ∈ N (A), A(γ0 + η) = Aγ0 + Aη = b + 0 = b, ¤ ± γ0 + η ∈ Sol(A; b). ‡ƒ, e γ ∈ Sol(A; b), K- η := γ − γ0 Œ• Aη = Aγ − Aγ0 = b − b = 0, = η ∈ N (A). ù`² Sol(A; b) •¹u {γ0 + η | η ∈ N (A)}. ½n 2.2.14 ¢Sþ´'unØþXÛ O˜‡‚5•§|)85 o(. {ü/`, ¦)˜ ‡‚5•§|¤ 9 Ì‡óŠÃš±eA‡: ˜´ÏL' XêÝ ÚO2Ý •5(½´Ä k), ´éÑ˜‡A), n´(½éA àg‚5•§| ˜‡Ä:)X, , ± (2.2.14.1) ª /ª ¤•§| Ï). ¢Sþ, ùA‘óŠŒ±ÏLpd–e ž {Óž?1, äN öŠÚ½ Ù¢·‚31˜Ù § 1.1.3 !¥Ò®²?ØL . •óƒ (ë„·K 1.1.19), ÏLpd–e ž { ·‚kòXêÝ z•1•{/, ÏL1•{/é n − r ‡gdCþ (n •C êþ, r •XêÝ ÚO2Ý Ó •), ÏL-gdCþ?¿ K ¥ Š k1 , . . . , kr , “\ 1•{/ •§|= Œ•˜(½Ù¦Cþ Š. dž Ï)Lˆª•¹ëê k1 , . . . , kn−r . ÏL©lCþ{Œ±é ƒAàg‚5•§| ˜|Ä:)X±9˜‡A). ·‚2Þ˜‡~fES˜epd–e ~ 2.2.15. ¦‚5•§| AX = b  1  A = 2 3 ž {, ^B•Ïn)#CÚ\ Ï), Ù¥ −1 0 −1 0 1 −1 1 0 −1  −1  −1 , −1   1   b = 2 . 0 ˜ Vg. 2.2 ˜ A^ 89 ˜‡ë•‰YXe: ˜‡A) γ0 = 1 3 3 , − , , 0, 0 . 4 4 2 àg‚5•§| AX = 0 ˜‡Ä:)X: 1 3 1 1 3 , , −1 , 1 , 0 , η2 = , − , , 0, 1 . η1 = 2 2 4 4 2 Ï)Lˆª: γ= 1 3 3 1 3 1 3 1 , − , , 0 , 0 + k1 , , −1 , 1 , 0 + k2 , − , , 0, 1 4 4 2 2 2 4 4 2 Ù¥ k1 , k2 Œ3 K ¥?¿ äNOŽL§lÑ. 2.2.3 • Š. 8†‹IAÛ þ˜ !¥·‚w , ˜‡àg‚5•§| AX = 0 )8 N (A) ´˜‡•þf˜m. XJ ´šàg ‚5•§| AX = b, Ù)8 Sol(A; b) ¥ ˜„ ƒ/X γ0 + η (ë„ (2.2.14.2)), Ù¥ γ0 ´ Sol(A; b) ¥ ½ ƒ, η Œ±3XêÝ A "z˜m N (A) ¥ HÙ¥¤k •þ. • ,Ø2´•þf˜m, Ööé¯¬w ù«/ª 8ÜÙ¢•š~Š ·‚•§;€·¶. ½ Â 2.2.16. L, M • K n ¥ š˜f8. 1. ½Â L + M = {u + v | u ∈ L , v ∈ M }. =, ? L ¥•þÚ M ¥•þ‰\{, ¤kŒU ( J˜3˜å ¤ 8ÜP• L + M . XJ L ••¹˜‡•þ α (= L = {α}), ·‚ò {α} + M {P• α + M . 2. e λ ∈ K, ½Â λL = {λu | u ∈ L}. 3. XJ•3•þ α ∈ K n Úf˜m U ⊆ K n ¦ L = α + U , ·‚¡ L ´ K n ¥ ˜‡ (‚5) • 8 ((linear) affine set), ½ö K n ˜‡ (‚5) • f8 ((linear) affine subset). dž, L Ú f˜m U 'X£ã• L f5²1u (rigidly parallel to) U , α ¡• L ˜‡ (•þ) “L (representative). ¦^þã½ÂŒ±òšàg‚5•§| AX = b )8 Sol(A; b) £ã• γ0 + N (A), Ù¥ γ0 ´ AX = b ˜‡A). •Ò´`, Sol(A; b) ´f5²1uf˜m N (A) ˜‡• 8. ~ 2.2.17. þ²L : U = {(t, 2t) ∈ R2 | t ∈ R}. K U ´ R2 ¥•þ (1, 2) Ü¤ Ç• 2 †‚. • 8 f˜m, ã”þwÒ´²¡ L := (4, 5) + U = {(4 + t, 5 + 2t) | t ∈ R} ´²¡S²L: (4, 5) Œ Ú••²£ƒ Ç• 2 †‚. , †‚. ˜«*:5w, L Œ±@•´†‚ U ÷X•þ (4, 5) (2.2.18) ~ 2.2.17 ¢Sþ´ÏL• 8 /ª‰Ñ ²¡S˜^†‚ `², ²¡S ?¿˜^†‚ÑŒ±”~ 2.2.17 @ ¤• 8 /ª. Äk·‚£ ˜e²¡S†‚ ˆ«•§Lˆ/ª. ·‚• , † ´Œ±ê¡/ª •§5£ã: (2.2.18.1) ax + by + c = 0 , Ù¥ a , b , c •~ê, … a , b Ø ~f. ·‚y35{‡/ ‹I²¡ R2 ¥ •". †‚o 1 90 Ù •þ˜m9Ùf˜m Ï~ò/X (2.2.18.1) ª •§¡•²¡S†‚ ˜„•§ (general equation)!ÊÏ•§ (ordinary equation) ½Ûª•§ (implicit equation). 5¿, (2.2.18.1) Œ±À•=¹˜‡•§ ‚5•§|, Ù ¥Xê a, b Ø •". †ƒƒéA àg‚5•§• ax + by = 0. XJ- (A, B)T •Tàg•§ ˜‡š") (~X (A, B) = (−b, a)), (x0 , y0 )T ••§ ax + by + c = 0 ˜‡A), K (2.2.18.1) Ï)/X  x = x + tA 0 (2.2.18.2) Ù¥ t ∈ R ´Œ±?¿ Š ëê, A , B ´~ê. y = y0 + tB XJæ^•þ (2.2.18.3) /ª, (2.2.18.2) Œ± ¤ x x0 A , Ù¥ t ∈ R ´Œ±?¿ = +t y y0 B Š ëê, A , B ´~ê. (ùÙ¢´ (2.2.14.2) ˜‡•{üA~.) Ï~ò (2.2.18.2) ªÚ (2.2.18.3) ª©O¡•ƒA†‚ ëê•§ (parametric equation) Ú•þª•§ (vector equation). XJ- U ••þ (A, B)T Ü¤ f˜m, @o (2.2.18.3) ªL« †‚Ò´f5²1u U ˜ ‡• 8. ,, Ø ê AÏœ¹, ~^ †‚•§Lˆ/ª„k±eA«: • wª•§ (explicit equation): (2.2.18.4) y = ax + b , Ù¥ a , b ∈ R •®‰~ê. ù«/ª A:´, XJò y À• x ¼ê, KT•§²w‰Ñ /ª •§k‡"€´Ã{L«Ñ† y ¶²1 †‚. • : T¼ê Lˆª. ,, ù« ª (point slope form) •§: (2.2.18.5) Ù¥ x0 , y0 , α ∈ R •®‰~ê. y = α(x − x0 ) + y0 , ù‡•§¥, (x0 , y0 ) Œ±@•´†‚þ‰½˜: •§w,•Ø·û† y ¶²1 †‚. ‹I, α ∈ R K´T†‚ Ç. : ª • : • ª (point direction form) • §! I O • § (standard equation) ½ é ¡ • § (symmetric equation): (2.2.18.6) y − y0 x − x0 = , A B Ù¥ x0 , y0 , A, B ∈ R •®‰~ê. ùp, Ù¥ (x0 , y0 ) Œ±@•´†‚þ‰½˜: ‹I, •þ, ¡•T†‚ ˜‡•••þ (direction vector). w,, XJ®• (x0 , y0 ) Ú (x1 , y1 ) ´†‚þØÓ †‚ ˜‡•••þ. ¤±·‚kXe†‚•§ (A, B) ∈ R2 KÍ£T†‚•• ü‡:, @o•þ (x1 − x0 , y1 − y0 ) ´T (2.2.18.7) y − y0 x − x0 = , x1 − x0 y1 − y0 ù«/ª•¤•†‚ ü:ª (two point form) •§. Ù¥ x0 , y0 , x1 , y1 ∈ R •®‰~ê. ¦^:•ª•§ (2.2.18.6) ½ü:ª•§ (2.2.18.7) žI‡5¿, XJ, ©1•", ·‚AT @•éA ©f•ð u" (ÏdƒA Cþ•U ˜‡~Š), , ˜‡C KŒ±?¿ Š. (žÖög••Ÿo.) 2.2 ˜ A^ 91 • :{ª (point normal form) •§: (2.2.18.8) α(x − x0 ) + β(y − y0 ) = 0 , Ù¥ x0 , y0 , α, β ∈ R •®‰~ê. ù‡•§¥, (x0 , y0 ) •†‚þ‰½: ‹I, (α, β) ∈ R •R†uT†‚ ••þ˜‡‰ ½ •þ, ¡•T†‚ ˜‡{•þ (normal vector). ,, (2.2.18.8) ª¥ ~ê α, β ØUÑ ´ 0 (={•þØU´"•þ), ÄKT•§ØUL«˜^†‚. ±þ•§¶¡«a„õ, ¢SþÖöATØI‡ kPM ù ¶¡. éäNAÛ¯K, ƒ&Ö ö•‡ŠâI‡Ñ\g•ATØJé Ü· •§/ª5¦^. XJÖöú k7‡XÚ/ÆS‹ IAÛ, ØJgCé Ü· ë•Ö (~X [29], ±9 [31] Ú [33] k'Ù!). ~ 2.2.19. c¡·‚?Ø ²¡S †‚Ú• 8ƒm éA'X. Ù¢·‚Œ± aq/5? Ø˜m¥ ²¡Ú†‚. ~X, XJ U = {(t, 2t, 3t) ∈ R3 | t ∈ R}, K U ´ R3 ¥•þ (1, 2, 3) Ü¤ f˜m, ã”þwÒ ´˜m¥²L :!± (1, 2, 3) • (˜‡) •••þ †‚. • 8 L := (0, 4, 5) + U = {(t, 4 + t, 5 + 2t) | t ∈ R} ´˜m¥²L: (0, 4, 5)!± (1, 2, 3) ••••þ †‚. , ˜«*:5w, L Œ±@•´†‚ U ÷X•þ (0, 4, 5) Œ Ú••²£ƒ †‚. 3 2'X, e W := {(x, y, 0) ∈ R | x, y ∈ R}, KAÛþw W ´ x ¶Ú y ¶Ü¤ ²¡. • 8 Π := (0, 4, 5) + W = {(x, 4 + y, 5) | x, y ∈ R} ´˜m¥²L: (0, 4, 5)!²1u W •²£ƒ ²¡. ²¡, •Œ±@•´²¡ W ÷X•þ (0, 4, 5) (2.2.20) Ú (2.2.18) ˜ã¥ (Øaq, ˜m¥ ?¿²¡ Π ÑŒ± ª, Ù¥ α ∈ R3 • ½ •þ, W ´ R3 ¥ 2 ‘f˜m. ¯¢þ, ˜m¥ ²¡˜„•§• Ú• 8 Π = α+W / Ù¥ a, b, c, d •®‰~ê, … a, b, c Ø (2.2.20.1) ax + by + cz + d = 0 , ÏL)•§ØJ ˜m¥²¡ ëê•§    x = x0 + tA1 + sA2 (2.2.20.2) ¤• Œ •". y = y0 + tB1 + sB2    z = z0 + tC1 + sC2 Ù¥ x0 , y0 , z0 , Ai , Bi , Ci •®‰~ê, t, s ∈ R ´Œ±?¿ Š ëê t, s ∈ R ´Œ±?¿ Š ëê ½ö•þª•§ (2.2.20.3)         x x0 A1 A2         y  =  y0  + t B1  + s B2  , z z0 C1 C2 Ù¥ x0 , y0 , z0 , Ai , Bi , Ci •®‰~ê, ùp, v1 := (A1 , B1 , C1 )T Ú v2 := (A2 , B2 , C2 )T ´† (2.2.20.1) éA àg•§ ax + by + cz = 0 ˜ ‡ Ä : ) X, f ˜ m W := span(v1 , v2 ) Ò ´ à g • § ax + by + cz = 0 ) ˜ m. Ï d, 1 92 Ù •þ˜m9Ùf˜m l (2.2.20.3) ª·‚Œ±wÑ, ˜m¥ ?¿²¡ÑŒ± ¤• 8 α + W /ª, Ù¥ α ∈ R3 , W ⊆ R3 • 2 ‘f˜m. d ·‚• , ˜m¥ÏL‰½ ˜: (x0 , y0 , z0 ) …†˜‡‰½š"•þ (a, b, c) R† ²¡ ´•˜(½ . •þ (a, b, c) ¡•T²¡ ˜‡{•þ. ¢Sþ, T²¡S?¿˜: ‹I÷v a(x − x0 ) + b(y − y0 ) + c(z − z0 ) = 0 . (2.2.20.4) ù«/ª •§¡•²¡ :{ª•§. ,, ·‚• , 3˜m¥(½˜‡²¡ •ª„k±eA«: −−→ • ² L å : 3 Ó ˜ ‡ ‰ ½ : P (x0 , y0 , z0 ), … Ø ² 1 ü ‡ š " A Û • þ P Q = (a1 , b1 , c1 ) −→ Ú P R = (a2 , b2 , c2 ), ½ ö ` ² L : P … † ü ‡ Ø ² 1 š " • þ v1 = (a1 , b1 , c1 ) Ú v2 = (a2 , b2 , c2 ) ²1; • ²L‰½ ü‡: (x0 , y0 , z0 ) Ú (x1 , y1 , z1 ), ¿…²1u˜‡‰½ (a, b, c) † (x1 − x0 , y1 − y0 , z1 − z0 ) ‚5Ã'); • ²Ln‡Ø ‚ š"•þ (a, b, c) (b : (x0 , y0 , z0 ), (x1 , y1 , z1 ) Ú (x2 , y2 , z2 ). 3±þA«œ¹eÑŒ± Ñ²¡ ˜‡šëêª•§, ©O¡•²¡ ˜:ü•þª (onepoint two-vector form)!ü:˜•þª (two-point one-vector form) Ún:ª (three point form) •§. ØLù •§ LÎ‡^ 1 ªù‡Vg, ¤±·‚ 1oÙ (4.1.7) ˜ã25•[?Øù‡ ÆK. (2.2.21) ²¡ (2.2.21.1) e5·‚?Ø˜m¥ †‚. w,, ˜m¥ 8. ¤±, ˜m¥˜^†‚ ˜„•§¶ •  a x + b y + c z + d = 0 1 1 1 1 a2 x + b2 y + c2 z + d2 = 0 †‚o´Œ±À•ü‡Ø²1 (•Ø-Ü) Ù¥ ai , bi , ci , di •®‰~ê, …Ý a1 a2 b1 b2 c1 c2 ! •• 2 . (žÖög•ùpb XêÝ • u 2 ¿ÂÛ3.) ÏL)•§|Œ± ˜m¥†‚ ëê•§    x = x0 + tA1 (2.2.21.2) y = y0 + tB1    z = z0 + tC1 Ù¥ x0 , y0 , z0 , A1 , B1 , C1 •®‰~ê, t ∈ R ´Œ±?¿ Š ëê t ∈ R ´Œ±?¿ Š ëê. ½ö•þª•§ (2.2.21.3)       x x0 A1       y  =  y0  + t B1  z z0 C1 Ù¥ x0 , y0 , z0 , A1 , B1 , C1 •®‰~ê, ¶ •,ù´‡•§|, <‚„´S.òÙ¡•˜m†‚ “•§”. 2.2 ˜ A^ 93 l (2.2.21.3) ª·‚Œ±wÑ, ˜m¥ U ⊆ R3 •˜‘f˜m. l (2.2.21.2) ª½ (2.2.21.3) ª„Œ± (2.2.21.4) ˜m†‚ x − x0 y − y0 z − y0 = = , A B C e®•†‚þ (2.2.21.5) ?¿†‚ÑŒ± ¤• /ª, Ù¥ α ∈ R3 , :•ª•§ (•¡IO•§½é¡•§) Ù¥ x0 , y0 , z0 , A, B , C ∈ R •®‰~ê. ü‡: (x0 , y0 , z0 ) Ú (x1 , y1 , z1 ), KŒ± x − x0 y − y0 z − y0 = = , x1 − x0 y1 − y0 z1 − z0 8 α+U Ñ†‚ ü:ª•§ Ù¥ x0 , y0 , z0 , x1 , y1 , z1 ∈ R •®‰~ê. Ú²¡¥ œ/˜ , 3:•ª½ü:ª•§¥, XJ,‡©1•", KA@•éA u". XJ (2.2.21.4) ½ 2.2.21.5 ¥kü‡©1•", KéA ü‡Cþ•U ~Š, þŒ±?¿ Š. ©f•ð 1n‡C 5P 2.2.22. Q,˜m¥ †‚Ú²¡ÑŒ±^‚5•§½•§|5L«, §‚ƒm ˜'X• ÑŒ±ÏL‚5•§| ƒ'nØ5©Û. Þ~5`, ˜m¥ü‡²¡ ˜'XÃšn«: -Ü (coincident)!²1 (parallel) Ø-Ü! ƒ (intersecting) Ø-Ü (½=, Ø²1•Ø-Ü). Ù¥• ˜«œ¹ d^‡´ùü‡²¡ 8•˜^†‚. äN ó, XJ Πi , i = 1, 2 ´•§• ai x + bi y + ci z + di = 0 ²¡ (Ù¥é ½ i, Xê ai , bi , ci Ø •"), @o ! a1 b1 c1 d1 • Π1 † Π2 -Ü¿›XÝ •• 1. a2 b2 c2 d2 ! a1 b1 c1 a1 • Π1 † Π2 ²1 Ø-Ü …= Ý •• 1 a2 b2 c2 a2 ! a1 b1 c1 • Π1 † Π2 ƒ u˜^†‚ …= Ý •• 2. a2 b2 c2 b1 b2 c1 c2 d1 d2 ! •• 2. XJ L ´˜m¥ ˜^†‚, Π ´˜m¥ ˜‡²¡, @o öƒm ˜'Xkn«: L • ¹u²¡ Π S, L ²1u Π Ø•¹u Π, ½ö L † Π ƒ u•˜˜:. XJ L1 , L2 ´˜m¥ ü^†‚, @o§‚ƒm 'X ko«ŒU: L1 † L2 -Ü, L1 † L2 ²1 Ø-Ü, L1 † L2 ´ü^É¡†‚ (skew lines), ½ö L1 † L2 ƒ u•˜˜:. ±þ Ñ ˆ« ˜'X (†‚†²¡ƒm, ½ö†‚††‚ƒm) ÑŒ±Ú· •§|X êÝ ½O2Ý A5éXå5, äN OOKØńóüŠŒ±` , •ýØ´(J óŠ. Ïd, ·‚3ùpŽÑ˜ i. ÏLŒþ ¢~OŽ!@ý nØ©Û, ±97‡ž¿© k'ë •] ½ö•<¦ , ƒ&Öö˜½Œ±gC•gCÖþù˜ iŽÑ SN. ·‚ƒ&, ÏLù ˜‡gCg•Ú&¢ L§, Öö Â¼¬'† Ö·‚˜˜•[ Ñ •£:‡´L, ék' SN n)•¬•\ •. ±þ·‚?Ø Ún 2.2.23: ~fAT®²v U, W • K n õ . e¡·‚5Øy• f˜m, α, β • K n ¥ 1. α + (β + U ) = (α + β) + U = β + (α + U ). 2. (α + U ) + (β + W ) = (α + β) + (U + W ). 8 ˜ •þ, a, b ∈ K. 5Ÿ. 1 94 Ù •þ˜m9Ùf˜m 3. a(bU ) = (ab)U = b(aU ), a(α + U ) = aα + aU . … 4. U + U = U , aU ⊆ U , a 6= 0 ž, aU = U . 5. U + W E´ K n f˜m, §•¹ U ∪ W , ¿…é?¿f˜m V ⊆ K n , U + W ⊆ V ‡^‡´ U ∪ W ⊆ V . •Ò´`, U + W ´ K n ¥Óž•¹ U Ú W (sum). • ¿©7 f˜m, ·‚¡ƒ•f˜m U Ú W Ú y². Øä (1) , (2) Ú (3) † Uì½Â y=Œ. Øä (4) ¤á nd´f˜m U 'u\{Úê¦$Žµ4, ¿…•¹"•þ (ù˜:Œ± y U ⊆ U + U ). (5) U½ÂŒ± y U + W ´f˜m. Ï•"•þ 0 áuf˜m W , ?¿ u ∈ U Œ± ¤ u = u + 0, l áu U + W . ¤± U ⊆ U + W . aqŒy W ⊆ U + W . b V ´ K n f˜m. XJ U + W ⊆ V , KÏ• U ∪ W ⊆ V , k U ∪ W ⊆ V . ‡ƒ, bX U ∪ W ⊆ V , Ké?¿ u ∈ U Ú w ∈ W , þk u ∈ V, w ∈ V . u´Šâf˜m'u•þ\{ µ4 5, u + w •áu V . ùÒy² U + W ⊆ V . ·K 2.2.24: 1. ±e•ã U, W • K n f˜m, α, β ∈ K n •?¿•þ. d: (a) α − β ∈ U . (b) α + U = β + U . (c) α + U † β + U 2. α + U = U …= 8š˜. …= α ∈ U, 3. •¹'X α + U ⊆ β + W ¤á …= (α + U ) ∩ U 6= ∅. α − β ∈ W … U ⊆ W. Ïd, XJ α + U = β + W , K7, U = W . ù`², z‡• 1. 8•¬Ú•˜˜‡f˜mf5² y². (1) (a) ⇒ (b). b u := α − β ∈ U . éu?¿ v ∈ U , ·‚k α + v = (u + β) + v = β + (u + v). Ï• U 'u\{µ4, ¤± u + v ∈ U . ¤± α + v = β + (u + v) ∈ β + U . ùÒy² α + U ⊆ β + U . ,˜•¡, Ï• β − α = −u •áu U (nd´f˜m¥?¿•þ \{_•áuTf˜m), ¤±a qŒ• β + U ⊆ α + U . Ïd α + U = β + U . (b) ⇒ (c). w,, ^‡ (b) `² (α + U ) ∩ (β + U ) = α + U . Ï• α = α + 0 ∈ α + U , ¤±^‡ (b) %¹ (α + U ) ∩ (β + U ) š˜. (c) ⇒ (a). b •3 γ ∈ (α + U ) ∩ (β + U ), @o•3 u, v ∈ U ¦ γ = α + u = β + v. u´ α − β = v − u ∈ U , Ï•f˜m U 'u~{µ4. (2) 3(Ø (1) ¥ β •"•þ=Œ. (3) XJ α + U ⊆ β + W , Kd α ∈ α + U Œ• α ∈ β + W . ù`² α − β ∈ W . d , éu?¿ u ∈ U , ·‚k α + u ∈ α + U ⊆ β + W , ¤± α + u − β ∈ W . c¡®²w α − β ∈ W , ¤±|^f ˜m W 'u~{ µ45=• (α + u − β) − (α − β) ∈ W , ½= u ∈ W . ùÒy² U ⊆ W. 2.2 ˜ A^ 95 ‡L5, b α − β ∈ W … U ⊆ W . K α − β + U ⊆ W + W = W . u´ α + U = β + (α − β + U ) ⊆ β + W . (, ˜«g´´: d α − β ∈ W (Ü(Ø (1) Œ• α + W = β + W . ¤±2\þ U ⊆ W ù˜b Ò • α + U ⊆ α + W = β + W .) –d·‚y² 1˜‡äó. 1 ‡äów,´1˜‡äó † íØ. Šâ·K 2.2.24 (3), ·‚Œ±ò˜‡• 8 L ‘ê½Â•†ƒf5²1 @‡f˜m ‘ ê, ¿P• dim L. nÜ·K 2.2.24 (1) Ú (3) (ØŒ±• : XJü‡• 8©Of5²1uf˜m U Ú W , ¿…§‚©O±•þ α Ú β •˜‡“L , @oùü‡• 8ƒ ¿©7‡^‡´ U = W ¿… ü‡•þ“L α − β áuf˜m U . (2.2.25) L = α + U Ú M = β + W ´ K n ¥ • 8, Ù¥ U, W •f˜m. XJ U ⊆ W ½ö W ⊆ U , ·‚¡ L † M ²1. XJ U = W , ·‚¡ L † M f5²1. dim L = dim M ž, L Ú M ²1 1. (ž¯Öö•ŸoQ?) žÖög1 y±e(Ø: • ü‡• 8 L Ú M f5²1 …= •3•þ γ ¦ du öf5² L = γ + M. • b L1 , L2 , L3 ´ K n ¥ n‡• 8. XJ L1 † L2 f5²1, L2 † L3 f5²1, @o L1 † L3 •f5²1. (ddŒ•, “ f5²1”´˜« d'X.) ´, XJ•• L1 † L2 ²1, … L2 † L3 ²1, @o˜„5`ØUä½ L1 Ú L3 •²1. (žÖöÞ~`²ù˜:.) ~ 2.2.26. U = span((1, 2, 0)) ⊆ R3 , Kéu α = (0, 0, 1) Ú β = (2, 4, 1), ·‚k α + U = β + U . qe W = {(x, y, 0) | x, y ∈ R}, γ = (−1, 0, 2), K U ⊆ W . Uì (2.2.25) ¥ ½Â, α + U ²1 u γ + W . ù†·‚ AÛ†*´¬Ü , Ï•XJxÑã”, ·‚¬w α + U ´˜^²1u²¡ γ+W †‚. XJ γ 0 = (−1, 0, 1), K α + U ⊆ γ 0 + W , =†‚ α + U u²¡ γ 0 + W ƒS. 2.2.4 SK S K 2.2.1. OŽe Ý •   0 0   1 0 1 2 1 −1 1 −2 0 −1 −1 −2 1 1  2 0   , −1 1  1  0  0   1 4 0 1 0 2 5  3 0  0 3  6 0  ,  −1 0 1 1 1  −1  2   1 0 2 −2 4 −2 0 0 0 1 3 6  4  5  6  ,  32 77 1 2 3 14 32  1  1  0   0 1 0 1 1 0 0  14 12 6  6 104 21   7 6 3 22 −80 −9 1 0 1 1 0 0 0 0 1 1  0  0  0   0 1  8 2 9 17    , 4 1  7 −13 1 96 S K 2.2.2. ¦±e n • •:  1  1   0  . . .   .. . 0 S K 2.2.3. Šâëê λ S K 2.2.4. Ý B ∈ Mn×s (K) ¦ 0 0 ··· 1 0 0 1 1 0 .. . ··· 1 .. 0 .. . .. . Š5?ØÝ A ∈ Mm×n (K) AB = 0. S K 2.2.5. A ∈ Mn (K) rank(A) •ŒŒU Š. n2 ‡Ý . 0 .. 1 . .. . .. . 1 1   0  ..  .  ..   .   0 ··· 0 1   1 λ −1 2   2 −1 λ 5 1 10 −6 1 •XÛ ƒ¥– Ý k n2 − n + 1 ‡•". y² rank(A) < n, ¿¦ S K 2.2.7. A ∈ Mm×n (K), B ´l A ¥ÀÑ, s 1 y² rank(B) ≥ rank(A) + s − m, ¿Þ~`² > Ú = A ∈ Mm×n (K) Š. • • r. y ²: é ? ¿ s ≥ n − r, • 3 • • n − r S K 2.2.6. A ∈ Mn (K) •Œ_Ý , i1 , · · · , ir ´d [[1, n]] ¥î‚üO B ´d A 1 i1 , · · · , ir 1 ¤ r × n fÝ . ¦àg‚5•§| BX = 0 ˜‡Ä:)X. S K 2.2.8. Ù •þ˜m9Ùf˜m r ‡ê (Ù¥ 1 ≤ r ≤ n). s×n Ý . œ¹þkŒUÑy. • ≤ 1. y²: •3 a1 , · · · , am ∈ K Ú b1 , · · · , bn ∈ K ¦   a1 b1 a1 b2 · · · a1 bn    a2 b1 a2 b2 · · · a2 bn   A= . ..  .. ..  .  .. .  . am b1 · · · · · · am bn S K 2.2.9. •e ˆ‡àg‚5•§|éÑ˜‡Ä:)X:   x2 − x3 + x4 = 0      − 7x2 + 3x3 + x4 = 0 1.  x1 + 3x2 − 3x4 = 0      x − 2x + 3x − 4x = 0 1 2 3 4   x1 + x2 + x3 + x4 + x5 = 0      3x1 + 2x2 + x3 + x4 − 3x5 = 0 2.  x2 + 2x3 + 2x4 + 6x5 = 0      5x + 4x + 3x + 3x − x = 0 1 2 3 4 5   2x1 + 6x2 − x3 + 5x4 = 0      3x1 − x2 + 2x3 − 7x4 = 0 3.  4x1 + x2 − 3x3 + 6x4 = 0      x − 2x + 4x − 7x = 0 1 2 3 4 2.2 ˜ A^ 97    x1 − 2x2 + x3 + x4 − x5 = 0     2x1 + x2 − x3 − x4 − x5 = 0 4.  x1 + 7x2 − 5x3 − 5x4 + 5x5 = 0      3x − x − 2x + x − x = 0 1 2 3 4 5   x1 − 2x2 + x3 + x4 − x5 = 0      2x1 + x2 − x3 + 2x4 − 3x5 = 0 5.  2x1 − 5x2 + x3 − 2x4 + 2x5 = 0      3x − 2x − x + x − 2x = 0 1 2 3 4 5   2x1 + x2 − x3 − x4 + x5 = 0      3x1 + 3x2 − 3x3 − 3x4 + 4x5 = 0 6.  x1 − x2 + x3 + x4 − 2x5 = 0      4x + 5x − 5x − 5x + 7x = 0 1 2 3 4 5   3x + 2x2 − 5x3 + 4x4 = 0   1 7. 3x1 − x2 + 3x3 − 3x4 = 0    3x1 + 5x2 − 13x3 + 11x4 = 0 S K 2.2.10. ² β áu A A ∈ Mm×n (K), β ∈ K 1×n . b 1˜m R(A). S K 2.2.11. äe¡ (ùp•§| àg‚5•§| AX = 0 àg‚5•§|´Äkš"):   x2 + · · · + xn−1 + xn       x1 + x3 + · · · + xn   ··· ··· ··· ··· ··· ··· ··· ···     x1 + x2 + · · · + + xn      4x + x + · · · + x 1 2 n−1 n ‡•§, n ‡™•ê, …1 i ‡•§ = 0 = 0 )þ÷v βX = 0. y ··· = 0 = 0 Ò†>Ø¹™•ê xi .) S K 2.2.12. •þ η1 , · · · , ηr Ñ´‚5•§| AX = b y²: c1 η1 + · · · + cr ηr •´Ó˜‚5•§| AX = b ), c1 , · · · , cr •~ê, … c1 +· · ·+cr = 1. ). ! A b n×1 S K 2.2.13. A ∈ Mm×n (K), b ∈ K . •Ä (m + 1) × (n + 1) Ý B = T . b 0 y²: XJ rank(A) = rank(B), K‚5•§| AX = b ˜½k). S K 2.2.14. ée :)X. 1. z‡‚5•§|, éÑT•§| Ï), ±9ƒA   2x1 − 2x2 + x3 − x4 + x5      x1 + 2x2 − x3 + x4 − 2x5  4x1 − 10x2 + 5x3 − 5x4 + 7x5      2x1 − 14x2 + 7x3 − 7x4 + 11x5 àg‚5•§| = 1 = 1 = 1 = −1 ˜‡Ä 1 98 2.   9x1 + 12x2 + 17x4       6x + 3x2 + 3x3 + 8x4   1 3x1 + 6x2 + 6x3 + 13x4     6x1 + 5x2 − x3 + 7x4      x + 6x + 2x + 9x 1 2 3 4 = 4 = 4 = 2 = 3 = 0 Ù •þ˜m9Ùf˜m S K 2.2.15. γ0 ´‚5•§| AX = b ˜‡A), η1 , · · · , ηs ´àg‚5•§| AX = 0 ‡Ä:)X. γ1 = γ0 + η1 , · · · , γs = γ0 + ηs . y²: AX = b ?¿˜‡) γ þŒ ¤±e/ª: Ù¥ c0 , c1 , · · · , cs ∈ K … c0 + c1 + · · · + cs = 1 . γ = c0 γ0 + c1 γ1 + · · · + cs γs S K 2.2.16. ˜ A = (aij ) ∈ Mn (R) ÷v é?¿ 1 ≤ i, j ≤ n, i 6= j þk aij = −aji , aii = ajj . y²: eàg‚5•§| AX = 0 kš"), K a11 = · · · = ann = 0. S K 2.2.17. ˜m¥ †‚ L ²L: A(2, 3, −5), ¿…†±e†‚ l ²1: l : ¦†‚ L •§. S K 2.2.18. þ²1. ¦ Π ˜m¥ ²¡ Π ²L: A(1, 2, 3), ¿…† α = (1, −2, 1) Ú β = (0, 1, 2) ùü‡• ëê•§Ú˜„•§. S K 2.2.19. ˜m¥ ˜‡:{ª•§. S K 2.2.20. b y z+1 x−2 = = . −1 3 4 ²¡ Π ˜„•§• 3x − 2y + 5z − 1 = 0. ¦ Π a ∈ R •ëê, La L«˜m¥˜„•§• La † z ¶ƒ . ¦ëê a ŒU ˜‡{•þ¿  3x − y + 2z − 6 = 0 Š. Š, ?Ø†‚ L Ú ²¡ Πk x−5 y+4 z−1 = = 1 −2 3 Πk : x + ky − 5z − 10 = 0 ¡, ¿¦§‚¤3²¡ x−3 y−2 z+1 = = 3 2 −2 x+2 y−3 z+3 L2 : = = 2 −3 4 L1 : ²¡•§. ˜'X, Ù¥ L Ú Πk ©O•: L : S K 2.2.23. y²±eü^†‚ L1 Ú L2 †‚. x + 4y − z + a = 0 S K 2.2.21. ¦˜m¥²L: M (2, −3, 1) …†²¡ 2x + 3y + z + 1 = 0 ²1 S K 2.2.22. Šâëê k ∈ R ÑΠ ˜„•§. 2.3 Ä–•þ˜m S K 2.2.24. 99 m •š"¢ê. Šâ m Š, ?Ø±eü^†‚ L1 Ú L2 ˜'X: x−3 y−1 z−7 = = m 4 2 x+2 y z−1 L2 : = = 2 −3 4 L1 : S K 2.2.25. ˜m¥ †‚ l Ú²¡ Π •§Xe: x−5 y+3 z−1 = = , 2 −2 3 Π : x + 2y − 5z − 11 = 0 . l : älÚΠ elÚΠ ˜'X. Xk•˜ •§U•: :, ž¦ÑT : ‹I. y−3 z−4 x − 13 = = , 8 2 3 Π : x + 2y − 4z + 1 = 0 . l : Ó ¯K‰YXÛ ? S K 2.2.26. ²¡† ‹IX¥‰½ n‡: A(a1 , a2 ), B(b1 , b2 ), C(c1 , c2 ).   a1 a2 1   ¿©7‡^‡´Ý  b1 b2 1 •• 3. c1 c2 1 1. y²: A, B, C n:Ø ‚ 2. b ‚, P (α, β) ń A, B, C n:Ø / ABC %. y²: e a1 , a2 , b1 , b2 , c1 , c2 þ•knê, K α, β •þ•knê. S K 2.2.27. b A, B ∈ Mm×n (K) ˜½kú š"). S K 2.2.28. m × (n + s) Ý •Ñ < n2 . y² AX = 0 Ú BX = 0 ùü‡àg‚5•§| A ∈ Mm×n (K), B ∈ Mm×s (K). . C = (A B) ´ò A, B †m¿ü˜˜ 1. y² max{rank(A) , rank(B)} ≤ rank(C) ≤ rank(A) + rank(B). 2. Þ~`² max{rank(A) , rank(B)} < rank(C) Ú max{rank(A) , rank(B)} = rank(C) kŒUÑy. 3. Þ~`² rank(C) < rank(A) + rank(B) Ú rank(C) = rank(A) + rank(B) œ¹þ œ¹þkŒUÑy. S K 2.2.29. y²: éu?¿ü‡Œ ƒÓ Ý A, B, rank(A + B) ≤ rank(A) + rank(B). 2Þ~`² rank(A + B) < rank(A) + rank(B) Ú rank(A + B) = rank(A) + rank(B) œ¹þk ŒUÑy. 2.3 Ä–•þ˜m ªu´žÿÚ?˜„ •þ˜m½Â . Ù¢·‚3 Ùcü!¥y² Nõ(Ø, ÑŒ±† í2 ˜„ k•‘•þ˜m¥. ¤±, éuNõù (Ø·‚¬† ŽÑy², ½ö•-ãy ²¥ ˜ '…gŽ, ò•äN [!3‰ÖöÖv. ÚL ˜ , K o´L« C ˜‡f•. 1 100 2.3.1 Ù •þ˜m9Ùf˜m Ä–B, Ä– ! ÖöØ”ëì·K 2.1.2 5n)Xe½Â: ½ Â 2.3.1. V •˜‡š˜8Ü. 2b : • 3V ?¿ü‡ ƒƒm®²½Â ˜«$Ž, ¡•\{ (addition). =, éu V ¥?¿ ƒ α Ú β, ÑU,˜²( {KéA V ƒS ˜‡•˜(½ ƒ, T ƒP• α + β. (žÖ ö5¿, ùp PÒ + ==´• Ä–/5L«ù«$Ž, §8c•´˜‡/ªþ “ ŽÎ”, Ù?ÛäN ¹Â„vk7‡²(•Ñ. ·‚•I‡38 ‘ z‡äN~f¥2äN) º·‚Ž^ù‡PÒ“LŸo¹Â.) • 3 K ?¿ ƒÚ V ?¿ ƒƒm®²½Â ˜«$Ž, Ù$Ž(Jo´ V ¥ ƒ, ·‚ ¡T$Ž•ê¦ (scalar multiplication). =, éu?¿ k ∈ K Ú?¿ α ∈ V , ÑUì,˜²( {KéA V ¥˜‡•˜(½ ƒ, T ƒP• k · α ½ kα. XJþã (V ¥?¿ü‡ ƒƒm ) \{Ú (V ¥ ƒÚ K ¥~êƒm ) ê¦$Ž÷v ±e¤k5Ÿ, ·‚Ò¡ V ëÓ¤‰ \{Úê¦$Ž´ K þ ˜‡•þ˜m (vector space over K) ½‚5˜m (linear space over K). 1. \{(ÜÆ: é?¿ α, β, γ ∈ V , (α + β) + γ = α + (β + γ). 2. \{ †Æ: é?¿ α, β ∈ V , α + β = β + α. 3. \{kð : •3 ƒ 0 ∈ V (¡• V ¥\{ α ∈ V þk α + 0 = α = 0 + α. (8 ÖöÙGù‡Vgƒ , ·‚•¬²~† Ù ¤ 0 ù«çN f.) ˜‡ð (identity element)) ¦ ± 0 5L« V ¥ \{ð , é¤k Ø72ò 4. \{_o•3: é?¿ α ∈ V , •3˜‡ ƒ α0 ∈ V ¦ α + α0 Ú α0 + α Ñ´˜‡\{ð . ?Û÷vd^‡ α0 Œ±¡• α ˜‡\{_ (additive inverse) ½öK (negative). Xƒc3·K 2.1.2 ¥¤`, ±þ 4 ‡5ŸŒ±o(•ù †+. 5. ê¦kð ˜é{: V 'u\{$Ž/¤˜‡ : K ¥~ê 1 éuê¦$Ž÷v: 1 · α = α é¤k α ∈ V ¤á. 6. ê¦†~ê¦{÷v(ÜÆ: é?¿ c, k ∈ K Ú?¿ α ∈ V , (ck)α = c(kα). 7. ê¦Ú~ê\{÷v© Æ: é?¿ c, k ∈ K Ú?¿ α ∈ V , (c + k)α = cα + kα. 8. ê¦Ú•þ\{÷v© Æ: é?¿ c ∈ K Ú?¿ α, β ∈ V , c(α + β) = cα + cβ. ·‚2grN, î‚5`, “•þ˜m”ù‡Vg•¹o‡‡ƒ: V ù‡8Ü, K ù‡•, ±9®² ‰½ \{$ŽÚê¦$Ž. • ŠéLã•B, kžÿ·‚•¬æê¡˜ `{: V 'u‰½ \{Úê¦$Ž´ K þ •þ˜m, ½ö‰½ \{Úê¦$ŽU¦ V ¤• K þ •þ˜m, . ,, ·‚•~~æ^˜«{z `{, =, † ` V ´ K þ ˜‡•þ˜m. ù«`{ Ø 2J9\{Úê¦$Ž, ´·‚7L˜Ù, \{Úê¦´“•þ˜m”ù‡Vg ˜Ü©. éÓ V Ú K kŒUŒ±‰ÑØÓ \{Úê¦, ÑU¦ V ¤• K þ •þ˜m. V ´ K þ•þ˜mž, V ƒ•~ ¡••þ (vector), K þ •þ˜mkž• ¡ • K-•þ˜m. XJ K ©ª´ ½ ½ö3þe©´²w , ·‚~~Òr `{“K-•þ˜ m” {•“•þ˜m”. ·‚^“¢•þ˜m” (real vector space) Ú“E•þ˜m” (complex vector space) ©O•“ R Ú C þ •þ˜m. 2.3 Ä–•þ˜m 101 (2.3.2) ±þ½Â¥\{Úê¦ ½Â´±Ä– /ª‰Ñ , §‚ŒU‹·‚ÙG ê½ K n ¥ •þƒm \{Úê¦ƒ $ . Ïd, ˜ ·‚®²S±•~ ¯¢, éu˜„ •þ˜m5` ŒUÑI‡\±y²(@Ù (5. ±e·‚Þ˜ •-‡ ~f: 1. •þ˜m V ¥ \{ð ˜½´•˜ . ¯¢þ, XJ 0 Ú 00 Ñ´ V ¥ \{ð , @ocö 0 ð 5Ÿq`² 0 + 0 = 0. ¤±7, 0 = 00 . 8 ·‚~¡ V ¥ ð •Ù¥ 2. •þ˜m V ¥?¿•þ α 0 ù´Ï•, e β, β Ñ´ α 5Ÿ`² 0 + 00 = 00 , ð ö "•þ (zero vector). K (=\{_) ´•˜ K , @oŠâ"•þ . 5ŸÚ\{(ÜÆŒ β = β + 0 = β + (α + β 0 ) = (β + α) + β 0 = 0 + β 0 = β 0 . 8 ·‚ò α 3. •þ\{kž K P• −α. Æ (cancellation law), =, é?¿•þ α, β , γ, l α + β = α + γ Œ±íÑ β + γ. ¯¢þ, ò ª α + β = α + γ 5Ÿ=Œ β = γ. †mü>Óž\þ α _ , 2ÏL\{ (ÜÆÚ"•þ 4. •þ\{Œ±£‘, =, é?¿•þ α, β, γ, l α + β = γ Œ±íÑ α = γ + (−β) (•{Ò´ ªü>Óž\þ −β). 5. éu?¿•þ α ∈ V Ú?¿~ê k ∈ K, 0 · α = 0 , k · 0 = 0 , (−1) · α = −α . (ùp·‚A¿«© éu1˜‡ þ 5Ÿ: êi 0 Ú"•þ 0.) ª, ·‚|^êiƒm ð ª 0 + 0 = 0 !ê¦Ú~ê\{ © Æ±9"• 0 · α + 0 = 0 · α = (0 + 0) · α = 0 · α + 0 · α . é±þ 1 m •†àÚ•mà¦^\{ž Æ= 0 = 0 · α. ‡ ª y²•aq. Ì‡ «O´ùg·‚Ø´^êim ª 0 + 0 = 0, ´^•þ ª 0 + 0 = 0, ê¦†~ê\{ © Æùg‡^ê¦Ú•þ\{ © Æ“O. ùp äNØyL§·‚3‰ÖööS. éu1n‡ ±91˜‡ ª, ·‚I‡^ êi 1 3ê¦$Ž¥ ª (J. ¯¢þ, ù 5Ÿ y ð 5Ÿ, ê¦Ú~ê\{ © Æ, 0 = 0 · α = (−1 + 1) · α = (−1) · α + 1 · α = (−1) · α + α . ù‡ ª (ëÓ\{ • (−1) · α = −α. †Æ) ¿›X (−1) · α ´ α ˜‡\{_. Šâ\{_ •˜5, ·‚ l±þ 1n‡ ª·‚• , éu?¿•þ α, β, ± α + (−1)β Ú α + (−β) ü«•ª •þ´˜ . ù‡$Ž(JŒ±{P• α − β. ù , V ¥ ƒƒmÒk ~{ (subtraction) ù«$Ž. g • K 2.7. V ´K þ •þ˜m, α ∈ V , c ∈ K. y²: 1 102 1. −(−α) = α, =, −α K Ù •þ˜m9Ùf˜m ´ α. 2. e cα = 0, K‡o c = 0 ‡o α = 0. (J«: ù‡(Ø ^5Ÿ.) y²¬^•þ˜m½Â¥ 1 5, 6 ü (2.3.3) V ´ K þ •þ˜m, α1 , . . . , αr ´ V ¥ •þ|. Úƒcaq/Œ±½Â‚5|Ü Vg. =, •þ| α1 , . . . , αr ˜‡ K-‚5|Ü´•Xe•ª ˜‡•þ: k1 α1 + · · · + kr αr Ù¥ k1 , . . . , kr ∈ K . ÚL ˜ , ·‚^ spanK (α1 , . . . , αr ) ½ö{z PÒ span(α1 , . . . , αr ) 5L« α1 , . . . , αr ¤k K-‚5|Ü ¤ 8Ü. XJ•þ β ∈ V áu span(α1 , . . . , αr ), ·‚¡ β Œ± α1 , . . . , αr ‚5L Ñ. ½˜•þ|U…=U‚5LÑ"•þ. ±e½Â Ÿþ´3ìˆ½Â 2.1.5. ½ Â 2.3.4. W ´ K þ •þ˜m, V ´ W f8. XJ±e¤k^‡¤á, ·‚Ò¡ V ´ W ˜‡‚5f˜m!•þf˜m½ö (•{Ñ/`¤) f˜m: 1. W ¥ \{ð 2. V 'u W ¥ (="•þ) 0 áu V . \{µ4, =, éu?¿ u, v ∈ V þk u + v ∈ V . Šâù‡^‡, ·‚Œ±r W þ®k \{$Ž•› V ¥ ƒ5•Ä. ¤±·‚Œ±` V l W @pU« ˜‡\{$Ž, ½ö` W þ \{p V þ ˜‡\{. 3. V 'u W Ú K ¥~ê ê¦$Žµ4, =, éu?¿ v ∈ V Ú?¿ λ ∈ K, þk λv ∈ V . Šâù‡^‡, ·‚Œ±r W † K ¥~êƒm ê¦$Ž•› V ¥ ƒ5•Ä. ¤±·‚ Œ±` V l W @pU« ˜‡ê¦$Ž, ½ö` W þ ê¦p V þ ˜‡ê¦. XJ V ´ W f˜m¿…´ W ýf8 (=, V 6= W ), @o·‚` V ´ W ˜‡ýf˜m (proper subspace). ØJwÑ, e V ´ W f˜m, K V ¥?¿˜| ƒ ?¿ K-‚5|ÜE,áu V . X· K 2.1.7 ¤ã, f˜m V él W U« \{Úê¦$Ž•´˜‡•þ˜m. (2.3.5) S • K-•þ˜m W ¥ ˜‡•þ| (ÚL ˜ , ·‚¦^“•þ|”˜cž%@Ù¥ =¹k•õ‡ (Œ±´"‡) •þ), K span(S) ´ W ˜‡f˜m, ¡•d S )¤ (½Ü¤) f˜ m. XJ S ´˜•þ|, ½ span(S) •"•þ˜m {0}. XJ F ´ W ¥ ˜‡•þx (“•þx”˜c#NÙ¥ÑyÃ•õ‡•þ), ·‚Œ±½Â [ span(F ) = span(S) S ùp S H F f • þ | (=, S ´ Ï L À F ¥k•õ‡•þ • þ |). Ø J y, span(F ) E´ W f˜m, ·‚òÙ¡• F )¤ f˜m, Ù¥ ƒ¡• F ¥•þ ˜‡ K-‚ 5|Ü. e β ∈ span(F ), •¡ β Œ± F ‚5LÑ. XƒcÚn 2.1.13 ¤`, 3?¿•þ˜m W ¥ ˜‡•þx F ¥¤¹ •þ õ, Ù)¤ f ˜m• Œ. , , Ún 2.1.19 (9ÙíØ 2.1.20) •kƒA í2: 3?¿•þ˜m W ¥, span(F ) ´•¹•þx F • •þf˜m. ,, ½Â 2.1.21 ¥‚5 d Vg•Œ±í2 ˜„ •þ ˜m¥?¿•þx œ¹. (žÖögC ÑƒA ½Â !) 2.3 Ä–•þ˜m 103 éu W ˜‡f˜m V , XJ•3˜‡ (=¹k•õ‡•þ ) •þ| S ¦ V = span(S), K ¡ V ´k•)¤ (finitely generated), ¿…ò S ¡• V ˜‡)¤|½Ü¤|. Uì½Â 2.1.22 (3) ¥ •ªŒ±Ó /½Â V 4 )¤|. Ó /, éu˜‡•þx F , XJf˜m V u span(F ), ·‚•Œ±¡ F • V ˜‡)¤ x (generating family). XJ F ´ V )¤x, ¿…l F ¥?¿ K˜‡•þƒ •{ •þx Ø2´ V )¤x, K¡ F ´ V 4 )¤x (minimal generating family). AO/, Ï• W Œ±À•gC f˜m, ¤±·‚Œ±!Ø W ´Äk•)¤, ±9 W )¤ | (x) Ú4 )¤| (x) ù Vg. (2.3.6) 3?¿ K-•þ˜m W ¥, •þ| ‚5ƒ'5ŸÚ‚5Ã'5Ÿ•Œ±•ì½Â 2.1.11 5½Â, ·‚ùpØ2Kã. e S ••þ˜m W ¥ •þ|, •ì½Â 2.1.22 (1) Œ±½Â S ¥ 4Œ‚5Ã'|. •˜„/, F ´ W ¥ •þx. XJ F ¥ ?¿f•þ|Ñ‚5Ã', ·‚Ò` F ‚5 Ã', ÄK¡ F ‚5ƒ'. †óƒ, XJ F ¥•¹˜‡‚5ƒ' f•þ|, @o F Ò´‚5ƒ' , ÄK´‚5Ã' . N´wÑ, XJ F ‚5Ã', @oÙ¥Ø¬k-EÑy •þ. b V ´W ˜‡f˜m. XJ˜‡•þx F =´f˜m V )¤x, q´˜‡‚5Ã' x, @o·‚` F ´ V ˜|Ä. AO/, éu•þ˜m W , ·‚Œ±!ØÄ Vg. X J ˜ ‡ • þ x F0 ¥ • þ Ñ 3 F ¥ Ñ y – ˜ g, · ‚ ¡ F0 ´ F ˜‡f•þx (subfamily of vectors). XJ F ˜‡f•þx F0 ‚5Ã', ¿…éu F ¥ ?¿ ƒ γ, ò γ V\ F0 ƒ Ñ¬¦*¿ •þxC¤‚5ƒ' , @o·‚` F0 ´ F ¥ ˜‡4Œ‚5Ã'x (maximal linearly independent family). AO/, ·‚Œ± F •˜‡f˜m V , l V ¥4Œ‚5Ã'x Vg. g • K 2.8. W • K-•þ˜m, α1 , . . . , αr •Ù¥ •þ|. y²e •ã d: 1. •þ| α1 , . . . , αr ‚5Ã'. 2. Lˆª 0 = 0.α1 + · · · + 0.αr ´"•þ 0 3. é?¿•þ β ∈ W , XJ β Œ± Ü •ª´•˜ . ¤ α1 , . . . , αr ù|•þ‚5|Ü •˜•ª. α1 , . . . , αr ‚5LÑ, @oò β L«¤ α1 , . . . , αr ƒ‚5| g • K 2.9. W • K-•þ˜m, F , F0 •Ù¥•þ (Ø (§‚©O´·K 2.1.14, íØ 2.1.15 Ú·K 2.1.17 ¤ •þx. žÖöc[@ý/y²±e í2): 1. b F š˜. K F ‚5ƒ' ¿©7‡^‡´•3 γ ∈ F ¦ F ƒ¥í ˜g γ f•þx G Œ±‚5LÑ γ. (5¿: XJ γ 3 F Ñygê‡L˜g, @o γ E¬3f•þx G ¥Ñy.) 2. b F0 ´ F f•þx. Kd F0 ‚5ƒ'Œ±íÑ F ‚5ƒ', d F ‚5Ã'Œ±í Ñ F0 ‚5Ã'. 3. b F ‚5Ã'. Kéu?¿ β ∈ W , V\ β ^‡´ β ØU F ‚5LÑ. Ún 2.3.7: V ´ K-•þ˜m, F ´ V ¥•þ F ƒ ¤ •þxE,‚5Ã' •þx. Ke •ã d: ¿©7‡ 1 104 1. F ´ V ¥ Ù •þ˜m9Ùf˜m 4Œ‚5Ã'x. 2. F ´ V ˜|Ä, = F ‚5Ã'¿…´ V 3. F ´ V ˜‡4 ˜‡)¤x. )¤x. y². žÖö U=Œ. y 5Ún 2.1.25 íØ 2.3.8: V • K-•þ˜m. y²g´3ùpE,s , äN[!•‡3©i„`þ· F • V ¥ ?¿•þx, F0 ´ F f•þx. K F0 ´ F ¥ F0 ‚5Ã'¿…† F ‚5 d (= span(F0 ) = span(F )). 1. S • V ¥ ?¿ (=¹k•õ‡•þ 4Œ‚5Ã'|. 2. y². •íØ 2.1.26 íØ 2.3.9: ˜|Ä. 4Œ‚5Ã'x ? …= ) •þ|. K S ¥˜½•3,‡f•þ|´ S ¥ y²=Œ. V • K-•þ˜m. XJ V ´k•)¤ , K•3 V ¥ ,‡•þ| T ¦ T ´ V y². Šâ½Â, V k˜‡)¤| S. ŠâíØ 2.3.8, S ¥Œ±ÀÑ˜‡4Œ‚5Ã'| T , §÷v span(T ) = span(S) = V . ¤±d½Â=• T ´ V ˜|Ä. žÖög1 ·K 2.3.10: 1. b y: •ì·K 2.1.27 ÚíØ 2.1.28 V ´k•)¤ y²Œ±yÑe¡ (Ø: K-•þ˜m. V k˜‡)¤|T¹ r ‡•þ, K V ¥?¿‚5Ã'|¤¹ 2. V ?¿ü|Ä¤¹ •þê8ÑƒÓ. ù‡ê¡• V 3 K þ Ø¬ÚåÜÂ žÿ{P• dim V . •þê8Ø¬‡L r. ‘ê, P• dimK V , ½ö3 (2.3.11) V ´ K-•þ˜m. XJ V k˜|Ä´=¹k•õ‡•þ •þ|, ·‚¡ V ´k• ‘ (finite dimensional). ÄK¡ V ´Ã•‘ (infinite dimensional). Ï•Ä´)¤|, ¤±±þ½ÂgÄ`²k•‘•þ˜m´k•)¤ . ‡L5, íØ 2.3.9 ` ² k•)¤ •þ˜mÑ´k•‘ . d , ·K 2.3.10 wŠ·‚: k•‘•þ˜m ?¿Ääk ƒÓ •þê8, Tê8 uT•þ˜m ‘ê. V ´Ã•‘•þ˜mž, ·‚P dim V = ∞, ÄK·‚Œ±P dim V < ∞. (2.3.12) W • K-•þ˜m, S • W ¥ •þ|. ŠâíØ 2.3.8, S f•þ| T ´ S ¥ 4Œ‚5Ã'| …= T ´f˜m V = span(S) ˜|Ä. ¤±, ·K 2.3.10 (2) `² S ? Û4Œ‚5Ã'|Ñ¹kƒÓê8 •þ. ù‡ê¡• S •, P• rank(S). ù‡½Â•`²: rank(S) = dim span(S). e¡´·K 2.1.31 -#Qã. ·K 2.3.13: W ´k•‘ K-•þ˜m, F ´ W ¥?¿•þx, S0 ´l F ¥ Ñ ˜ •õ‡) •þ ¤ •þ|. XJ S0 ‚5Ã', @o˜½Œ±l F ¥2À eZ (k•õ‡!ŒU´"‡) •þV\ ¥, ¦ *¿ •þ| S ´ F ¥ 4Œ‚5Ã'|. (k S0 2.3 Ä–•þ˜m íØ 2.3.14: 105 W ´k•‘ K-•þ˜m. U, V ´ W 1. éu V ¥?¿‚5Ã'•þ| S0 , ˜½Œ±· *¿¤ •þ| S ´ V ˜|Ä. ù`², k•‘•þ˜m f˜m•Ñ´k•‘ 2. XJ U ⊆ V , K dim U ≤ dim V ¿… Ò¤á f˜m. V\k•õ‡ (Œ±´"‡) •þ S0 ¥¦ . …= U =V. y². (1) 3·K 2.3.13 ¥ F = V =Œ (Óžë„Ún 2.3.7). (2) Ø ª dim U ≤ dim V y²Ú·K 2.1.34 y² ˜ . b S0 ´ V ¥f˜m U ˜|Ä. Šâ (1), S0 Œ±*¿• V ˜|Ä S. XJ U 6= V , K S0 Ø´ V )¤|, Ï Ø´ V Ä. ¤± S ¥ •þêþ7Lî‚Œu S0 ¥•þ êþ, = dim V = |S| > |S0 | = dim U . íØ 2.3.15: V ´k•‘ K-•þ˜m, α1 , . . . , αr • V ¥•þ 1. α1 , . . . , αr ´ V ¤ •þ|. Ke ^‡ d: ˜|Ä; 2. r = dim V ¿… α1 , . . . , αr ‚5Ã'; 3. r = dim V ¿… α1 , . . . , αr ´ V )¤|. y². UÄÚ‘ê ½ÂŒ• (1) U íÑ (2) Ú (3). (3) ⇒ (1): ŠâÚn 2.3.7 •I‡y² α1 , . . . , αr ´ V 4 )¤|. Ï•®• r = dim V , ¤ ± V k˜|Ä¹k r ‡•þ. ·K 2.3.10 (Ø (1) `² V ?¿)¤|¤¹•þê8ØU u r. ¤±)¤| α1 , . . . , αr ¥ K˜‡•þ Ø¬2´ V )¤|. •Ò´`, α1 , . . . , αr ´ V 4 )¤|. (2) ⇒ (1): Ï•®• α1 , . . . , αr ‚5Ã', ¤±f˜m U = span(α1 , . . . , αr ) ‘ê´ r. y3 r = dim V . ¤±díØ 2.3.14 (2) Œ• U = V . ù`² α1 , . . . , αr •´ V )¤|, Ïd´ V ˜ |Ä. 5P 2.3.16. k, ÖöØJuy, íØ 2.3.8 (Ø (2) y²ØU{ü/í2 ¹kÃ•õ•þ •þxœ¹. Ïd, ·‚Ã{^ 5 •{y²Ã•‘ •þ˜mokÄ. ¢Sþ, ‡y²Ã•‘ •þ˜mokÄ, 7L^ ˜^ • 8ÜØún——ÀJún (Axiom of choice), ½ö†ƒ d (Ø (Ï~Œ[¬À^ Zorn Ún (Zorn’s lemma)). Ø=Xd, Š•·K 2.3.10 (2) í2, 3Ã•‘ •þ˜m¥, ?¿ü|Äƒm•˜½•3XV . éù (Ø Øy, k, ÖöŒ±ëw [16] ¥ ½n 1.9 Ú½n 1.12. ˜„ ó, ‚5“ê ÌZSN¥¿ØI‡^ 'uÃ•‘•þ˜mÄ (Ø, Ö6ž•Ø ‹Žéd \?Ø. ÏdÖö•Œ±Ø7y3Ò á˜Ùù (Ø. 2.3.2 ˜ Ä–? ~fõõÃõ ·‚ffÚ\ Ä– •þ˜m˜„Vg, ïá # •þ˜m ¢~. ~ 2.3.17. éu?¿ ê m Ú n, Xê g K ê¦´˜‡ K-•þ˜m. Ù¥¤k Ä "˜Ý m ˜|Ä. Ïd dimK Mm×n (K) = mn. e- U • Mn (K) ¥¤k þn Ý ¤ Ñ U ˜|Ä¿¦Ñ dimK U . •þ˜m ˜ Ä nØ. y3·‚I‡0 m × n Ý 8Ü Mm×n (K) UìÝ \{Ú (m×n) Eij , 1 ≤ i ≤ m, 1 ≤ j ≤ n ¤ù‡•þ˜ f8, K U ´ Mn (K) ˜‡f˜m. žÖöé 1 106 Ù •þ˜m9Ùf˜m ~ 2.3.18. L ´ K 3 C ¥ ˜‡*• (field extension), = K ⊆ L … L •´ C f•. ±Ï~ê \{Š• L \{, òÏ~ê ¦{Š• K † L ¥ ƒ ê¦, K L ´˜‡ K-•þ˜m. AO /, L Œ± • K , Ïd K gCŒ±Š• K-þ •þ˜m. ~X, ¢ê• R ÚEê• C ÑŒ±Š•knê• Q þ •þ˜m, C qŒ±Š• R þ •þ ˜m. ùpI‡5¿, eò C À• R þ •þ˜m, K 1, i ùü‡ C ¥ ƒ ¤ C Š• R-•þ˜m ˜|Ä. Ïd dimR C = 2. ´, XJò C ÀŠ C þ •þ˜m, K dimC C = 1. ¤±, •, ·‚ƒcvkAOrN, ¢Sþzg!Ø•þ˜mž, •O( `{I‡J ´äN=‡•þ •þ˜m. •Œ±`, • K À •´“•þ˜m”ù‡Vg½Â¥ ˜Ü©&E, ØO(•Ñù‡ • {¢SþØUŽ´ “•þ˜m”½Â. ~ 2.3.19. I • R š˜f8, - V = Map(I, R) L«¤kl I R ¼ê8Ü. f, g ∈ V ž, ÏLÅ: Šƒ\ •ªŒ±½ÂÑ¼ê f + g : x 7→ f (x) + g(x). Ó /, é?¿ λ ∈ R, λf Œ± ½Â•¼ê x 7→ λ · f (x). 'uù ½Â \{Úê¦, V ´¢ê• R þ •þ˜m, Ù¥ "•þ • Š• 0 ~Š¼ê. •˜„/, b A ´?¿ š˜8Ü, K •?¿ê•, U = Map(A, K) •¤kl A K N ¤ 8Ü. @oE,Œ±ÏLÅ: Šƒ\ •ª½Â U ¥?¿ü‡ ƒƒm \{, ±9 U ¥ ƒÚ K ¥~êƒm ê¦. ù U Òä K-•þ˜m ( . 8 eÃƒ‡ (², Map(A, K) Š••þ˜mž, Ù\{Úê¦o´@•Uþã•ª½Â. ~ 2.3.20. b I •R¥ ˜‡«m, ¢•þ˜m V = Map(I, R) U~ 2.3.19 ¤ã•ª½Â. - B : = {f : I → R •k.¼ê} = f : I → R •3~ê M > 0 ¦ é¤k x ∈ I þk |f (x)| ≤ M C : = {f : I → R •ëY¼ê} . K B Ú C Ñ´ V f˜m. ŠâêÆ©Û‘§ Ä •£, XJ I ´‡k.4«m, K C ⊆ B. ~ 2.3.21. K N •¤kl 0 m©?Ò…z˜‘ g K ¤kê an n∈N •ª½Â K N þ \{±9 K N † K ƒm ê¦: an n∈N + bn n∈N := an + bn n∈N ; é?¿ λ ∈ K , λ · an n∈N := λan n∈N . ¤ 8Ü. UìXe Œ± y, ù \{Úê¦¦ K N ¤•˜‡ K-•þ˜m. ·‚¡˜‡ê an n∈N A ??•" (vanish almost everywhere), XJ•3 N ∈ N ¦ é¤ k n ≥ N Ñk an = 0. - Z • K N ¥¤kA ??•" ê ¤ f8. K Z ´ K N ˜‡f ˜m. g • K 2.10. ÖöUÄ•~ 2.3.21 ¥ ÏdØïÆÖö}ÁÏé K N Ä.) f˜m Z éÑ˜|Ä ? (•þ˜m K N (2.3.22) 3± êÆ‘§¥, ŒUÖön) õ‘ª (polynomial) Vg•õ 5•Ä¯K . •Ò´`, XJ±i1 x “L˜‡¼ê gCþ, @o'u x (polynomial function) ´•Œ±kXeLˆª ˜‡¼ê f : x 7−→ f (x) = an xn + an−1 xn−1 + · · · + a1 x + a0 Ù¥ n ´,‡g,ê, a0 , . . . , an Ñ´~ê. Ä´J±£ã , ´l¼ê Ý ˜‡õ‘ª¼ê 2.3 Ä–•þ˜m 107 Ù¢, éuõ‘ª ˜‡• Ÿ n)´, Ø‡r x Š,‡ê8þ½Â ¼ê gCþ, ´==òÙÀŠ˜‡/ªÎÒ. • âÑù«*g, ·‚†^Œ i1 X 5L«ù ˜‡Ä–Î Ò, ¿¡ƒ•˜‡C (variable) ½™½ (indeterminate). éz‡g,ê n ∈ N, ·‚2Ú\˜‡ƒ 'é ÎÒ X n . ˜‡± X •™½ !Xê g K õ‘ª/ª (nonzero formal polynomial with coefficients in K) ½Â•˜‡/ªLˆª f (X) = (2.3.22.1) +∞ X ai X i = a0 X 0 + a1 X 1 + a2 X 2 + · · · + · · · , i=0 Ù¥z‡ ai ∈ K …•3 n ∈ N ¦ i ≥ n žþk ai = 0 . Ï~·‚•ò a0 X 0 ù˜‘{ • a0 , ò a1 X 1 { • a1 X. • {zâŠ, ·‚ò“õ‘ª/ª”{ ¡•õ‘ª. 3Ø 7‡•Ñ™½ ž½öF"{zPÒž, ·‚„¬ò f (X) {P• f . Lˆª¥ ~ê ai ¡•õ‘ª f Xê (coefficients). XJ (2.3.22.1) ¥z‡ ai u 0, ·‚¡Tõ‘ª f •"õ‘ª (zero polynomial), P• f = 0. XJ f ´š"õ‘ª, @o·‚Œ±ò (2.3.22.1) ¥Xê•" ‘ŽÑK. u´ f (X) Œ±•˜ / ¤Xe/ª: (2.3.22.2) f (X) = an X n +an−1 X n−1 +· · ·+a1 X +a0 , Ù¥ n ∈ N , a0 , a1 , . . . , an ∈ K … an 6= 0 . dž·‚¡ f gê (degree) • n, P• deg(f ) = n. dž an ¡• f Ä‘Xê (leading coefficient). Ä‘Xê• 1 õ‘ª¡•Ä˜õ‘ª (monic polynomial). 5¿, gê•" õ‘ªŒ± Óu K ¥ š"~ê. ·‚Ö¿5½"õ‘ª gê• −∞, "õ‘ª Ä‘Xê5½• 0. "õ‘ªÚgê•" õ‘ª¡•~Šõ‘ª (constant polynomial). Xê g K ~Šõ‘ ª 8ÜŒ± Óu K. ^ K[X] L«¤kXê g K ™½ • X õ‘ª ¤ 8Ü. P+∞ P+∞ éu K[X] ¥ ü‡õ‘ª f = i=0 ai X i Ú g = i=0 bi X i , ·‚½Â§‚ Ú• (2.3.22.3) f + g := +∞ X (ai + bi )X i . i=0 ù ‰Ñ K[X] þ ˜‡\{$Ž. (2.3.22.4) XJ λ ∈ K, ·‚½Â λ Ú f λf := ê¦• +∞ X (λai )X i . i=0 ÖöØJ y, 'uþ¡½Â \{Úê¦, K[X] ´ K-þ •þ˜m, Ù¥ "•þÒ´"õ‘ ª. Ù¥ •þx F = {X n | n ∈ N} ´ K[X] ˜|Ä. Ïd, dimK K[X] = ∞. éu?¿ d ∈ N, - K[X]≤d L« K[X] ¥gêØ‡L d õ‘ª ¤ f8, K K[X]≤d ´ K[X] f˜m (Uì.~, ·‚@• −∞ ≤ d, ¤±"õ‘ªáu K[X]≤d ). õ‘ª| 1, X, . . . , X d ´ K[X]≤d ˜|Ä, ¤± dimK K[X]≤d = d + 1. ,, õ‘ªƒm„Œ±½Â¦{. äN5`, XJ f, g ∈ K[X] ¤/X (2.3.22.2) k•‘/ ª• n m X X f= ai X i , g = bi X i i=0 K ö (2.3.22.5) i=0 ¦È½Â• f g := m+n X i X i=0 k=0 ! ak bi−k Xi . 1 108 ØJ y, 'uõ‘ªgê Ù •þ˜m9Ùf˜m ª deg(f g) = deg(f ) + deg(g) o¤á. (Uì.~, ·‚@•éu?¿ θ ∈ N ∪ {−∞}, −∞ + θ = −∞.) 'uõ‘ª Ú, Ø ª deg(f + g) ≤ max{deg(f ), deg(g)} o¤á. g • K 2.11. U •õ‘ª˜m K[X] ƒ¥"õ‘ª±9gê•óê U ´Ä´ K[X] f˜mQ ? 2.3.3 õ‘ª ¤ f8. @o * û˜m (2.3.23) 3c¡ 2.2.3 ˜!¥, ·‚w K n ¥ f8ƒmŒ±½Â\{, XJ U Ú W Ñ´ K n f˜m, @o U + W ´˜‡•Œ f˜m. ØJwÑ, Ó EéN´í2 ˜„ •þ˜m ¥. äN5`, V • K-•þ˜m, L, M • V ¥ š˜f8. ·‚”½Â 2.2.16 ¥˜ , ½Â (2.3.23.1) L + M := {u + v | u ∈ L , v ∈ M } . e λ ∈ K, Œ±½Â (2.3.23.2) λL := {λu | u ∈ L} . XJ•3 V ,‡f˜m U Ú,•þ α ∈ V ¦ L = α + U , ·‚¡ L ´˜‡f5²1u U • 8, α ´§ ˜‡•þ“L . žÖög1 y: 3Ún 2.2.23 Ú·K 2.2.24 ¥, ò K n †¤?¿ K-•þ˜m V ƒ 5 ¤k(ØE,¤á. AO/, XJ U Ú W Ñ´ V f˜m, K U + W ´•¹Óž•¹ U Ú W • f˜m, U 6= W ž, ØŒUk• 8ÓžÚ U, W f5²1. d , Ó˜‡• 8Œ±kN õØÓ •þ“L , ?¿ü‡•þ“L ˜½áuf5²1uT• 8 @‡f˜m. (2.3.24) U ´,‡•þ˜m V f˜m. XJÖöc[" Ún 2.2.23 •ã, ŒU¬uy± ey–: ü‡f5²1u U • 8Uìþã\{$Ž (JE,´˜‡f5²1u U • 8, éu?¿š"~ê a ∈ K Ú,‡f5²1u U • 8 α + U , ê¦ a(α + U ) (J aα + aU •´˜‡f5²1u U • 8, Ï• a 6= 0 ž aU = U (ë„Ún 2.2.23 (4)). Ø• Öö¬Ø¬Ž ù ¯K: XJrz‡f5²1u U • 8 Nþ Š˜‡ÔN, , ò¤kù ÔN˜3˜å ¤˜‡8Ü, P• V /U , @oÏLþ¡¤` \{Úê¦, UÄ¦ V /U ¤•˜‡•þ˜mQ ? 3UY?Øƒc, ·‚kÚ\˜‡PÒ. é?¿ α ∈ V , ·‚r• 8 α + U UP• [α]U , ± ¡ù«PÒ5rN [α]U y3 Š8Ü V /U ¥ ˜‡ ƒ5w–. ·‚•¡ [α]U ´ U ˜‡ 8 (coset), α ¡•ù‡ 8 ˜‡“L . k Ö (~X [22]) •ò [α]U ¡• α U Ó{a½ •{a (congruence class modulo U , residue class modulo U ). Šâ• 8 ½Â, V /U ¥ z‡ ƒ (=z‡f5²1u U • 8, ½ö` U z‡ 8) ÑŒ± ¤ [α]U ù«/ª. Ïd, XJ·‚F"`²,(Øé V /U ¥¤k ƒ¤á, •‡- α H V ¥•þ, y¤ã(Øéu [α]U o¤á=Œ. ²LF% y, ÖöØJuy, V /U 'u (2.3.23.1) ª‰Ñ \{ ( ¤˜‡ †+. äN 5`: 2.3 Ä–•þ˜m 109 (1) \{÷v(ÜÆ: é?¿ [α]U , [β]U , [γ]U ∈ V /U , [α]U + [β]U + [γ]U = [α]U + [β]U + [γ]U . (2) \{÷v †Æ: é?¿ [α]U , [β]U ∈ V /U , [α]U + [β]U = [β]U + [α]U . (3) \{kð : ƒ [0]U ∈ V /U Œ±÷v é?¿ [α]U ∈ V /U , [0]U + [α]U = [α]U = [α]U + [0]U . (4) \{_o•3: éu?¿ [α]U ∈ V /U , ƒ [−α]U ∈ V /U Œ±÷v [α]U + [−α]U = [0]U = [−α]U + [α]U . •þ˜m l^5Ÿ®² y ˜Œ, ù´‡ ¯<% žE ! e5·‚5wê¦$Ž´Ä÷v{eo^5Ÿ. ùž·‚I‡3¿: c¡ (2.3.23.2) ª½Â ê¦ØU Ün/ Š8Ü V /U þ ê¦, Ï•~ê 0 Ú• 8 α + U Uì (2.3.23.2) ª$Ž (J˜„Ø´ V /U ¥ ƒ (=f5²1u U • 8), ´"•þ˜m {0}. •Ò´`, Uì (2.3.23.2) ª½Âê¦ {, ~êÚ V /U ¥ ƒ$Ž (JØ´˜½á3 V /U ù‡8ÜSÜ. ù ,´¦ V /U ¤••þ˜m 1˜-æN. 3$/´, ù-æN´éN´â» . Ï••˜ ¯K3u~ê 0 ê¦, Šâ•þ˜m 5 Ÿ, ~ê 0 Ú?Û•þê¦ (J7L u"•þ (ë„ (2.3.2) ˜ã¥ Ñ 5Ÿ 5). ¤±·‚Œ ±<•/5½, ~ê 0 Ú V /U ¥ ƒ$Ž (J u V /U ¥ “"•þ” (c¡·‚®²w , V /U ¥ "•þ´ [0]U , = U g ‰Ñ • 8). rù‡ ½Ú (2.3.23.2) ª(Üå5, Œ±@•8Ü V /U þ¡äkXe•ªû½ ê¦$Ž: (2.3.24.1) λ · [α]U := [λα]U . ÖöŒ± y, XJ• 8 [α]U ¦^, •þ“L β, = [α]U = [β]U , @o [λα]U = [λβ]U . ¤ ± [λα]U =•6u• 8 [α]U , Ø•6u¤À •þ“L α. †óƒ, (2.3.24.1) ª´ûÐ ½Â . 5¿ (2.3.24.1) ªm>• ´• 8 λα + U . λ 6= 0 ž, U = λU . ¤±dž λα + U • u λα + λU , = λ Ú α + U Uì (2.3.23.2) ƒ¦ (J. e5·‚žÖögC y: (2.3.24.1) ª½Â ê¦$Ž±9c¡¤ã \{$Ž (U ¦ V /U ¤•˜‡•þ˜m. ½ Â 2.3.25. V • K-•þ˜m, U ´ V f˜m. ± U z‡ 8 (½=f5²1u U • 8) Š• ƒ ¤8Ü V /U . Uì (2.3.24) ˜ã£ã •ªŒ±½Â V /U þ \{Ú V /U ¥ ƒ† K ¥~êm ê¦, ¦ V /U ¤•˜‡ K-•þ˜m. ù‡•þ˜m¡• V U û˜m (quotient space of V modulo U ). 5P 2.3.26. ŒUkÖö¬¯: “û˜m”˜c¥ “û”Ú·‚Ï~n) êƒm‰Ø{¦ûkŸo éXí ? ‰Y´’½ . ´•[ )ºI‡ 9Öö cØ7:un) ˜ Ä–Vg. ¢Sþ, 3Ä–“êÆ¥k˜«˜„ “ÏL d'X5‰û” öŠ. <a1˜g‰ûŒUÒ ´ÏLü‡ êƒØ‰û5 knê. knê8Ü Q ˜«únz£ã¢Sþ´,‡8ÜÏ L,‡ d'X‰û 8Ü. ·‚ùp £ãš~oÑ, Ì‡´Ï•8c·‚ØF" \ L 1 110 Ù •þ˜m9Ùf˜m õ Ä–[! ¥. XJÖöš‡·‚•²˜‡ë•©z {, Œ± w [27, 1nÙ, §5] ½ö [9, §5.9, §29]. ·‚þ¡0 û˜m, Ù¢• 9 ˜« d'X, ¡• U Ó{ (congruent modulo U ). ä N5`, ·‚Œ±r V ¥ü‡ ƒ α, β ¡• U Ó{, XJ α + U = β + U (ù du α − β ∈ U ). ò V Uì U Ó{ù‡ d'X‰û (ë„ [27, 1"Ù §1]), 8ÜÒ´ V /U . (2.3.27) E U ´,•þ˜m V ¥ f˜m. f˜m U Š^ØI‡AOrNž, • {zP Ò, <‚~~P V = V /U , ¿…r•þ α ∈ V éA 8 [α]U = α + U {P• α. ÏLò˜‡•þ ˜‡N α ∈ V éA § 8 α ∈ V , ·‚Œ± π : V −→ V = V /U ; (2.3.27.1) α 7−→ α = α + U . ù‡N ¡• V û˜m V /U ûN (quotient map) ½ög,ÝK (natural projection)!;‰ ÝK (canonical projection). Ï• U z‡ 8Ñk•þ“L , ¤±ûN π ´÷ . Ó˜‡ 8 Œ±kØÓ •þ“L , ¤± π ˜„5`Ø´ü . Šâ•þ˜m V ¥\{Úê¦ ½Â (= (2.3.23.1) Ú (2.3.24.1)), N´ y: é?¿ α, β ∈ V Ú?¿ λ ∈ K, þk (2.3.27.2) λ·α=λ·α. α+β =α+β , Uì·‚31nÙò‡Ú\ ‚5N . ½Â, (2.3.27.2) ¿Â¢Sþ´` (2.3.27.1) ª¥ ûN π ´˜‡ (2.3.28) ·‚Q²! , éu˜‡Ý A ∈ Mm×n (K), †ƒƒ' k A ˜m C (A) Ú A " m m×1 n n×1 z˜m N (A). cö´ K = K ¥ f˜m, öK´ K = K ¥ f˜m. ·‚y3æ^ û˜m *:5ïá N (A) ¤k 8Ú ˜m C (A) ¥¤k•þƒm ˜‡éA'X. Äk, éu?¿ b ∈ C (A), ·‚• ‚5•§| AX = b k), …Ù)8 Sol(A; b) ´"z˜m N (A) ˜‡ 8, •Ò´û˜m K n /N (A) ¥ ˜‡ ƒ. ¤±·‚kù ˜‡N (2.3.28.1) σ : C (A) −→ K n /N (A) ; b 7−→ Sol(A; b) . ‡L5, XJ L ∈ K n /N (A), ·‚Œ±ÀJ˜‡•þ“L ¦ L = α + N (A), , •Ä Aα ∈ K n . ŠâÝ ¦{ 5Ÿ·‚• Aα ˜½áu A ˜m. , , XJ β ´ L ,˜‡•þ“L , = α + N (A) = L = β + N (A), @o α − β ∈ N (A), = A(α − β) = 0. ¤± Aα = Aβ + A(α − β) = Aβ. Ïd, Aα ••6u• 8 L , Ø•6u§ •þ“L À . ù ·‚Œ±½Â˜‡N (2.3.28.2) η : K n /N (A) −→ C (A) ; α + N (A) 7−→ Aα . žÖö y: σ Ú η ùü‡N p•_N . Uì1nÙ nØ, ¢Sþ σ Ú η Ñ´•þ˜mƒm ‚5N . Ï•§‚Ñ´V , •Œ±`§‚´•þ˜mƒm Ó . Šâ1nÙ nØ, Ó •þ˜mäkƒÓ ‘ê. Ïd (2.3.28.1) ª (½ö (2.3.28.2) ª) Œ n ±`² dim K /N (A) = dim C (A). ·‚• , ˜m C (A) ‘êÒ´Ý A • rank(A). "z˜m N (A) ‘ê u n − rank(A) (½n 2.2.12 (1)). ¤±, û˜m K n /N (A) ‘ê u˜ m K n ‘ê~ f˜m N (A) ‘ê. ·‚y35y², ù‡y–Œ±í2 •˜„ œ¹. ·K 2.3.29: V ´k•‘ K-•þ˜m, U ´ V dim V − dim U . f˜m. K V /U •´k•‘ , ¿… dim(V /U ) = 2.3 Ä–•þ˜m 111 y². f˜m U ˜|Ä e1 , . . . , em . ù´ V ¥ e1 , . . . , em , em+1 , . . . , en . y3·‚•I‡`² ˜‡‚5Ã'|, ÏdŒ±*¿• V ˜|Ä em+1 = em+1 + U , · · · , en = en + U (2.3.29.1) ´û˜m V = V /U ˜|Ä. k5`² (2.3.29.1) ‰Ñ V ˜‡)¤|. û˜m V ¥ ?˜ ª. |^ V Ä e1 , . . . , em , em+1 , . . . , en Œ±ò V ¥ •þ α ¤ ƒŒ± α = λ1 e1 + · · · + λm em + λm+1 em+1 + · · · + λn en , (2.3.29.2) ò (2.3.27.1) ¥ ûN ÓžŠû (2.3.29.2) / Ù¥ λi ∈ K . ü>, 2|^ (2.3.27.2) ªŒ± α = λ1 e1 + · · · + λm em + λm+1 em+1 + · · · + λn en , (2.3.29.3) ¤ α = α+U Ù¥ λi ∈ K . 5¿ e1 , . . . , em áu U , Ïd•þ α0 := λ1 e1 + · · · + λm em áu U . ¤±3û˜m V = V /U ¥ α0 = 0. •Ò´` λ1 e1 + · · · + λm em = 0 = •þ˜m V ¥ "•þ. ùÚ (2.3.29.3) (Üå5Œ±`² α = 0 + λm+1 em+1 + · · · + λn en = λm+1 em+1 + · · · + λn en . Ïd α ∈ V ´ (2.3.29.1) ¥ ƒ| ‚5|Ü. e5, ·‚I‡`² (2.3.29.1) ´ V ¥ ‚5Ã'|. •d·‚b λm+1 , . . . , λn ∈ K U¦ λm+1 em+1 + · · · + λn en = 0 3û˜m V = V /U ¥¤á. (2.3.29.4) Šâ (2.3.27.2), þª†>¢Sþ u´ (2.3.29.4) ª`² β + U † ±•3~ê c1 , . . . , cm ∈ K ¦ u•þ β := λm+1 em+1 + · · · + λn en ∈ V ¤“L 8 β = β + U. 8 U = 0 + U ƒ , ¤± β ∈ U . U ± e1 , . . . , em •˜|Ä, ¤ λm+1 em+1 + · · · + λn en = β = c1 e1 + · · · + cn em . u´3 V ¥·‚ (2.3.29.5) (−c1 )e1 + · · · + (−cm )em + λm+1 em+1 + · · · + λn en = 0 . Ï• e1 , . . . , em , em+1 , . . . , en ´ V ¥ ‚5Ã'|, ¤± (2.3.29.5) ª¥ O/, λm+1 = · · · = λn = 0. ù ´·‚¤Iy² (Ø. 2.3.4 ¤k~êXê u 0. A †Ú††È ƒ c · ‚ Q ² J , X J U Ú W Ñ ´ • þ ˜ m V ¥ f ˜ m, @ o § ‚ Ú U + W E ´ V ¥ f ˜ m (Ú n 2.2.23 (5)). ˜ ‡ g , ¯ K ´, X J U Ú W ‘ ê ® •, U Ø U † Ï L dim U + dim W ù ¦Úû½ U + W ‘ ê Q ? @ ý g ¢ ƒ Ö ö A T Ø J u y, dim(U + W ) = dim U + dim W ù‡úª˜„5`´Øé . 'X, 3n‘˜m K 3 ¥, XJ U Ú W ´ ü ‡ 2 ‘ f ˜ m, @ o dim(U + W ) Ø Œ U u 4, Ï • U + W E ´ K 3 f ˜ m, dim(U + W ) ≤ 3 = dim K 3 . ·‚ ! 1˜‡8I´‡`²XÛOŽÚ˜m U +W ‘ê, ±9Ûž ª dim(U +W ) = dim U + dim W U ¤á. 1 112 ½n 2.3.30: U, W ´•þ˜m V ¥ f˜m. 1. Ú˜m U + W ´k•‘ …= 2. b . KkXe‘êúª (dimension formula) U Ú W Ñ´k•‘ Ù •þ˜m9Ùf˜m U Ú W Ñ´k•‘ . dim(U + W ) = dim U + dim W − dim(U ∩ W ) . y². (1) Ï• U Ú W Ñ•¹u U + W , ¤± U + W ‘êk•ž, U Ú W 7,•Ñ‘êk•. ‡ ƒ, XJ U Ú W Ñ´k•‘ , @oŒ±©O §‚ ˜‡)¤| S Ú T . dž S ∪ T ´ U + W ˜‡)¤| (žÖö(@gCn)ù‡Øä), ¤± U + W ´k•)¤ , Ïd´k•‘ . (2) k f˜m U ∩ W ˜|Ä u1 , . . . , um . §Œ±*¿• U ˜|Ä u1 , . . . , um , v1 , . . . , vj , •Œ±*¿• W ˜|Ä u1 , . . . , um , w1 , . . . , wk . ·‚•Iy²: u1 , · · · , um , v1 , · · · , vj , w1 , · · · , wk (2.3.30.1) ´ U +W ˜|Ä. ·‚3‰Öögy: (2.3.30.1) ´ U + W ˜‡)¤|. ±e•I2y² (2.3.30.1) ´‚5Ã'|. •d, b ~ê| a1 , . . . , am , b1 , . . . , bj , c1 , . . . , ck ∈ K ¦ a1 u1 + · · · + am um + b1 v1 + · · · + bj vj + c1 w1 + · · · + ck wk = 0 . dª£‘Œ a1 u1 + · · · + am um + b1 v1 + · · · + bj vj = −c1 w1 − · · · − ck wk . (2.3.30.2) 5¿ ¦ (2.3.30.2) †>áu U , m>áu W . Ïdm>áu U ∩ W . Ïd•3~ê d1 , . . . , dm ∈ K − c1 w1 − · · · − ck wk = d1 u1 + · · · + dm um . (2.3.30.3) du u1 , . . . , um , w1 , . . . , wk ´ W ˜|Ä, Ïd´‚5Ã'|. ¤± (2.3.30.3) ª¥¤kXê• 0. AO/ c1 = · · · = ck = 0. “£ (2.3.30.2) ª=• a1 u1 + · · · + am um + b1 v1 + · · · + bj vj = 0 . y3|^ u1 , . . . , um , v1 , . . . , vj ù|•þ –dy.. (2.3.31) Šâ½n 2.3.30 (2) ¥ ‚5Ã'5Œ ‘êúª, éu V a1 = · · · = am = b1 = · · · = bj = 0. ½n ü‡k•‘f˜m U Ú W , dim(U + W ) = dim U + dim W ⇐⇒ U ∩ W = 0 . (2.3.31.1) (ùp·‚#NgCE^PÒ, ¦ 0 Œ±L«"•þ˜m.) ·‚Ž}Áò±þ?Ø‰í2, 5•Ä?¿k•õ‡f˜m Ú. Äk, éu•þ˜m V ¥ k•õ‡f8 L1 , . . . , Lr (Ù¥ r ≥ 2), Œ±ÏL4í •ªò L1 + · · · + Lr ½Â• L1 + · · · + Lr := L0 + Lr , ½ö, ·‚•Œ±† (2.3.31.2) Ù¥ L0 := L1 + · · · + Lr−1 U8B{Œ±b ®²k½Â . ½Â L1 + · · · + Lr := {α1 + · · · + αr | Ù¥éz‡ αi ∈ Li Œ±?¿À }. 2.3 Ä–•þ˜m 113 •{zPÒ, ·‚~~P r X Li = L1 + · · · + Lr . i=1 • ØJ y: XJ U1 , . . . , Ur Ñ´ V f˜m, K U1 + · · · + Ur ´ V ¥Óž•¹z‡ Ui , 1 ≤ i ≤ r f˜m. b z‡ Ui Ñ´k•‘f˜m. Œ±y²: Ø ª (2.3.31.3) dim(U1 + · · · + Ur ) ≤ dim U1 + · · · + dim Ur o¤á (g•K 2.12). y3·‚Ž• ª dim(U1 + · · · + Ur ) = dim U1 + · · · + dim Ur ÛžU ¹•~. |^ (2.3.31.1) Ú (2.3.31.3) ØJuy (2.3.31.4) ¤á. ± r = 3 œ dim(U1 + U2 + U3 ) = dim(U1 ) + dim(U2 ) + dim(U3 )  dim(U + U ) = dim(U ) + dim(U ) 1 2 1 2 ⇐⇒ dim(U1 + U2 + U3 ) = dim(U1 + U2 ) + dim(U3 ) ⇐⇒ U1 ∩ U2 = (U1 + U2 ) ∩ U3 = 0 . 3 (2.3.31.4) ¥ • ˜‡^‡ “U1 ∩ U2 = (U1 + U2 ) ∩ U3 = 0” ƒ¥, U1 , U2 Ú U3 n‡f˜m / q Ø´ ² . Œ´ ª dim(U1 + U2 + U3 ) = dim(U1 ) + dim(U2 ) + dim(U3 ) ¥, ùn‡ f˜m / ´ ² . ùq w Ø g,. ØL, Öö ¤g•K 2.13 ƒ Ò¬²xÙ¥ n . g • K 2.12. U1 , . . . , Ur (r ≥ 2) ´•þ˜m V ¥ r ‡k•‘f˜m. y²Ø g • K 2.13. U1 , U2 , U3 ´•þ˜m V ¥ n‡f˜m. y²e (1) U1 ∩ U2 = (U1 + U2 ) ∩ U3 = 0; (2) U2 ∩ U3 = (U2 + U3 ) ∩ U1 = 0; (3) U1 ∩ U3 = (U1 + U3 ) ∩ U2 = 0; (4) U1 ∩ (U2 + U3 ) = U2 ∩ (U1 + U3 ) = (U1 + U2 ) ∩ U3 = 0. (5¿: ùp·‚Øb½ Ui ´k•‘ ˜m !) ^‡ Ž‡?˜Ú ² ª dim(U1 + · · · + Ur ) = dim U1 + · · · + dim Ur ¤á ê¡ ½Â´•B$ å». ½ Â 2.3.32. ˜«•ªò α (2.3.32.1) U1 , · · · , Ur ••þ˜m V ¤ ª (2.3.31.3). d: ˜„œ¹, ·‚@•¦ f˜m. XJéu?¿ α ∈ U1 + · · · + Ur , =k•˜ α = α1 + α2 + · · · + αr Ù¥z‡ αi ∈ Ui , @o·‚` U1 + · · · + Ur ´˜‡†Ú (direct sum). XJ˜‡f˜m W U ¤k•õ‡f˜m W1 , . . . , Wr Ú (= W = W1 + · · · + Wr ), … W1 + · · · + Wr ´†Ú, ·‚` W ´ W1 , · · · , Wr †Ú, P• W = W1 ⊕ · · · ⊕ Wr . •{zPÒ, · ‚~~P r M W = Wi . i=1 kž·‚•¡ W = W1 ⊕· · ·⊕Wr ù«Lˆª´ W into subspaces). ˜‡f˜m†Ú©) (direct sum decomposition 1 114 ·K 2.3.33: U1 , U2 ´•þ˜m V f˜m. Ke •ã Ù •þ˜m9Ùf˜m d: 1. U1 + U2 ´†Ú. 2. "•þ 0 =U± 0 = 0 + 0 •ª ¤Úª α1 + α2 /ª, Ù¥ α1 ∈ U1 , α2 ∈ U2 . •Ò´`, XJ α1 ∈ U1 , α2 ∈ U2 , … α1 + α2 = 0, @o7, α1 = α2 = 0. 3. U1 ∩ U2 = 0. XJ U1 Ú U2 Ñ´k•‘ , K±þ•ã•Úe¡ ª d: dim(U1 + U2 ) = dim(U1 ) + dim(U2 ) . y². 3‰Öö. (½‘,Ç‘žù).) |^8B{Œ±ò·K 2.3.33 í2•: ·K 2.3.34: U1 , · · · , Ur ´•þ˜m V f˜m. Ke •ã d: 1. U1 + U2 + · · · + Ur ´†Ú. 2. "•þ 0 =U± 0 = 0 + 0 + · · · + 0 •ª ¤Úª α1 + α2 + · · · + αr /ª, Ù¥z‡ αi ∈ Ui . •Ò´`, XJz‡ αi ∈ Ui , … α1 + α2 + · · · + αr = 0, @o7, α1 = α2 = · · · = αr = 0. 3. éuz‡ i = 1, . . . , r, þk ! (2.3.34.1) Ui \ X Uj =0. 1≤j≤r j6=i XJz‡ Ui Ñ´k•‘ (2.3.34.2) , K±þ•ã•Úe¡ dim(U1 + · · · + Ur ) = ª r X d: dim(Ui ) . i=1 y². 3‰Öö. (½‘,Ç‘žù).) ~ 2.3.35. ·‚Þ˜~`²: ·‚•Än‡f˜m U1 , U2 , U3 ž, =‡¦ùn‡f˜müü • 0 Øv±`² U1 + U2 + U3 ´†Ú. ·‚ 8 U1 = {(x, y, 0) ∈ K 3 | x, y ∈ K} , U2 = {(0, 0, z) ∈ K 3 | z ∈ K} , U3 = {(0, y, y) ∈ K 3 | y ∈ K} . K U1 + U2 + U3 Ø´†Ú, ´ U1 ∩ U2 = U1 ∩ U3 = U2 ∩ U3 = 0. íØ 2.3.36: V •k•‘ K-•þ˜m, U1 , . . . , Ur • V K±e•ã d: 1. V = U1 ⊕ · · · ⊕ Ur . f˜m. 2.3 Ä–•þ˜m 115 2. éz‡ i = 1, . . . , r, ? Ui αi1 , . . . , αini ˜|Ä, •þ| α11 , α12 , · · · , α1n1 ; α21 , · · · α2n2 ; · · · · · · ; αr1 , · · · , αrnr ˜½´ V ˜|Ä. 3. éz‡ i = 1, . . . , r, •3 Ui αi1 , . . . , αini ˜|Ä, ¦ •þ| α11 , α12 , · · · , α1n1 ; α21 , · · · α2n2 ; · · · · · · ; αr1 , · · · , αrnr ´V ˜|Ä. y². e αi1 , . . . , αini ´ Ui U1 + · · · + Ur = ˜|Ä, @o Ui = span(αi1 , · · · , αini ). u´ r X span(αi1 , · · · , αini ) i=1 = span(α11 , α12 , · · · , α1n1 ; α21 , · · · α2n2 ; · · · · · · ; αr1 , · · · , αrnr ) ƒ XJ V = U1 ⊕ · · · ⊕ Ur , K˜•¡ V = U1 + · · · + Ur , þª`²Ü¿ αi1 , . . . , αini ù ´V ˜‡)¤|. ,˜•¡, Šâ·K 2.3.34 ¥ ^‡ (2) Œ±`²•þ| •þ| α11 , α12 , · · · , α1n1 ; α21 , · · · α2n2 ; · · · · · · ; αr1 , · · · , αrnr ‚ 5 Ã ', l ´ V ˜ | Ä. (½ ö Ö ö • Œ ± | ^ í Ø 2.3.15 5 y ² ù ˜ :.) ù Ò y ² (1)⇒(2). ‡ L 5, X J ® • ± þ Ï L Ü ¿ αi1 , . . . , αini •þ|´ V Ä, K Œ í Ñ dim V = P dim Ui ±9 V = U1 + · · · + Ur . ¤±Uì½Â, ùÒ´` V = U1 ⊕ · · · ⊕ Ur . Ïd, (3)⇒(1). Ï•z‡ Ui o•3Ä, ¤± (2)⇒(3) ´w, . g • K 2.14. XJ3íØ 2.3.36 ¥b V ´Ã•‘ ek, s@•ƒA (ØATXÛO(•ã¿y² ? e520 ˜‡-‡ , Öö´Ä@•Ekaq (ØŒ±¤á ? ¯¢. ·K 2.3.37: V ´k•‘•þ˜m, U ´ V f˜m. ˜½•3f˜m W ⊆ V ¦ V = U ⊕ W . ù W ¡• U ˜‡†ÚÖ (direct sum complement) ½Ö˜mk (complementary subspace). y². f˜m U ˜|Ä α1 , . . . , αr , (ŠâíØ 2.3.14) Œ±òÙ*¿• V ˜|Ä α1 , . . . , αr , αr+1 , . . . , αn . - W = span(αr+1 , . . . , αn ), K αr+1 , . . . , αn ´ W g • K 2.15. Þ~`²: ˜‡f˜m ˜|Ä. díØ 2.3.36 Œ• V = U ⊕ W . †ÚÖ˜„Ø´•˜ c¡·‚?Ø ˜‡•þ˜m©)•f˜m†Ú m“© ”Ñ˜‡•Œ˜m •ª. kq k Ör÷v V = U + W f˜mÑ¡• U . œ¹. e¡·‚0 ˜«l?¿ü‡•þ˜ Ö˜m. ¤±, ·‚¦þ¦^“†ÚÖ”ù‡c±;•· . 1 116 Ù •þ˜m9Ùf˜m (2.3.38) V1 , . . . , Vn • K-•þ˜m. 3(k È (Cartesian product) V := V1 × · · · × Vn ù‡8Ü þ, ÏLÅ‡©þþéAƒ\Úê¦Œ± V þ \{Úê¦. =, é?¿ ui , vi ∈ Vi , i = 1, . . . , n Ú?¿ λ ∈ K, ½Â (u1 , . . . , un ) + (v1 , . . . , vn ) := (u1 + v1 , · · · , un + vn ) , λ(u1 , . . . , un ) := (λu1 , · · · , λun ) . ØJ y, Uìþã\{Úê¦ V1 × · · · × Vn ¤˜‡•þ˜m, ¡ƒ• V1 , . . . , Vn product). - V = V1 × · · · × Vn . •Ä V ¥ e f8 †È (direct Ṽ1 : = {(u1 , 0, · · · , 0) ∈ V | u1 ∈ V1 } = {(u1 , . . . , un ) ∈ V | éz‡ j 6= 1 , uj = 0} , ·················· (2.3.38.1) Ṽn : = {(0, · · · , 0 , un ) ∈ V | un ∈ Vn } = {(u1 , . . . , un ) ∈ V | éz‡ j 6= n , uj = 0} . ¯¢þ, Ṽi , i = 1, . . . , n þ• V f˜m, … V = Ṽ1 ⊕ · · · ⊕ Ṽn (2.3.38.2) b V1 , . . . , Vn þ•k•‘ K-•þ˜m. éz‡ i = 1, . . . , n, αi1 , . . . , αimi • Vi ˜|Ä. K (2.3.38.3) (α11 , 0, 0, · · · , 0) , · · · , (α1m1 , 0, 0, · · · , 0) ´ Ṽ1 ˜|Ä , (0, α21 , 0 · · · , 0) , · · · , (0, α2m2 , 0, · · · , 0) ´ Ṽ2 ˜|Ä , ···································· (0, 0, · · · , 0 , αn1 ) , · · · , (0, , 0, · · · , 0 , αnmn ) ´ Ṽn ±þØä äN ˜|Ä , y3‰ÖööS. ·K 2.3.39: V1 , . . . , Vn •k•‘ K-•þ˜m. éz‡ i = 1, . . . , n, Ä. KXe•þ| αi1 , . . . , αimi • Vi ˜| (α11 , 0, 0, · · · , 0) , · · · , (α1m1 , 0, 0, · · · , 0) (0, α21 , 0 · · · , 0) , · · · , (0, α2m2 , 0, · · · , 0) ···································· (0, 0, · · · , 0 , αn1 ) , · · · , (0, , 0, · · · , 0 , αnmn ) ´ V1 × · · · × Vn ˜|Ä. ¤±, dim(V1 × · · · × Vn ) = dim(V1 ) + · · · + dim(Vn ) y². nÜ(2.3.38.2), (2.3.38.3) ÚíØ 2.3.36 Œ 2.3.5 c˜‡Øä. ˜‡'u‘ê äó‘=w,. SK S K 2.3.1. ¢•þ˜m: é±eˆ‡œ¹, ä8Ü V UÄ^•½ $Ž Š•\{!$Ž Š•ê¦ ¤ 2.3 Ä–•þ˜m 117 1. V = R2 , é?¿ (a1 , b1 ), (a2 , b2 ) ∈ V Ú k ∈ R, ½Â (a1 , b1 ) (a2 , b2 ) = (a1 + a2 , b1 + b2 ) , k 2. V • (a1 , b1 ) = (a1 , b1 ) . ¢ê8, é?¿ a, b ∈ V Ú k ∈ R, ½Â a b = ab , k a = ak . S K 2.3.2. V ´ K-•þ˜m, U±e•ª½Â K ¥ ê† V ¥•þ $Ž : v ∈ V , k ∈ K. e k = 0, K- k v = 0, e k 6= 0, K- k v = k1 v. Á¯: ± V ¥ 5 \{Š•\{, ±#½Â $Ž Š•ê¦, ù UÄ¦ þ˜m ? V ¤• K-• S K 2.3.3. - V = R2 . é?¿ (a1 , b1 ), (a2 , b2 ) ∈ V Ú k ∈ R, ½Â (a1 , b1 ) (a2 , b2 ) = (a1 + a2 , b1 + b2 + a1 a2 ) , k(k − 1) 2 a . k (a1 , b1 ) = ka , kb + 2 1. y²: ò Š• V þ \{! Š• V Ú R ƒm 2. - M = {(a, 0) | a ∈ R}, N = {(0, b) | b ∈ R}. ¦´ ) f˜m. S K 2.3.4. b (a) \{ é˜‡š˜8Ü V ®²½Â ê¦, Œ±¦ V ¤•˜‡¢•þ˜m. ä M, N ´Ä´þã•þ˜m V (\{´ , ê ˜‡\{$Ž, §÷v±e5Ÿ: (ÜÆ¤á. (b) •3˜‡ ƒ0∈V ¦ (c) éu˜‡ ½ é?Û v ∈ V þk v + 0 = v. ÷vc˜^‡ ƒ 0 ∈ V , ±9?¿ v ∈ V , •3 u ∈ V ¦ v + u = 0. y²: 1. é?¿ α, β ∈ V , XJ α + β = 0, K β + α = 0. (J«: β0 ∈ V ¦ β + β 0 = 0. ±ØÓ•ªOŽ ((β + α) + β) + β 0 .) 2. é?¿ v ∈ V , þk 0 + v = v. S K 2.3.5. V ´š˜8Ü, b 3 V þ®²½Â ˜‡\{$Ž, 3 K Ú V ¥ ƒƒm® ²½Â ˜‡ê¦$Ž. b ®•¤‰½ \{Úê¦÷v•þ˜m ½Â (=½Â 2.3.1) ¥Ø \{ †Æƒ 7 ^5Ÿ. 1. y (2.3.2) ˜ã?Ø 5 ‡5Ÿ y²¿™^ \{ †Æ, Ïd§‚3y3 ¤á. AO/, éu?¿ α ∈ V , ÙK −α ∈ V k(ƒ ½Â. 2. y²: Ù¦ †Æéu V þ¤‰ 7 ^5ŸíÑ.) (J«: é?¿ α, β ∈ V , ±· \{•¤á. (ù`²: 3•þ˜m ½Â¥, \{ •ªOŽ (α + β) + (−1) · (β + α).) b eE, †ÆŒ±d 1 118 Ù •þ˜m9Ùf˜m S K 2.3.6. V = R2 , K = R, U ì Ï ~ • þ \ { ½ Â V ¥ ƒ m \ {. é u c ∈ K Ú v = (x1 , x2 ), Uì c · v := (c2 x1 , c2 x2 ) •ª5½Âê¦. Á¯•þ˜m ½Â (=½Â 2.3.1) ¥1 5–8 ^5Ÿ= ¤á, = Ø¤á ? S K 2.3.7. V ´š˜8Ü, b 3 V þ®²½Â ˜‡\{$Ž, 3 K Ú V ¥ ƒƒm® ²½Â ˜‡ê¦$Ž. b ®•¤‰½ \{Úê¦÷v•þ˜m ½Â (=½Â 2.3.1) ¥Ø 1 4 ^ ('u\{_ •35) ƒ 7 ^5Ÿ. y²: 1 4 ^5Ÿ•¤á ¿©7‡^‡´éu?¿ α ∈ V þk 0 · α = 0. S K 2.3.8. V ´ K-•þ˜m, A ´˜‡8Ü, f : A → V ´˜‡V , Ù_N P• f −1 : V → A. é?¿ a, b ∈ A ½Â a b = f −1 (f (a) + f (b)). é?¿ k ∈ K Ú?¿ a ∈ A, ½Â k a = f −1 (kf (a)). y² A 'u\{ Úê¦ ¤ K-•þ˜m. S K 2.3.9. ò R2 À• R-•þ˜m. ‰Ñ R2 ¥˜‡š˜f8 U U Ø´ R2 f˜m. S K 2.3.10. ˜m ? ~f, ¦ U 'uê¦´µ4 K = Q(i) = {a + bi | a, b ∈ Q}. ò C À• K-•þ˜m. ž¯f8 R ´Ä´ C S K 2.3.11. 3 K n ¥e f8Ü´Ä´f˜m? XJ´f˜m, ž(½§ XJØ´f˜m, ž Ñ§¤)¤ f˜m, ¿‰ÑTf˜m ˜|Ä. 1. W = {(a1 , a2 , · · · , an ) ∈ K n : a1 + a2 + · · · + an = 0}; 2. U = (a1 , a2 , · · · , an ) ∈ K n : a1 , a2 , · · · , an ØÓžŒu", ½ØÓž 3. V = (a1 , a2 , · · · , an ) ∈ K n : k,‡ i, ¦ ai > 0 . S K 2.3.12. y²½‰Ñ‡~: v1 , v2 , , v3 , v4 ´•þ˜m V v1 , v 2 ∈ U , v3 ∈ / U, v4 ∈ / U , K v1 , v2 ´ U ˜|Ä. S K 2.3.13. U, V Ú W Ñ´,•þ˜m , f ‘ê, ¿‰Ñ˜|Ä; u" ; ˜|Ä, U ´ V f˜m. XJ f˜m. 1. ª U ∩ (V + W ) = (U ∩ V ) + (U ∩ W ) ´Ä˜½¤á ? e´, ž‰Ñy², eÄ, ž‰Ñ‡~. 2. ª U ∩ (V + (U ∩ W )) = (U ∩ V ) + (U ∩ W ) ´Ä˜½¤á ? e´, ž‰Ñy², eÄ, ž‰Ñ ‡~. S K 2.3.14. U Ú W Ñ´•þ˜m V f˜m. y²±eØä d: 1. U ∪ W = U + W . 2. U ⊆ W ½ W ⊆ U . 3. U ∪ W ´‡f˜m. S K 2.3.15. U, V Ú W Ñ´,•þ˜m f˜m. y²: (U + V ) ∩ (U + W ) = U + (U + V ) ∩ W . S K 2.3.16. α, β ´•þ˜m V ¥ •þ, W ´ V • U , {β} ∪ W )¤ f˜mP• M . y²: XJ β ∈ U , β∈ / W, K α ∈ M. S K 2.3.17. √ f˜m. f8 {α} ∪ W )¤ ω = −1+2 3 i ∈ C. ½Â Q(ω) = {a + bω | a, b ∈ Q}. f˜m P 2.3 Ä–•þ˜m 119 √ Ý, =, ω = −1−2 3 i .) 1. y²: Q(ω) ´ Q ˜‡*•, … ω ∈ Q(ω). (ùp ω L«Eê ω 2. Uì~ 2.3.18 ¥ •ªò Q(ω) À• Q-•þ˜m. ¦ Q(ω) Š• Q-•þ˜m 3. òe ƒ|À• Q-•þ˜m Q(ω) ¥ f•þ|. ˜|Ä. •þ|. •±ez‡•þ|éÑ˜‡4Œ‚5Ã' (a) 21 , −3, 4. (b) 1, ω, ω 2 , ω 3 , ω 4 . √ (c) ω, ω, 3. S K 2.3.18. √ ω = −1+2 3i ∈ C. •ÄEÝ  1  A=  ω ω2  , ½Â R[A] := {f (A) | f ∈ R[X]} . y² R[A] 'uÝ \{Úê¦ ¤¢ê•þ •þ˜m, ¿¦Ñ§ ˜|ÄÚ‘ê. S K 2.3.19. V = Map([−π, π] , R), Uì~ 2.3.19 ¤ã •ªò V À•¢•þ˜m. òe |À• V ¥ •þ|. ž ä±eˆ‡•þ|´Ä‚5ƒ', ¿¦§‚ •. ¼ê 1. (cos x)2 , (sin x)2 . √ 2. sin x , sin( 2x). 3. 1, sin x , sin(2x), sin(3x), · · · , sin(nx). (ùp Š V ¥ ƒ.) 1 Œ±@•´ Š• 1 ~Š¼ê, l Œ± 4. 1, sin x , (sin x)2 , · · · , (sin x)n . 5. sin(ax), cos(bx) Ù¥ a, b •‰½ š"¢ê. S K 2.3.20. •Ä¢•þ˜m R[X]≤4 ¥ Xef8 U := {f ∈ R[X]≤4 | f (2) = f (5)} . y² U ´ R[X]≤4 f˜m, ¿éÑ U S K 2.3.21. m ∈ N∗ . b ´ K[X]≤m ˜|Ä. S K 2.3.22. ˜|Ä. f0 , . . . , fm ∈ K[X] …éuz‡ i ∈ [[0, m]], deg(fi ) = i. y² f0 , . . . , fm A ∈ Mn (K), CA := {B ∈ Mn (K) | AB = BA}. 1. y² CA ´ Mn (K)  1   2 2. A= ..  .  ˜‡f˜m.    . ¦ CA   ‘êÚ˜|Ä. n 3.  1  n = 3, A = 0 3 0 1 1  0  0. ¦ CA 2 ‘êÚ˜|Ä. 1 120 S K 2.3.23. M • K n×1 AX = 0 )˜m. f˜m. y²: •3Ý Ù •þ˜m9Ùf˜m A ∈ Mn (K) ¦ M uàg‚5•§| S K 2.3.24. V ´ K-•þ˜m. XJ M ⊆ V ´‡f˜m, … M 6= V , K¡ M ´ V ýf˜m. Sr y²: éu V ?¿k•õ‡ýf˜m U1 , · · · , Ur 7k i=1 Ui 6= V . (J«: α ∈ Ur \ (U1 ∪ · · · ∪ Ur−1 ), β ∈ U1 \ (U2 ∪ · · · ∪ Ur ). éz‡ i, •õ•k˜‡ ai ∈ K ¦ α + ai β ∈ Ui .) S K 2.3.25. 3¢•þ˜m V = Map([−π, π] , R) ¥•Ä•þ| 1, cos x, cos(2x), cos(3x) )¤ ˜m W . ¦ W ‘êÚ˜|Ä. S K 2.3.26. V ´l R 1. y² V ´Ã•‘ R ¤këY¼ê ¤ f ¢•þ˜m. . 2. éz‡ ê n, - Wn := {f ∈ V | f (x + n) = f (x) é¤k x ∈ R ¤á}. (=, Wn ´ V ¥± n •˜‡±Ï ¤k±Ï¼ê ¤ f8.) y²: z‡ Wn Ñ´ V 3. - W = S f˜m. n≥1 Wn , =, W •±þ¤k Wn ¿8. ž¯ W ´Ä´ V S K 2.3.27. é±eˆ‡œ¹, ¦•þ| α1 , α2 , α3 , α4 3 K 4 ¥)¤ f˜m ? •Ÿo ? f˜m ‘ê: 1. α1 = (2, 1, 3, 1), α2 = (1, 2, 0, 1), α3 = (−1, 1, −3, 0), α4 = (1, 1, 1, 1). 2. α1 = (2, 1, 3, −1), α2 = (−1, 1, −3, 1), α3 = (4, 5, 3, −1), α4 = (1, 5, −3, 1). S K 2.3.28. é±eˆ‡œ¹, ©O¦Ñ U ∩ W Ú U + W ˜|Ä: 1. U = span(α1 , α2 ), W = span(β1 , β2 ), Ù¥ α1 = (1, 2, 1, 0), α2 = (−1, 1, 1, 1), β1 = (2, −1, 0, 1), β2 = (1, −1, 3, 7) . 2. U = span(α1 , α2 ), W = span(β1 , β2 ), Ù¥ α1 = (1, 1, 0, 0), α2 = (1, 0, 1, 1), β1 = (0, 0, 1, 1), β2 = (0, 1, 1, 0) . S K 2.3.29. U = span(α1 , α2 , α3 ), W = span(β1 , β2 ), Ù¥ α1 = (1, 2, −1, −2), α2 = (3, 1, 1, 1), α3 = (−1, 0, 1, −1) , β1 = (2, 5, −6, −5), β2 = (−1, 2, −7, 3) . ¦ U ∩W Ú U +W ˜|Ä. S K 2.3.30. α1 , α2 , α3 , α4 Ñ´ K 4 ¥ •þ, K 4 ¥d•þ| α1 , α2 Ú α3 , α4 )¤ OP• V Ú W . ée z«œ¹, ©O ä K 4 = V ⊕ W ´Ä¤á. 1. α1 = (0, 1, 0, 1), α2 = (0, 0, 1, 0), α3 = (1, 0, 1, 0), α4 = (1, 1, 0, 0); 2. α1 = (−1, 1, 1, 0), α2 = (0, 1, −1, 1), α3 = (1, 0, 0, 0), α4 = (0, 0, 0, 1); 3. α1 = (1, 0, 0, 1), α2 = (0, 1, 1, 0), α3 = (1, 0, 1, 0), α4 = (0, 1, 0, 1). f˜m© 2.3 Ä–•þ˜m 121 S K 2.3.31. 3•þ˜m K 2n ¥, •Ä±eü‡f˜m: V = (a1 , a2 , · · · , a2n ) ∈ K 2n : ai = an+i , 1 ≤ i ≤ n , W = (a1 , a2 , · · · , a2n ) ∈ K 2n : ai = −an+i , 1 ≤ i ≤ n . y² K 2n = V ⊕ W . S K 2.3.32. V = Mn (K), UÝ ¤ f8, W ⊆ V ´d‡é¡Ý 1. y²: U, W Ñ´ V \{Úê¦ò V À• K-•þ˜m. ¤ f8. U ⊆ V ´dé¡Ý f˜m, ¿©O• U, W éÑ˜|Ä. 2. y²: é?¿ A ∈ V þk A + AT ∈ U, A − AT ∈ W . 3. y²: V = U ⊕ W . S K 2.3.33. U • Mn (K) ¥ þn ¤ f˜m, L • Mn (K) ¥ m. Á¯: Mn (K) = U + L ´Ä¤á ? Mn (K) = U ⊕ L ´Ä¤á ? S K 2.3.34. M, N, M1 , M2 þ••þ˜m V ¥ ²: V = M1 ⊕ M2 ⊕ N . S K 2.3.35. f˜m. b en ¤ f˜ V = M ⊕ N, M = M1 ⊕ M2 . y M1 ´àg‚5•§ x1 + x2 + · · · + xn = 0 )˜m, M2 ´àg‚5•§|   x1 − x2 = 0    x − x = 0 2 3   ·········     xn−1 − xn = 0 )˜m. y²: K n×1 = M1 ⊕ M2 . S K 2.3.36. A ∈ Mn (K) ´Œ_ , Ùc r 1|¤Ý B ∈ Mr×n (K), n − r 1|¤Ý C ∈ M(n−r)×n (K). M, N ©O•àg‚5•§| BX = 0 Ú CX = 0 )˜m. y²: K n×1 = M ⊕ N . S K 2.3.37. M, N ´•þ˜m V f˜m. L´M 3V ¥ ˜‡†ÚÖ˜m. 1. y²: e M ⊆ N , K N = (N ∩ M ) ⊕ (N ∩ L) = M ⊕ (N ∩ L). 2. eØb M ⊆ N , ´Ä N = (N ∩ M ) ⊕ (N ∩ L) ˜½¤á ? S K 2.3.38. U1 , · · · , Ur ´•þ˜m V Pi−1 i = 2, · · · , r þk Ui ∩ j=1 Uj = 0. f˜m. y²: U1 + · · · + Ur ´†Ú …= éu?¿ S K 2.3.39. U1 , · · · , Ur ´•þ˜m V •þ α ∈ V , k…=k˜«Lˆª f˜m. y²: U1 + · · · + Ur ´†Ú …= éu,‡ α = α1 + · · · + αr Ù¥z‡ αi ∈ Ui . 1 122 x S K 2.3.40. 3 M2 (K) ¥,¤k/X y 8ÜP• V2 . y²: V1 Ú V2 Ñ´ M2 (K) S K 2.3.41. 3Ý Ù •þ˜m9Ùf˜m ! ! −x a b Ý 8ÜP• V1 , ¤k/X Ý z −a c f˜m, ¿¦ dim V1 , dim V2 , dim(V1 + V2 ), dim(V1 ∩ V2 ). ˜m M2 (K) ¥éÑ˜|Ä A1 , A2 , A3 , A4 , ¦ éz‡ j, þk A2j = Aj . S K 2.3.42. 3õ‘ª ¤ •þ˜m K[X] ¥, ÷v f (−X) = f (X) ¤kõ‘ª f (X) 8ÜP• W . ÷v g(−X) = −g(X) ¤kõ‘ª g(X) ¤ 8ÜP• U . y²: W Ú U Ñ´ K[X] f˜m, ¿… K[X] = W ⊕ U . ¤ S K 2.3.43. •Ä R g ¼ê ¤ ¢•þ˜m V = Map(R, R). XJ¼ê f : R → R ÷ v f (x) = f (−x) é?¿ x ∈ R ¤á, K¡ f •ó¼ê (even function). XJ¼ê g : R → R ÷v g(−x) = −g(x) é¤k x ∈ R ¤á, K¡ g ´Û¼ê (odd function). ± V1 Ú V2 ©OL« V ¥ó¼ êÚÛ¼ê ¤ f8. y² V1 , V2 Ñ´ V f˜m, … V = V1 ⊕ V2 . S K 2.3.44. )¤ K[X]≤m . m ∈ N∗ . b õ‘ª f0 , . . . , fm ∈ K[X]≤m Ñ3 1 ? S K 2.3.45. •Ä¢•þ˜m R[X]≤4 ¥ f8 Z 1 U := Š• 0. y² f0 , . . . , fm ØU f ∈ R[X]≤4 f (x)dx = 0 … f (−1) = 0 . 0 1. y² U ´ R[X]≤4 f˜m. 2. ¦ U 3 R[X]≤4 ¥ ˜‡†ÚÖ. S K 2.3.46. α1 , α2 ´•þ˜m V1 ˜|Ä, β1 , β2 ´•þ˜m V2 ˜|Ä. •þ| (α1 + α2 , β1 + β2 ) , (α1 − α2 , β1 + β2 ) , (α1 + α2 , β1 − β2 ) , (α1 − α2 , β1 − β2 ) ´Ä´ V1 × V2 ˜|Ä ? e´, ž‰Ñy²; eÄ, ž)º•ŸoØ´. S K 2.3.47. f ∈ K[X] ´ 5 gõ‘ª. - U = {gf | g ∈ K[X]}. y²: U ´ K[X] f˜m, ¿…û˜m K[X]/U ´k•‘ . éÑ K[X]/U ˜|Ä. 1nÙ ‚5N 9ÙÝ L« ·‚• , 8Üƒm éXÌ‡‚§‚ƒm N 5• . •þ˜mØ==´8Ü, „N‘k‚ 5( (•\{Úê¦$ŽÚ§‚ ˆ«5Ÿ). Ïd, éu•þ˜m, Ï~<‚•'%@ U ± §‚‚5( N ——‚5N . Ù¥·‚o´ K • C ˜‡f•. 3.1 3.1.1 ‚5N Ä Vg “ê½Â†AÛ†* ½ Â 3.1.1. V Ú W Ñ´ K-•þ˜m. ˜‡l V W mapping) ´•÷v±eü‡^‡ ˜‡N f : V → W : K-‚5N (K-linear map, K-linear 1. f Ú•þ \{$Ž †. =, éu?¿ u, v ∈ V , 3 W ¥k ª f (u + v) = f (u) + f (v) ¤á. 2. f Ú•þ ¤á. ê¦$Ž †. =, éu?¿ u ∈ V Ú?¿ λ ∈ K, 3 W ¥k ª f (λu) = λf (u) Ï~3Ø–uÚåØ) œ¹e, ·‚Œ±ò“K-‚5”{¡•“‚5”. ‚5N kž•¤•Ó (homomorphism). l V W ¤k K-‚5N ¤ 8ÜP• HomK (V, W ), ½ö{P• Hom(V, W ). éu W = V œ¹, HomK (V, V ) Ï~UP• EndK (V ) ½ ö (•{ü/) End(V ). ù‡8Ü¥ ƒ•¡• V gÓ (endomorphism) ½ö V g ‚ 5C† (linear transformation) ½ö‚5Žf (linear operator). l“ê Ý`, \{Úê¦Ò´•þ˜mØ 8Ü á5ƒ •˜ ( . Ïd, éu• þ˜m ‚5N , ·‚‡¦¿…•‡¦TN ±\{Úê¦ü«$Ž, ùATń¤A ‰ {. éu˜ d\{Úê¦ 5Ÿû½ ¯¢, ½ö˜ û) 5 Vg, Ï~·‚Œ± y§‚3 ‚5N ŠêäkûÐ éA'X. ·K 3.1.2: f : V → W ´ K-•þ˜m†ƒm 1. f (0) = 0. =, f ò V ¥ "•þ (\{ð ‚5N )N . K: •W ¥ "•þ. 2. éu?¿ a, b ∈ K Ú?¿ u, v ∈ V , f (au + bv) = af (u) + bf (v). =, f ò‚5|ÜN |Xê ‚5|Ü. 3. e U ´ V f˜m, K f (U ) ´ W •Ó˜ f˜m. y². |^†\{ †5Œ• 0 + f (0) = f (0) = f (0 + 0) = f (0) + f (0). ü>ž f (0) = 0. 1 2 ‡Ú1 3 ‡Øä y²3‰Öö. 123 ˜‘ f (0) = 1nÙ ‚5N 124 kw•{ü ‚5N 9ÙÝ L« ~f. ~ 3.1.3. b K = R. e V = W = R, f : R → R •‚5N , a = f (1) ∈ R. @o½Â 3.1.2 ¥ 5Ÿ 2 L²: é?Û x ∈ R, 7, f (x) = ax. ‡L5, ? ~ê a ∈ R, ÏL½Â x 7→ ax Œ± R g ˜‡ R-‚5N . 2•Ä V = W = R2 œ¹. dž e1 = (1, 0), e2 = (0, 1) ´ R2 ˜|Ä. e f : R2 → R2 ´‚5 N , f (e1 ) = (a, c), f (e1 ) = (b, d), Ké?¿ (x, y) ∈ R2 , Ï• (x, y) = xe1 + ye2 , d·K 3.1.2 (2) Œ• (3.1.3.1) f (x, y) = (ax + by , cx + dy) . ‡L5, ?¿‰½~ê a, b, c, d ∈ R, ± (3.1.3.1) ªŠ•½ÂŒ± LOŽ yŒ• f •‚5N . ~ 3.1.4. ·‚5Þ~`²: ½Â 3.1.1 ¥ ü‡^‡´ƒpÕá (1) •Ä¢•þ˜m V = R2 Ú W = R. ½ÂN  0 ex=0 f (x, y) = y e x 6= 0 R2 g ˜‡N f. Ï . K f (λu) = λf (u) é?¿ λ ∈ R Ú?¿ u ∈ R2 ¤á. ´ù‡N f Ø´‚5N (2) •ÄE•þ˜m V = W = C. KE ÝN f : z 7→ z ÷v . é?Û u, v ∈ V = C , þk f (u + v) = f (u) + f (v) . ´ù‡N f Ø´ C-‚5N . (3.1.5) ég,/, ·‚½Â ‚5N A 3AÛþLyÑ“‚5” A . •Ä ‡¦´˜^† ‚ N ”A „´†‚ (½öòz•˜‡:). ±¢•þ˜m R2 •~. ·‚•#Nò R2 ƒÀ•²¡þ: ‹I. f : R2 → R2 •²¡ 2 R g ˜‡N . XJéu²¡S?¿n‡ ‚ : P, Q, S, ± f Š^ƒ ¤ n‡: f (P ), f (Q), f (S) E, ‚, @o·‚` f ´˜‡† (collineation). (ùp, XJn‡:¥k˜ ´ -Ü , KgÄ@•§‚ ‚.) XJ f : R2 → R2 ´‚5N , @o f ˜½´‡† . ¯¢þ, XJ P, Q, S ´ ‚ n‡:, K −→ −−→ •3 λ ∈ R ¦ P S = λP Q. P P 0 = f (P ), Q0 = f (Q) 9 S 0 = f (S). Ï• f ò :N • :… f ±\{, −− → −−→ −−→ −→ −−→ −→ P 0 S 0 = OS 0 − OP 0 = f (OS) − f (OP ) = f (P S) −−−→ −−→ Ón, P 0 Q0 = f (P Q). du f • ±ê¦, ¤±, −− → −−−→ −→ −−→ −−→ P 0 S 0 = f (P S) = f (λP Q) = λf (P Q) = λP 0 Q0 . ùÒ`² P 0 , Q0 , S 0 n: ‚. ‡L5, ²¡ R2 g ˜‡† ´Ä˜½´‚5N Q ? ‰Y´Ä½ ! Äk, XJ ½˜‡•þ, ~X α = (1, 0), ò¤k:÷X α Œ Ú••‰²£, ù ²£C †w,´‡† . §Ø´‚5N , Ï•§vkr :N • : (¢Sþ²£C†vk?ÛØ Ä:). 3.1 ‚5N Ä Vg Ùg, =¦´ ½ ½ †‚þ. ~X- 125 : † •Ø˜½´‚5N , Ï•·‚Œ±r1r¤k:N ˜^ f (x, y) = (x3 , 0) . ddŒ„, ·‚c¡é‚5N ‰Ñ “ê½Â, ‡'AÛ†*þÏL† Ú ± :ØÄù AÛ5Ÿ‰ ½Âäk•r •›5. Öö8 ¬²w/N¬ , ½Â 3.1.1 ù °(½ÂÑ ‚5 N , äk•\ õ nØNX, 3¢SA^ž••\B|Úk . ù•NyÑ“êÆ °(5ÚÄ –5¤‘5 êÆ¿Â. 5P 3.1.6. ¦+·‚3 (3.1.5) ˜ãff`L: ± :ØÄ † Ø˜½´‚5N . XJ·‚ ?˜Ú‡¦TN ´‡V , @o§˜½´‡‚5N . ù¢Sþ´˜‡š~ • ØN´ ½n, Ï~ ¡• (• ) AÛÄ ½n (fundamental theorem of (affine) geometry). O(5`, T½nŒ± ù •ã: XJ n ≥ 2, f : Rn → Rn ´ò :N • : † , ¿… f ´V , @o f ˜½´‡‚5N . ±þ½n y²Œ±3 [5, §2.6] é , 3Ø© [7] ¥„Œ±w ˜‡\rí2‡. (ØL·‚¿Ø ïÆÐÆö Öùü‡©z.) y3·‚ÏLAÛ†*†“êOŽ(Ü ˜ ‚5C†. •ª5?Ø˜ AÏ ‚5N , AO´²¡ g ~ 3.1.7. ?¿ ½~ê λ ∈ R. ·‚UXe•ª½Â˜‡N f : R2 → R2 : éu²¡ R2 S ?¿ −−→ −−→ ˜: P , f ò P N •: P 0 , ¦ OP 0 = λOP . lAÛ†*þ, ÖöØJ yù N f ´˜ ‡‚5N . 3ù‡N Šê, ²¡S?Û˜^‚ã¬C¤†ƒ••²1 •Ý• •Ý λ ˜ ^‚ã. ·‚rù‡N ¡•˜‡˜ (scaling), ~ê λ ¡•§ ˜ Ïf (scaling factor). XJ^ “êLˆª5 Ñ f , @o f (x, y) = (λx , λy) . ÏLù‡Lˆª, ,Œ±±“êOŽy² f (´ÎÜ½Â 3.1.1 ‚5N . •˜„/, éu?¿•þ˜m V Ú?¿~ê λ, N v 7→ λv ´ V g ‚5C†, ¡•˜ Ïf• λ ˜ . ˜ C†• ¡•ê¦C† (scalar multiplication transformation). ~ 3.1.8. ½²¡S ü‡: A, B. ·‚b O, A, B n:Ø ‚. ·‚UXe•ª½Â˜‡N 2 f : R → R2 : éu²¡ R2 S ?¿˜: P , L P :‰²1u OA †‚, - f (P ) •T†‚† †‚ OB : P 0 . ·‚E,3‰Öö y, lAÛ†*þŒ±wÑù N f ´˜‡‚5N . ù‡N N JŒ±Ž–•˜«ÝK. äN5`, b k²1u†‚ OA ´••ŠX•þ −→ OA ‡•• 1å3²¡Sì . ‚ã OP Ï•ñ ùå1‚ 3†‚ OB þ/¤˜ãÒK‚ã, ùãÒK‚ãÒ´ OP 0 , Ù¥ P 0 = f (P ). ·‚r±þN f ¡•²1u†‚ OA!Ý•†‚ OB ÝK (projection onto the line OB parallel to the line OA). −→ −−→ • ‰Ñ f “ê£ã, ·‚P u = OA, w = OB. 2P U = span(u), W = span(w). dž O, A, B n:Ø ‚ ^‡Ò´` •þ| u, w ‚5Ã' (ë„ (2.1.12)). Ï• R2 ‘ê´ 2, ù• ¿›X R2 Œ± ¤†Ú©) R2 = U ⊕ W /ª. éu?¿ v ∈ R2 , Œ±±•˜ •ªòÙ ¤ v = α + β, Ù¥ α ∈ U, β ∈ W . K f ò v N • β. Þ‡äN~f5`, XJ u = (1, 1), w = (−2, 1), K f ²wLˆª• 2x − 2y −x + y , f (x, y) = 3 3 (žÖö yù‡(Ø). 1nÙ ‚5N 126 9ÙÝ L« •˜„/, b •þ˜m V k˜‡†Ú©) V = U ⊕ W , Ù¥ U, W ´ V ¥ f˜m. @oŒ ±UXe•ª½Â˜‡ V g ‚5C† f : é?¿ v ∈ V , ±•˜ •ªò v ¤ v = u + w /ª, ½Â f (v) = w. Šâ“†Ú©)”Ú“‚5N ” Ä–½Â, Œ± yù N f ´‚5 . ·‚¡ù‡ f • V 'u†Ú©)ª V = U ⊕ W • W ÝK (projection onto W with respect to the decomposition V = U ⊕ W ), ½ö²1u U !Ý• W ÝK (projection onto W parallel to U ). 5¿: ù‡½ÂÓž•6u†Ú©)ª¥ U Ú W ü‡f˜m. éuþ¡£ã ÝKN f : V → V , Ï•éu?¿ v ∈ V , f (v) o´áuf˜m W , ¤±·‚ •Œ±@•UÓ •ªŒ± lV W ˜‡‚5N f 0 : V → W . ¦+ f 0 † f ŸÃ , ´“êÆ¥Ï~„´ò öÀ•ØÓ N . ¯¢þ, f 0 o´÷ , f KØ, (ž¯Öög•• Ÿo). ØL, <‚Ï~#N·‚E^âŠ, •r f 0 ¡•²1u U!Ý• W ÝK. −−→ ~ 3.1.9. α ∈ R. éu²¡S ?¿˜: P , ±lÝ• α Ýò•þ OP 7X : O _ž −−→ ^=, ¤ •þ• OP 0 . (e α < 0, “_ž ^= Ý α ”Œ±n)•“^ž ÀJ Ý |α|”.) ÏL P 7→ P 0 ù«•ª½ÂÑ˜‡N f : R2 → R2 . ÖöŒ±lAÛ Ý yù‡N ´‚5N . ·‚y3í f “êLˆª. •Ò´`, e: P ‹I• (x, y), ·‚F"¦Ñ: P 0 ‹I 0 0 (x , y ) XÛÏL x, y 5Lˆ. −−→ −−→ ·‚ •{´k± OP ••Š•,‡# † ‹IX ˜‡‹I¶••, ±_ž ^= OP −−→ −−→ 90◦ •••# † ‹IX,˜‡‹I¶••. äN5`, P OQ ••þ OP 7 :^=∗ 90◦ •þ. K −−→0 −−→ −−→ OP = (cos α)OP + (sin α)OQ . (3.1.9.1) (ò5Ö¿˜‡ã« ???) −−→ −−→ •þ OP ‹IÚ: P ‹IƒÓ. ¤±•‡¦Ñ OQ ‹I, , “\ (3.1.9.1) ª=Œ¦Ñ −−→ −−→ : P 0 (½ö`•þ OP 0 ) ‹I. {ü ã/©ÛŒ±• OQ = (−y, x). ¤±, þãN f ‹ ILˆª• f (x, y) = (x cos α − y sin α , x sin α + y cos α) . (3.1.9.2) ·‚ò±þN 3.1.2 f ¡•^= •α ^= (C†) (rotation by the angle α). •õ~f c¡·‚?Ø ˜ kAÛ¿Â ‚5N . y3·‚?Ø•˜„ ˜ ~f. ~ 3.1.10. éu?¿•þ˜m V , òz‡ v ∈ V N •"•þŒ± ˜‡N . ù‡N w,´ V g ‚5C†, ¡•"C†!"N (zero transformation, zero map). ·‚± 0 5L«"N . (y3·‚b½Öö®²é (‚5) “êÆ¥ˆ«E^PÒ 0 y–i˜„. .) •˜„˜ , éuü‡•þ˜m V, W , ò V ¥¤k ƒN • W "•þ, •Œ± ˜‡ "N . ù ,•´‡‚5N . AO/, éu?¿•þ˜m V , "N ´l V "•þ˜m 0 •˜‚5N ( ,, ¢Sþ• ´V "•þ˜m •Ñ ), kž·‚¬± V → 0 5L«ù‡N . aq/, "N ´l 0 V •˜‚5N , ·‚~rù‡N P• 0 → V . ~ 3.1.11. éu?¿•þ˜m V , l V g ð N (•¡ð IdV : v 7→ v w,´‡‚5N . ± 3Ø–uÚåÜÂž, ·‚•²~rð C†{üP• I. ∗ XJvkAO `², ·‚` “^=”o´•_ž •• ^=. C† identity transformation) 3.1 ‚5N ~ 3.1.12. Ä Vg 127 A ∈ Mm×n (K). N LA : Mn×p (K) −→ Mm×p (K) ; X 7−→ AX RA : Ms×m (K) −→ Ms×n (K) ; Y 7−→ Y A Ú þ•‚5N . AO/, N    x1 x1  .   .   .  7−→ A  .   .   .  xn xn  fA : K n×1 −→ K m×1 ; (3.1.12.1) Ú gA : K 1×m −→ K 1×n ; (3.1.12.2) þ•‚5N (y1 , . . . , ym ) 7−→ (y1 , . . . , ym )A . ~ 3.1.13. ò C À• R þ R-‚5N . N •þ˜m (ë„~ 2.3.18). E C −→ R ; •´ R-‚5N ÝN z = x + yi 7→ z = x − yi Ñ´ z = x + yi 7−→ 2x = z + z̄ . ~ 3.1.14. L • K ˜‡*•. K V = L Œ±À• K •þ˜m (ë„~ 2.3.18). ?¿ ÏLÏ~ êƒm ¦{½ÂN L −→ L ; ù´˜‡ K-‚5N ½ α ∈ L, x 7−→ αx . . ~ 3.1.15. •Ä K þ± X •™½ õ‘ª˜m K[X]. éuz‡ f = 0 /ª ê (formal derivative) f •Xeõ‘ª f 0 := X P i≥0 ai X i ∈ K[X], ½ÂÙ iai X i−1 = a1 + 2a2 X + 3a3 X 2 + · · · . i≥1 N´ y: N ~ 3.1.16. f 7→ f 0 ´ K[X] g ‚5C†. g ∈ K[X]. ÏLõ‘ª¦{½ÂN K[X] −→ K[X] ; ù´˜‡ K-‚5N f 7−→ f g . . ~ 3.1.17. A •š˜8Ü, a ∈ A, V = Map(A, K) (ë„~ 2.3.19). ÏLòz‡ f ∈ Map(A, K) 3: a ? ŠŒ± ˜‡N eva : V = Map(A, K) −→ K ; ÖöA Œ±g1 y: ±þN ´‡‚5N . f 7−→ f (a) . 1nÙ ‚5N 128 ~ 3.1.18. N C[0, 1] •4«m [0, 1] þ½Â ¤k¢ŠëY¼ê ¤ 9ÙÝ L« •þ˜m (ë„~ 2.3.20). Z 1 C[0, 1] 7−→ R ; f 7−→ f (x)dx 0 ´‡‚5N . ~ 3.1.19. U •¼ê| I •R¥ m«m, C(I) • I þ ¤k¢ŠŒ ¼ê ¤ n ∈ N∗ . - •þ˜m. 1 , sin(x) , sin(2x) , · · · , sin(nx) 3 C(I) ¥)¤ f˜m, V •¼ê| 1 , cos(x) , cos(2x) , · · · , cos(nx) 3 C(I) ¥)¤ f˜m. ÏL¼ê¦ N U −→ V ; ´‡‚5N f 7−→ f 0 . ~ 3.1.20. •Äl 0 m©?Ò…z˜‘5g K ¤kê ¤ K-•þ˜m K N (ë„~ 2.3.21). N f : K N −→ K N ; (a0 , a1 , . . . , ) 7−→ (a1 , a2 , · · · ) Ú g : K N −→ K N ; Ñ´‚5N (a0 , a1 , . . . , ) 7−→ (0, a0 , a1 , a2 , · · · ) . ~ 3.1.21. V • K-•þ˜m, U ´ V V /U ; α 7→ α ´‚5N (ë„ (2.3.27.2)). ‚5N o(3e¡ ·K 3.1.22: Ú‚5N ·K¥: f˜m. K (2.3.27.1) ª¥‰Ñ ƒmkžŒ±ÏL· öŠû)Ñ# ‚5N ûN π:V →V = . A«~„ öŠ•ª U, V, W • K-•þ˜m. 1. é u ? ¿ ‚ 5 N f : U → V Ú g : U → V , Ï L u ∈ U 7−→ f (u) + g(u) f + g : U → V •´‚5N . =, f, g ∈ HomK (U, V ) %¹ f + g ∈ HomK (U, V ). N 2. éu?¿‚5N f : U → V Ú~ê λ ∈ K, ÏL u ∈ U 7−→ λf (u) N λf : U → V • ´‚5N . =, XJ f ∈ HomK (U, V ), @oéu?Û~ê λ ∈ K, ¼ê λf áu HomK (U, V ). 3~ 3.1.10 ¥·‚JL, HomK (U, V ) „k‡"N . ¤±nþŒ•, HomK (U, V ) Uìþã\ {Úê¦Œ±À• K-•þ˜m Map(U, V ) f˜m. 3. e g : U → V Ú h : V → W Ñ´‚5N , KEÜN g h h◦g : U − →V − → W •´‚5N . y². 3ŠSK. (½3‘,Ç‘žù).) (3.1.23) 8 ·‚²~rü‡N g Ú h EÜN h ◦ g ( TEÜk¿Âž) {üP• hg. Š•˜‡•-‡ AÏœ¹, ·‚²~¬•Ä˜‡•þ˜m V g ‚5C† A : V → V . n ˜•¡, éu?¿ ê n, Œ±ò A n)•˜‡EÜN A1 A2 · · · An , Ù¥ A1 = A2 = · · · = An = A ; éu n = 0, 2 ½ A 0 u V þ ð C† IdV . ,˜•¡, EndK (V ) = HomK (V, V ) ´ 3.1 ‚5N Ä Vg 129 ‡•þ˜m (·K 3.1.22 (2)). éu?¿~ê a0 , a1 , · · · an ∈ K, Ï• A 0 , A = A 1 , A 2 , . . . , A n Ñ Œ±À••þ˜m EndK (V ) ¥ ƒ, ¤±‚5|Ü a0 A 0 + a1 A + · · · + an A n ´k¿Â , §“L EndK (V ) ¥ ˜‡ ƒ. XJ·‚•Äõ‘ª f (X) = a0 + a1 X + · · · + an X n ∈ K[X] @oŒ±½Â f (A ) := a0 Id +a1 A + · · · + an A n . ùÒ´`, ˜‡‚5C†Œ±“\ ˜‡õ‘ª¥, ¦ õ‘ª Š¡•˜‡# ‚5C†. 8 Öö¬²x, ùo˜‡ØJn) ¯¢éun)‚5N NõS%5Ÿäk9Ù-‡ ¿Â, § •´õ‘ªnØ3‚5“ê¥U užãŒŠ^ nØÄ:. 3.1.3 Ø†”!‚5˜m Ó éu8Üm N , ·‚Œ±?Ø´Ä´ü !÷ ½öV . •þ˜mƒm ‚5N , •´8Üm N , ¤±·‚Œ±?ØÓ ¯K. ùž·‚¬uy, Ï•k “‚5” ‡¦, ü ½ö÷ ^‡~~k•\•B O•{. ·‚kÚ\7‡ ½Â. ½ Â 3.1.24. f : V → W ••þ˜mƒm ‚5N . 1. ·‚ò V 3 f Šê ”8 f (V ) P• Im(f ), {¡• f ” (image). ù´ W ˜‡f˜m (·K 3.1.2 (3)), k Ö (~X [3]) •¡ƒ• f Š• (range), Ïd•òÙP• range(f ). 2. V ¥ f8 {v ∈ V | f (v) = 0}, ¡•‚5N f Ø (kernel), P• Ker(f ). ÖöŒ± y, Ker(f ) ´ V ˜‡f˜m. k Ö (~X [3]) •¡ƒ• f "z˜m (null space), Ïd•ò ÙP• null(f ). ·K 3.1.25: f : V → W ••þ˜mƒm ‚5N . 1. f ´÷ dim W . …= Im(f ) = W . XJ W ´k•‘•þ˜m, K f ´÷ 2. f ´ü …= Ker(f ) = 0, …= …= dim Im(f ) = dim Ker(f ) = 0. y². (1) c˜‡ØäÃš•´=ã´“÷ ”ù‡Vg ½Â. Ï• Im(f ) ´ W f˜m, ¤±Šâ íØ 2.3.14 (2) Œ•: Im(f ) = W du dim Im(f ) = dim W . (2) ˜‡“ …= ”´Ï••k"•þ˜m ‘êâ¬ u 0. c˜‡äó y²3ŠSK. (½3‘,Ç‘žù).) ~ 3.1.26. ‰½Ý A ∈ Mm×n (K), •Ä (3.1.12.1) ª‰Ñ fA : K n×1 −→ K m×1 ; ‚5N X 7−→ fA (X) := AX . ŠâƒA ½ÂŒ• Ker(fA ) = N (A). ,˜•¡, éu?¿ X = (x1 , . . . , xn )T ∈ K n×1 , Ý ¦È AX ¢SþÒ´ò A ± x1 , . . . , xn •Xê‰‚5|Ü. ¤±, Im(fA ) = C (A). Ïd, ±þ‚5N fA ØÒ´Ý A "z˜m, fA ”Ò´ A ˜m. AO/, Ý A • u dim Im(fA ) (Ï•·‚• rank(A) = dim C (A)). 1nÙ ‚5N 130 ·K 3.1.27: f : V → W ••þ˜mƒm ‚5N 9ÙÝ L« . u1 , . . . , um • V ¥•þ|, U = span(u1 , . . . , um ). K f (U ) = span(f (u1 ) , . . . , f (um )). 1. 2. b dim V < +∞. Ke (a) f •÷ •ã d: . (b) éu V ?¿)¤| v1 , . . . , vm , •þ| f (v1 ), . . . , f (vm ) ´ W )¤|. (c) •3 V )¤| v1 , . . . , vm , ¦ )¤|. •þ| f (v1 ), . . . , f (vm ) ´ W y². (1) Uì“Ü¤f˜m” ½Â, 2Šâ“‚5N Ú‚5|ÜöŠŒ (2) (a)⇒(b). |^ (1) (Ø. (b)⇒(c). Ï• V ´k•‘ , ¤k V Äo•3. (b) %¹ (c). (c)⇒(a). E^ (1) (Ø=Œ. ·K 3.1.28: f : V → W ••þ˜mƒm ‚5N . †^S” 5Ÿ=Œ y. u1 , . . . , um • V ¥•þ|. 1. e u1 , . . . , um ‚5ƒ', K•þ| f (u1 ) , . . . , f (um ) •‚5ƒ'. 2. e f •ü , K•þ| u1 , . . . , um ‚5ƒ' (½Ã') ƒ' (½Ã'). 3. e •ã d: (a) f •ü . (b) éu V …= •þ| f (u1 ) , . . . , f (um ) ‚5 ?¿‚5Ã'| v1 , . . . , vr , •þ| f (v1 ), . . . , f (vr ) ´ W ‚5Ã'|. y². (1) Š Ï´‚5N ±‚5|Ü, = f (a1 u1 + · · · + am un ) = a1 f (u1 ) + · · · + am f (um ) é ?¿~ê ai ∈ K Ñ¤á. (2) d u f ´ ü , ‚ 5 | Ü a1 u1 + · · · + am um • ", …= f (a1 u1 + · · · + am un ) = a1 f (u1 ) + · · · + am f (um ) = 0. (3) (a)⇒(b). d (2) (Ø=•. (b)⇒(a). e f šü , K Ker(f ) 6= 0 (·K 3.1.25 (2)). •3 v ∈ V, v 6= 0 ¦ f (v) = 0. dž v1 = v ¤˜‡‚5Ã' •þ|, f (v1 ) = 0 (Š•=k˜‡•þ •þ|) ´‚5ƒ' . ½ Â 3.1.29. V, W ••þ˜m. 1. XJ f : V → W ´˜‡V (ù du`, Š•8Üm N f ´Œ_N ), … f ´‚5 N , @o·‚` f ´•þ˜m V Ú W ƒm ˜‡Ó N , ½{¡Ó (isomorphism). lV Ó g Ó N ¡• V (‚5) gÓ ((linear) automorphism). Ï~ò V ¤ 8ÜP• Aut(V ), ½ö3I‡•²• K ž, P• AutK (V ). ¤kg 2. XJ•3 V Ú W ƒm ˜‡Ó N , ·‚`•þ˜m V Ú W ´Ó (isomorphic), ½ö {¡ V Ú W Ó . (ÖöI‡5¿: ¥©Lˆ¥“Ó ”˜cQŒ±Š•¶cqŒ±Š•/N c, cöéA =©´ isomorphism, öéA K´ isomorphic.) Ï~·‚^ V ∼ = W 5L« V † W Ó . ln) “Ó ” ½Âm©, ·‚F"ÖöATÅìN¬ ´d\{Úê¦ (±9‚5|Ü) û½ 5Ÿ, Ñ¬ Ó N ù ±! ¯¢: éu•þ˜m ó, … 3.1 ‚5N Ä Vg g • K 3.1. f : V → W ´•þ˜mƒm 131 ‚5N . y²: 1. e f ´÷ … V ´k•‘ , K W •´k•‘ , … dim V ≥ dim W . 2. e f ´ü … W ´k•‘ , K V •´k•‘ , … dim V ≤ dim W . 3. e f ´Ó , K§ g • K 3.2. ‚5N ? f −1 : W → V •´‚5N f : V → W ´•þ˜mƒm éuk•‘ ~ 3.1.30. _N ‚5N , ¿…´ü •þ˜m, ÏLÓ Œ±ò§‚‰š~{ü V • n ‘ K-•þ˜m. v1 , . . . , v n ´ V f = fv1 ,...,vn : K n −→ V ; …´Ó , N . . ´Ä f ?Û†_Ñ˜½´ ©a. ˜|Ä. KN (a1 , . . . , an ) 7−→ a1 v1 + · · · + an vn . ´ ˜ ‡ Ó . (ù p, · ‚ ¦ ^ fv1 ,...,vn ù « ‘ k e I P Ò ž ´ • r N T N ½Â•6u v1 , . . . , vn ù|Ä À .) ¯¢þ, ±þÓ •ØL3-ã˜‡{ü -‡ ¯¢: e v1 , · · · , vn • V ˜|Ä, @o V ¥?˜•þ v Œ±•˜/ ¤ v1 , · · · , vn ‚5|Ü ! Šâ±þ~f, •‡À ÄØÓž, Ó À n ‘˜m V ¥ N •ØÓ. g ˜|Ä, ÒŒ± g • K 3.3. Á Ñ¢•þ˜m R2 n‡ØÓ gÓ ·K 3.1.31: V, W •k•‘•þ˜m, f : V → W •‚5N ˜‡l K n V Ó . w,, . . Ke ^‡ 1. f ´Ó N 2. f ´ü , ¿… dim V = dim W . 3. f ´÷ , ¿… dim V = dim W . 4. éu V ?¿˜|Ä v1 , . . . , vn , •þ| f (v1 ), . . . , f (vn ) ´ W ˜|Ä. 5. •3 V ˜|Ä v1 , . . . , vn , ¦ ˜|Ä. d: . •þ| f (v1 ), . . . , f (vn ) ´ W y². 3‰ÖööS. ¿… e5·‚•ã˜^-‡(Ø, §Œ±lnØþ• yù« E Ün5. ·‚XÛ E÷vA½‡¦ ‚5N , ·K 3.1.32: V, W • K-•þ˜m. b v1 , . . . , vn • V ˜|Ä (l dim V = n < +∞). Kéu W ¥ ?¿•þ| w1 , . . . , wn , •3•˜ ‚5N f : V → W ÷v: é¤k i ∈ [[1, n]] þk f (vi ) = wi . y². k5y²÷vI¦ ‚5N f ´•3 . éu?¿ v ∈ V , ·‚Œ±UXe•ª½Â f (v): •þ v Œ±•˜/ ‚5|Ü (~ 3.1.30), {• (3.1.32.1) v = a1 v1 + · · · + an vn . ¤Ä v1 , . . . , vn ƒ¥•þ 1nÙ ‚5N 132 Ïd·‚Œ±½Â f (v) = a1 w1 + · · · + an wn . Šâù˜½Â, v a1 = 1, a2 = · · · = an = 0. 9ÙÝ v = v1 ž, (3.1.32.1) ª¥ L« Xê÷ f (v1 ) = 1 · w1 + 0 · w2 + · · · + 0 · wn = w1 . Ón, f (vi ) = wi éz‡ i ∈ [[1, n]] Ñ¤á. d , e u ∈ V ´,˜•þ, §L«• v1 , . . . , vn u + v L«• v1 , . . . , vn ƒ‚5|Ü •ª• ‚5|Üž ¤ u = b1 v1 + · · · + bn vn , @o u + v = (b1 + a1 )v1 + · · · + (bn + an )vn . Ïd, U½Â f (u + v) = (b1 + a1 )w1 + · · · + (bn + an )wn = (b1 w1 + · · · + bn wn ) + (a1 w1 + · · · + an wn ) = f (u) + f (v) . ù Ò y þ¡ E N f †\{Œ †. ÖöŒ±g1 y, f ÷v‚5N ½Â¥Ù¦ 5Ÿ. Ïd f ´÷v¤k¤I^‡ ‚5N . e5y²·K¥•˜5 Øä. •d, b g : V → W ´˜‡‚5N , …éz‡ i ∈ [[1, n]] þk g(vi ) = wi . ·‚‡y² g Úþ¡½Â N f ƒ . •d, ? V ¥•þ v. b v L«• v1 , · · · , vn ƒ‚5|ÜžŒ± ¤ (3.1.32.1) ª /ª. K, Uìþ¡ ½Â f (v) = a1 w1 + · · · + an wn . éu g, ·‚|^‚5N 5ŸŒ±• g(v) = g(a1 v1 + · · · + an vn ) = a1 g(v1 ) + · · · + an g(vn ) . Ï•k g(vi ) = wi ‡¦, ¤±þªm> 5 y . ¤± f = g. ·Ky.. 3.1.4 * ‚5N Ä !•ÀùSN, ± 3.1.5 * ½n ·žÖ¿. Ü !•ÀùSN, ± 3.1.6 u a1 w1 + · · · + an wn , = g(v) = f (v). ù´é?¿ v ∈ V ·žÖ¿. SK S K 3.1.1. f : V → W ´ K-•þ˜m V Ú W ƒm éu?¿ a, b ∈ K Ú?¿ u, v ∈ V þk ˜‡N . y²: f ´‚5N f (au + bv) = af (u) + bf (v) . S K 3.1.2. V = K[X]. é?¿ f (X) ∈ V , - A (f (X)) = f (X + 1) − f (X). 1. ¦~Šõ‘ª 2 3N A Šê ”. …= 3.1 ‚5N Ä Vg 2. y²: A ´ V g 3. A ´Ä´ü 133 ‚5C†. ? •Ÿo ? S K 3.1.3. V ´ S K 2.3.3 ¥ ½ Â ¢ • þ ˜ m (Ù \ { Ú ê ¦ $ Ž X T S K ¥ ½ Â). 2 W = R , UìÏ~ •þ\{Úê¦ò W À•¢•þ˜m. Ñ V W ˜‡Ó N . √ √ S K 3.1.4. •Ä Q *• K = Q(i) = {a + bi | a, b ∈ Q} Ú K 0 = Q( 2) = {a + b 2 | a, b ∈ Q}, (Ï LÏ~ê \{Ú¦{) ò§‚ÑÀ• Q-•þ˜m. Ñl K K 0 ˜‡Ó N . S K 3.1.5. A, B ∈ Mm×n (K) ÷v rank(A) = rank(B). y²: •3 K-•þ˜mƒm Ó ∼ f : N (A) − → N (B), §Œ± ¤ f (X) = P X /ª, Ù¥ P ´ Mn (K) ¥,‡Œ_Ý . N S K 3.1.6. V ´ 1 ‘ K-• þ ˜ m. y ² é u ? ¿ f ∈ EndK (V ), • 3 ~ ê λ ∈ K ¦ f (v) = λv é?¿ v ∈ V ¤á. S K 3.1.7. A ´ n ‘ K-•þ˜m V þ ‚5C†. y²e •ã d: 1. A ´Œ_C†. 2. é?¿š"•þ α ∈ V , A α 6= 0. 3. é V ?¿˜|Ä ε1 , · · · , εn , •þ| A ε1 , · · · , A εn •´ V 4. éu?¿†Ú©) V = U ⊕ W , Ù¥ U, W ´ V ˜|Ä. f˜m, ok V = A (U ) ⊕ A (W ) ¤á. S K 3.1.8. u1 = (1, −1, 1) , u2 = (1, 1, 1) , α1 = (1, −1) , α2 = (2, −1) , α3 = (−3, 1) , β1 = (1, 0) , β2 = (0, 1) , β3 = (1, 1) . 1. ´Ä•3 R-‚5N A : R3 → R2 , ¦ A (u1 ) = (1, 0) , A (u2 ) = (2, 0) ? e•3, ž ù ‚5N . eØ•3, ž)º•ŸoØ•3. Ñ˜‡ 2. ´Ä•3 R-‚5N A : R2 → R2 , ¦ éz‡ j = 1, 2, 3 þk A (αj ) = βj ? e•3, ž ˜‡ù ‚5N . eØ•3, ž)º•ŸoØ•3. Ñ S K 3.1.9. V ´ K-•þ˜m, A •?¿š˜8Ü. Ú~ 2.3.19 ˜ , ÏLÅ: Šƒ\½ê¦ •ª, Œ±½Â Map(A, V ) ¥?¿ü‡ ƒƒm \{Ú Map(A, V ) ¥ ƒ† K ¥ ƒ ê¦. 'uù \{Úê¦, 8Ü Map(A, V ) ´˜‡ K-•þ˜m. 1. U •´ K-•þ˜m. ½Â L(U, V ) := { N y² L(U, V ) ´ Map(A, V ) f : U → V | ∀ u, v ∈ U , f (u + v) = f (u) + f (v)} . f˜m. 2. y²: e K = Q, K L(U, V ) = HomK (U, V ). S K 3.1.10. f (A ) = 0. V ´ n ‘ K-• þ ˜ m, A ∈ EndK (V ). y ²: • 3 š " õ ‘ ª f ∈ K[X] ¦ 1nÙ ‚5N 134 9ÙÝ L« S K 3.1.11. •þ˜m V ¥ ü‡f˜m U, W ÷v V = U ⊕ W . P : V → V ´²1u U Ý •W ÝKN (ë„~ 3.1.8), Q : V → V ´²1u W Ý• U ÝKN . 1. y²: P 2 = P , P Q = QP = 0. 2. y²: e W 6= V , K P ØŒ_. S K 3.1.12. . V ´k•‘•þ˜m, U ´ V 1. y²: •3‚5N 2. ½ÂN T :V →W ¦ T (u) = S(u) é¤k u ∈ U ¤á. f : V → W Xe: f (v) = y²: XJ U 6= V … S Ø´"N S K 3.1.13.  S(v) ev∈U 0 ev∈ /U , @o f Ø´‚5N . V = K[X]. ½Âü‡N y²: A Ú B Ñ´ V þ A : V −→ V ; f (X) 7−→ f 0 (X) , B : V −→ V ; f (X) 7−→ Xf (X) . ‚5C†, … A B − BA = IdV . S K 3.1.14. A , B ´•þ˜m V þ ê k, þk A k B − BA k = kA k−1 . S K 3.1.15. f˜m, W ´?¿•þ˜m, S : U → W ´‚5N A ´•þ˜m V þ 1. y²: é?¿ α ∈ V , •3•˜ A α2 = 0. ü‡‚5C†, … A B = BA = IdV . y²: é?¿ ‚5C†, b §÷v A 2 = A . ˜«©)ª α = α1 +α2 , Ù¥ α1 , α2 ∈ V ©O÷v A α1 = α1 , 2. y²: e α ∈ V ÷v A α = −α, K α = 0. S K 3.1.16. A , B ´•þ˜m V þ y²: A B = 0. ‚5C†. b A 2 = A , B 2 = B, (A + B)2 = A + B. S K 3.1.17. A1 , · · · , Ar ´•þ˜m V k•õ‡üüØÓ ¦ •þ A1 α, · · · , Ar α üüØÓ. (J«: |^SK 2.3.24.) 3.2 ‚5N !¥·‚UY \ïÄk•‘•þ˜mƒm W o´L«k•‘ K-•þ˜m. Ý ‚5N ‚5C†. y²: •3•þ α ∈ V L« . eškƒ‡ (², 3 !¥ V Ú 3.2 ‚5N 3.2.1 Ý L« 135 ‹IÚ‹IC† (3.2.1) ± K n L« •þ˜m K n×1 . 3c¡~ 3.1.30 ¥·‚w À½ V ˜|Ä v1 , . . . , vn ƒ , Œ±•˜/(½˜‡Ó N (3.2.1.1) ∼ f = fv1 ,··· ,vn : K n − →V ¦ : é?Û n ‘•þ˜m V , ?¿ éz‡ i ∈ [[1, n]] , f (ei ) = vi . ùp e1 = (1, 0, · · · , 0)T , · · · , en = (0, · · · , 0, 1)T L« K n IOÄ. (¯¢þ, N (3.2.1.1) •3•˜5•´·K 3.1.32 A~.) c [ T ˜ e Ø J u y, N (3.2.1.1) ½ Â Ø = = • 6 u v1 , · · · , vn ù • þ ¤ 8 Ü, … ¢ S þ é § ‚ ü S • k • 6 5: - f (e1 ) = v1 , f (e2 ) = v2 ž N w,Ú f (e1 ) = v2 , f (e2 ) = v1 ž N ØÓ. ¤±, ·‚ùpI‡¦^kSÄ Vg. •Ò´`, éu V ˜ |Ä v1 , · · · , vn , ·‚rÙ¥•þ üS••Ä3Sž, ·‚Ò¡kS•þ| (v1 , · · · , vn ) • V ˜|kSÄ (ordered basis). (žÖö5¿: ·‚3ùpA¿¦^ ‘k )Ò PÒ“(v1 , · · · , vn )”5 rNkSÄÚÃSÄü‡VgéA PÒ O.) Šâ½Â, (v1 , v2 , v3 , · · · , vn ) Ú (v2 , v1 , v3 , · · · , vn ) ´ØÓ kSÄ. y3·‚Œ±`, N (3.2.1.1) ´d˜|kSÄ À •˜(½ . ·‚ò (3.2.1.1) ¥N _N P• M = M(v1 ,··· ,vn ) : V −→ K n ; (3.2.1.2) v 7−→ M(v) . éu?¿ v ∈ V , • þ M(v) = (a1 , · · · , an )T ∈ K n ˆ ‡ © þ ê Š ai ´ X Û l v 5 ( ½ Q ? ‰ Y „ ´ ‡ l ½ Â ¥ é Ï. ¯ ¢ þ, ò N f Š ^ u (a1 , . . . , an )T Œ •, a1 , . . . , an ù ê Ò ´ · ‚ r v L ˆ ¤ k S Ä (v1 , . . . , vn ) ¥ • þ ‚ 5 | Ü ž, ˆ ‡ Ä • þ vi X ê. † ó ƒ, M(v) = (a1 , · · · , an )T ´ÏL ª v = a1 v1 + · · · + an vn •˜û½Ñ •þ. þªU (3.2.1.3) ¤Ý /ª=•   a1 . . v = (v1 , · · · , vn ) ·   .  = (v1 , · · · , vn ) · M(v) . an Ï~, ·‚± α = (x1 , · · · , xn )T ù«‹I/ªL«˜‡ •þž, ¢SþŒ±@•3 K n (e1 , · · · , en ) e α Lˆª•   x1  .   α = (e1 , · · · , en ) ·  ..   xn dd‰a', ·‚Œ±ò IOÄ •þ M(v) = M(v1 ,··· ,vn ) (v) ∈ K n×1 ¡••þ v 3kSÄ (v1 , · · · , vn ) e ‹I (•þ) (coordinates, coordinate vector). kž• •B, ·‚ò M(v) À• n × 1 Ý , Ïd•Œ±¡ƒ• v 3kSÄ (v1 , · · · , vn ) e Ý (L«) (matrix (representation)). e± B L«kSÄ (v1 , · · · , vn ), K M(v1 ,··· ,vn ) (v) Œ±P• MB (v). 1nÙ ‚5N 9ÙÝ L« y: v 3kSÄ (v1 , v2 , v3 , v4 ) e ‹I 136 ~ 3.2.2. ÖöŒ± y: K 4 ¥ •þ|         1 −5 1 2 −6 0 −3 1         v1 =   , v 2 =   , v 3 =   , v 4 =   2 −1 2 0 −7 4 1 6 ´ K 4 ¥ ˜|Ä. (•‡ yT•þ| •• 4 =Œ.) •þ α = (8, 9, −5, 0)T 3kSÄ (v1 , · · · , v4 ) e ‹I• (3, −4, −1, 1)T . =   3 −4   α = (v1 , v2 , v3 , v4 ) ·   . −1 1 éu˜„ •þ v = (x1 , x2 , x3 , x4 )T ∈ K 4 , žÖö • 4 2 1 3 x1 − 3 x2 + 3 x3 − x4 − 2 x + 7 x + 31 x − 1 x   27 1 27 2 27 3 9 4   10  8 7  27 x1 − 27 x2 + 27 x3 − 49 x4  7 11 14 1 27 x1 − 27 x2 − 27 x3 − 9 x4  †óƒ, N (3.2.2.1)  M(v1 ,··· ,v4 ) : K 4 −→ K 4 ²wLˆª•    4    2 1 x1 x1 3 x1 − 3 x2 + 3 x3 − x4 x  − 2 x + 7 x + 31 x − 1 x  x   2  1 27 2 27 3 9 4 = Q  2   7−→  1027    8 7 x3   27 x1 − 27 x3  x2 + 27 x3 − 49 x4  7 11 14 1 x4 x4 27 x1 − 27 x2 − 27 x3 − 9 x4 Ù¥ 4 3 − 2  Q =  1027  27 7 27  (3.2.2.2) w,, (3.2.2.2) ª¥ Ý − 23 7 27 8 − 27 − 11 27 1 3 31 27 7 27 − 14 27   −1 36  −2 − 91  1    = − 94  27  10 1 9 Q •6ukSÄ (v1 , v2 , v3 , v4 ) 7  −18 9 −27 7 31 −3    −8 7 −12 −11 −14 −3 ÀJ. g • K 3.4. (v1 , · · · , vn ) • V ˜|kSÄ, α1 , . . . , αr • V ¥?˜•þ|. éz‡ i ∈ [[1, r]], βi = Mv1 ,··· ,vn (αi ) • αi 3kSÄ (v1 , · · · , vn ) e ‹I•þ. y²: V ¥•þ| α1 , · · · , αr ‚5ƒ' (½Ã') ¿©7‡^‡´ •þ| β1 , · · · , βr ‚5ƒ ' (½Ã'). XJ£Þ2w˜e~ 3.2.2 ¥?Ø kSÄ (v1 , v2 , v3 , v4 ), ŒU[% Öö¬uy: N M = M(v1 ,··· ,v4 ) : K 4 −→ K 4 Š^ J, ¢Sþƒ ur K 4 ¥z‡•þ v 3IOÄ (e1 , · · · , e4 ) e ‹IC†• v 3kSÄ (v1 , · · · , v4 ) e ‹I. Ø• Öö´ÄŽ ù ¯K: éu˜„ (k•‘) •þ˜m V , XJ‰ V ü|ØÓ kSÄ (v1 , . . . , vn ) Ú (v10 , . . . , vn0 ), ·‚XÛÏL•þ3Ù¥˜|kSÄe ‹I5¦ÑÓ˜•þ 3,˜|kSÄe ‹IQ ? • £‰ù‡¯K, ky²˜‡{ü Ún. 3.2 ‚5N Ý Ún 3.2.3: (ε1 , . . . , εn ) ´•þ˜m V 1. •3•˜ L« Ý 137 ˜|kSÄ, (η1 , . . . , ηn ) ´ V ¥ kS•þ|. P = (aij ) ∈ Mn (K) ¦ (η1 , · · · , ηn ) = (ε1 , · · · , εn ) · P 2. ±þ^‡¥ Ý P ´Œ_Ý …= η1 , · · · , ηn •´ V ˜|Ä. y². (1) ¯¢þ, P 1 j Ò´•þ ηj 3kSÄ (ε1 , · · · , εn ) e (2) e P Œ_, Kd (η1 , · · · , ηn ) = (ε1 , · · · , εn ) · P Œ ‹I•þ. (ε1 , · · · , εn ) = (η1 , · · · , ηn ) · P −1 . ù`² ε1 , · · · , εn ÑŒ± ¤ η1 , · · · , ηn ‚5|Ü. Ïd η1 , · · · , ηn ´ V )¤|. Ï•¤¹• þ‡ê u dim V , ¤± η1 , · · · , ηn ´ V ˜|Ä (íØ 2.3.15). ‡L5, b η1 , · · · , ηn ´ V ˜|Ä. K† (1) ¥(ØÓnŒ•, •3•˜ Ý Q ∈ Mn (K) ¦ (ε1 , · · · , εn ) = (η1 , · · · , ηn ) · Q . u´ (ε1 , · · · , εn ) = (η1 , · · · , ηn ) · Q = (ε1 , · · · , εn ) · P · Q = (ε1 , · · · , εn ) · (P Q) . ù`²Ý M = PQ U ÷v (ε1 , · · · , εn ) = (ε1 , · · · , εn ) · M . , , Šâ (1), ù ² P Œ_. Ý •Uk˜‡. ü Ý In Kw,÷vd^‡. ¤± P Q = M = In . ù` ½ Â 3.2.4. B = (v1 , . . . , vn ) Ú B 0 = (v10 , . . . , vn0 ) þ••þ˜m V •˜ Œ_Ý P = PB, B0 ∈ Mn (K) ¦ kSÄ. ŠâÚn 3.2.3, •3 B 0 = B · PB, B0 . ½=, (v10 , · · · , vn0 ) = (v1 , · · · , vn ) · P . ·‚¡dÝ matrix). P •kSÄ B B0 LÞÝ (transition matrix) ½ÄC†Ý Ún 3.2.5: B, B 0 Ú B 00 þ••þ˜m V kSÄ. LÞÝ . K P −1 ´ B 0 B LÞÝ , P P 0 ´l B B 00 B0 P •B LÞÝ LÞÝ (change of basis , P 0 • B0 B 00 . y². 3‰ÖööS. ½n 3.2.6 (‹IC†úª): . Kéu?¿ v ∈ V , B Ú B 0 þ••þ˜m V kSÄ, P = PB, B0 • B B0 LÞÝ ƒ¦ƒ ,¬ MB (v) = PB, B0 MB0 (v) . •Ò´`, ?¿•þ v 3# kSÄ B 0 e v 3 5kSÄ B e ‹IÝ . ‹IÝ †>˜˜ B B0 LÞÝ 1nÙ ‚5N 138 y². 9ÙÝ L« B = (v1 , . . . , vn ), B 0 = (v10 , . . . , vn0 ). l v = (v10 , · · · , vn0 )MB0 (v) Ú (v10 , · · · , vn0 ) = (v1 , · · · , vn )PB, B0 Œ v = (v1 , · · · , vn ) · (PB, B0 MB0 (v)) . ùÒ`² PB, B0 MB0 (v) u MB (v). ±þ‹IC†úª í ATØŽ(J, PÁù‡úªžI‡ [%, Ø‡r#Î‹I ˜á· . • ;•”Ø, ·‚ïÆÖö3¦^ž¦þ3M¥£Ž˜eínL§, ±•òúªá †. XJr1l/ªþo( MB (v) = PB, B0 MB0 (v) Ú B 0 = B.PB, B0 ùü‡úª, •N·‚Œ±o (Ñ˜^“ƒ ƒž” K: c˜‡úªm> eI¥, B 0 ëYƒ /Ñy ü‡, $Ž(JC¤†> fž, B 0 ÒleI¥ž” . 1 ‡úª•kaq y–. ,, ÖöŒ±`ù«y–´˜«| Ü. ·‚• ¿`, ù«y– ÑyÙ¢´ uêÆ[‚3ÀJêÆPÒž Õäú%. XJe¡ •üUkÏuÖö, ò´Šö #Œ!¤: ‹IC†‡ %, Ý LÞÎ #. ‡¦‹I„•Î, †>LÞm>#. (3.2.7) Š••~„ ~fƒ˜, ·‚Ž;€•äN/?Ø˜e K n ¥ü|Äƒm ¦Ñ. b kSÄ B = (v1 , · · · , vn ) ¥ ˆ‡•þ ¤Xe/ª:     b11 b1n     b  21   b2n   , · · · · · · , vn =  .  v1 =  .  .   .   .   .  bn1 bnn kSÄ B 0 = (v10 , · · · , vn0 ) ¥ 2 P •B B0 LÞÝ LÞÝ XÛ ˆ‡•þK/X:  0   0  b11 b1n  0   0   b21   b2n  0    v10 =   ..  , · · · · · · , vn =  ..   .   .  0 bn1 b0nn . K ª B 0 = B · P ¢SþÒ´±e'uÝ ¦È ª: (b0ij ) = (bij )P . • Ò ´ `, X J r k S Ä B ¥ ˆ ‡ • þ ¤ / ª, k S • þ | (v1 , · · · , vn ) Ò ‰ Ñ ˜ ‡ Ý 0 0 0 B = (bij ). aq/ B éA˜‡Ý B = (bij ). KLÞÝ P = PB, B0 Ò´d ª B 0 = BP •˜û ½ Ý . ¯¢þ, B Œ±@•ÍOÄ (e1 , · · · , en ) B LÞÝ , Ïd§´‡Œ_Ý . dž‡¦Ñ B 0 = BP ù‡ ª¥ P , ·‚Œ±æ^pd–e ž { ! äN5`, Œ±ò B, B 0 †m¿ ü¤ n × (2n) ©¬Ý (B B 0 ). , ±Ð 1C†ò†Œ> Ý B z•ü Ý In , džéA m>Ý Ò¬C¤ B −1 B 0 = P . 3.2 ‚5N Ý L« žÖö•ì±þ•{ 139 y: e3 K 3 ¥•Ä v1 = (1, 0, −1)T , v2 = (2, 1, 1)T , v3 = (1, 1, 1)T Ú v10 = (0, 1, 1)T , v20 = (−1, 1, 0)T , v30 = (1, 2, 1)T .  0 1  KlkSÄ B := (v1 , v2 , v3 ) B 0 := (v10 , v20 , v30 ) LÞÝ • P = −1 −3 2 4 g • K 3.5. E = (e1 , · · · , en ) • K n IOÄ, B = (v1 , · · · , vn ) • K n     b1n b11     b b  2n   21     v1 =   ..  , · · · , vn =  ..   .   .  bnn bn1 ÖöU Ø²?ÛOŽ† 3.2.2 ‚5N ¥ ÑLÞÝ  1  −2. 4 ,˜|kSÄ. b PE, B í ? Ý c˜ !·‚`² kSÄ À •þ. y3·‚5`², k •þ Œ±•Ï·‚±‹I ‹IL«ƒ ‚5N /ª£ã˜„ (k•‘) •þ˜m L«•Œ±‘ƒC •\äN. (3.2.8) k5£ ˜‡ÙG ~f. z ‰½ ˜‡Ý A ∈ Mm×n (K), d A Œ±½ÂÑ m Kn K m ˜‡‚5N (ë„ (3.1.12.1))     x1 x1  .   .  n m    fA : K 7−→ K ; X :=  ..  7−→ AX = A  ..   xn xn •þ˜ y3·‚5`², ?Û˜‡‚5N T : K n → K m ÑŒ±•˜/ Ñ±þ/ª. =, éu?¿‰½ ‚5N T ∈ Hom(K n , K m ), •3•˜˜‡Ý A ∈ Mm×n (K) ¦ T = fA . ½=, éu¤k X ∈ K n þk T (X) = AX. ·‚ò K n ¥ IOÄP• B = (e1 , · · · , en ), K m ¥ IOÄP• C = (ε1 , · · · , εm ). ò•þ T (e1 ), · · · , T (en ) ÏL§‚3kSÄ C e ‹IL«•     a11 a1n      a21   a2n     T (e1 ) =  .  = a11 ε1 + · · · + am1 εm , · · · · · · , T (en ) =  .   = a1n ε1 + · · · + amn εm  ..   ..  am1 amn ‚55Ÿ, •þ X = (x1 , · · · , xn ) ∈ K n , ·‚k X = x1 e1 + · · · + xn en . u´Šâ T   a11 x1 + a12 x2 + · · · + a1n xn    a21 x1 + a22 x2 + · · · + a2n xn    . T (X) = T (x1 e1 + · · · + xn en ) = x1 T (e1 ) + · · · + xn T (en ) =  ..    . am1 x1 + am2 x2 + · · · + amn xn éu?¿ 1nÙ ‚5N 140 9ÙÝ L« Ïd, eòþ¡Ñy ~ê aij , 1 ≤ i ≤ m, 1 ≤ j ≤ n ü¤Ý A = (aij ) ∈ Mm×n (K), KþªL² T X = AX = fA (X). ¤±, ÷v^‡ T = fA Ý A o´Œ±Šâ‚5N T 5À . ,˜•¡, XJ T = fA é,‡Ý A ∈ Mm×n (K) ¤á, 7, Aej = fA (ej ) = T (ej ) éz‡ j ∈ [[1, n]] ¤á. Ý ¦{wŠ·‚ Aej Ò´Ý A 1 j . ¤± A z˜ d T •˜(½. ¤±, ÷v T = fA Ý A d T •˜(½. ±þ£ã l T ∈ Hom(K n , K m ) éA A ∈ Mm×n (K) L§Œ±@•´½Â ˜‡N (3.2.8.1) M = MB, C : Hom(K n , K m ) −→ Mm×n (K) ; T 7−→ M(T ) . •Ò´`, éz‡ T ∈ Hom(K n , K m ), Ý M(T ) ½Â•÷v T = fA @‡Ý A. ·‚ò M(T ) ¡• T 3 B, C ùü|kSÄe Ý (L«) (matrix (representation)). ‡L5, ò˜‡Ý A ∈ Mm×n (K) éA ƒA ‚5N fA ù‡L§•Œ±@•´˜‡N (3.2.8.2) L = LB, C : Mm×n (K) −→ Hom(K n , K m ) ; N´wÑ, (3.2.8.1) Ú (3.2.8.2) ¥ N p•_N A 7−→ L(A) = fA . . g • K 3.6. ·‚• Mm×n (K) Ú Hom(K n , K m ) Ñ´ K-•þ˜m (ë„~ 2.3.17 Ú·K 3.1.22 (2)). žÖö y (3.2.8.1) Ú (3.2.8.2) ¥ N þ•‚5N , Ïd ö´p_ Ó N . ddí • Hom(K n , K m ) ‘ê. (3.2.9) c ¡ (3.2.8) ¥ ? Ø Œ ± ” t / í 2 ˜ „ k • ‘ • þ ˜ m œ ¹. ä N 5 `, V Ú W © O • n ‘ Ú m ‘ K-• þ ˜ m. 3 V Ú W ¥ © O À ½˜ | k S Ä B = (v1 , · · · , vn ) Ú C = (η1 , · · · ηm ). éu˜‡‚5N T : V → W , •3•˜(½ ˜|~ê aij ∈ K, 1 ≤ i ≤ m, 1 ≤ j ≤ n ¦     a11 a1n      a21   a2n     T (v1 ) = (η1 , · · · , ηm )  .  , · · · · · · , T (vn ) = (η1 , · · · , ηm )  .    ..   ..  am1 amn ·‚òÝ  (3.2.9.1) a11   a21 M(T ) = MB, C (T ) :=   ..  . am1 a12 ··· .. . ··· ··· .. . ··· ···  a1n  a2n  ..   .  amn ¡• T 3 B, C ùü|kSÄe Ý (L«) (matrix (representation)). ¯¢þ, XJ¦^ (3.2.1) ˜ã¥‰Ñ •þ‹I Vg, KU ?1©¬Œ±ò±þÝ ¤ Xe/ª: (3.2.9.2) MB, C (T ) = M(v1 ,··· ,vn ), (η1 ,··· ,ηm ) (T ) = Mη1 ,··· ,ηm T (v1 ) , · · · , Mη1 ,··· ,ηm T (vn ) . •Ò´`, ‡ Ñ T 3 B = (v1 , · · · , vn ) Ú C = (η1 , · · · , ηm ) e Ý MB, C (T ), •I‡P4TÝ 1 j Ò´ B ¥1 j ‡•þ3 T N e ”UìkSÄ C ¤‹I•þž . ±þ·‚¢Sþ£ã ˜‡N (3.2.9.3) M = MB, C : Hom(V, W ) −→ Mm×n (K) ; T 7−→ MB, C (T ) . 3.2 ‚5N Ý L« 141 Œ±ÏL±e•ª E (3.2.9.3) _N : ?‰Ý A = (aij ) ∈ Mm×n (K) ƒ •kSÄ C = (η1 , · · · , ηm ) ¥•þ ‚5|ÜXê, Œ± •þ ,±A 1j Š •þ vj •gN •    a1n a11      a2n   a21     , · · · · · · , w = (η , · · · , η ) w1 = (η1 , · · · , ηm )  n 1 m  .   ..   ..   .  amn am1  Šâ·K 3.1.32, •3•˜ wj . ÏLù éA·‚ (3.2.9.4) L = LB, C : Mm×n (K) −→ Hom(V, W ) ; ·‚žÖö@ýc[/ ïÄ‚5N T ∈ Hom(V, W ) òkSÄ B = (v1 , · · · , vn ) ¥ ˜‡N Ý y: (3.2.9.3) Ú (3.2.9.4) ¥ N A 7−→ LB, C (A) . p•_N . kŸo¿ÂQ? XJ £› (3.2.8) ˜ã Øã, ÖöŒU¬²x: ·‚• ˜‡‚5N T ∈ Hom(K n , K m ) n m 3 (K Ú K V• ) IOÄe Ý A, ÒŒ±ÏLÝ ƒ¦ •ªOŽÑ˜„ X ∈ K n 3 T Šê ” (ùÒ´ ž·‚?Ø ¯¢ T (X) = AX). y3·‚5`², Ó ¯œ3·‚y3? Ø ˜„œ¹•¤á. b MB, C (T ) / X (3.2.9.1). v ∈ V. ò v ¤ ‚ 5 | Ü v = x1 v1 + · · · + xn vn , =, T (x1 , · · · , xn ) u v 3kSÄ B = (v1 , · · · , vn ) e ‹I•þ MB (v). @o T (v) = T (x1 v1 + · · · + xn vn ) = x1 T (v1 ) + · · · + xn T (vn )     a11 a1n      a21   a2n     = x1 (η1 , · · · , ηm )  + · · · + x (η , · · · , η ) n 1 m  .   ..   .   ..  am1 amn     a1n xn a11 x1      a21 x1   a2n xn     = (η1 , · · · , ηm )  .  + · · · + (η1 , · · · , ηm )  .    ..   ..  am1 x1 amn xn     a11 a12 · · · a1n x1      a21 · · · · · · a2n   x2     = (η1 , · · · , ηm )  ..  .. ..  ..   .  . .  . .   ..  am1 · · · · · · amn xn ùÒ´` (3.2.9.5) MC T (v) = MB, C (T ) · MB (v) . (éŒJ, ù‡úª¥ eIØ÷v“ƒ ƒž” 5Æ. ØL, ù‡úªATØ N´P·. Œ±Nyù ˜« K: XJòl B C Ä=†L§n)•˜«“N ”, @o§é•þ 1öŠž, PÒþATr§ 3†>, ù X T (v) ù‡PÒ´òN T 3†>˜ .) ·‚²~„^, ˜«•ªLã (3.2.9.5) ¥ ª, =¡±eãL …, § ‹I– † (ë„N¹ (A.2.7) ˜ 1nÙ ‚5N 142 9ÙÝ L« ã) (3.2.9.6) /W T V MB Kn MC MB, C (T )• / Km Ù¥, “MB, C (T )•” ^5L«†¦Ý MB, C (T ) ‰Ñ N X 7→ MB, C (T ) · X. ÖöXJ8 \ÆS•õ “êÆ•£, ˜½¬ÅìN¬ “ãL †”ù«`{´˜«š~ B$ y“zLã•ª. ”g•K 3.6 ¥ œ/˜ , (3.2.9.3) Ú (3.2.9.4) ¥ . ·‚òù‡-‡ ¯¢P¹3±e·K¥. N Ñ´‚5N ·K 3.2.10: V, W •k•‘ K-•þ˜m, B, C ©O• V, W K (3.2.9.4) ¥ N ´ K-‚5N . =, éu?¿‚5N λ ∈ K, MB, C (T + S) = MB, C (T ) + MB, C (S) ; •Ò´`, 3Ó˜|kSÄƒe, T + S Ý †~ê λ ê¦. Ý uT , Ïd´p_ Ó N kSÄ. T, S ∈ HomK (V, W ) Ú?¿~ê MB, T (λT ) = λMB, C (T ) . Ý †S Ý ƒÚ, λT Ý uT y². 3‰Öö. ˜‡•þ˜m ÑXe½Â: g ‚5N (=‚5C†) ´•~‘ ã‚5N . Ïd·‚;€‰ ½ Â 3.2.11. V •k•‘•þ˜m, B ´ V ˜|kSÄ. éu?¿‚5C† A : V → V , ·‚ Œ±é†mü>Ñy ˜m V ÑÀJ B ù|kSÄ. dž, ·‚òÝ MB, B (A ) {P• MB (A ), ¿{¡§• A 3kSÄ B e Ý (L«). ·‚I‡ÏL˜ ¢~ OŽ5\ éc¡nØ ~ 3.2.12. V = W = K 2 , B = (e1 , e2 ) • K 2 ~ 3.2.13. B = (e1 , e2 ) • R2 ¥ n). IOÄ, C = (η1 , η2 ), Ù¥ η1!= (1, 1), η2 = (1, 0). 0 1 Kð C† I = IdK 2 : V → W 3kSÄ B, C e Ý • MB, C (I) = . 1 −1 XJò C À• V!¥À kSÄ, B KÀ• W ¥À kSÄ, K I 3 kSÄ C, B e Ý 1 1 • MC, B (I) = . 1 0 IOÄ, η1 = (1, 0), η2 = (1, 1). Šâ·K 3.1.22, ! •3•˜ 1 1 ‚5C† A ∈ EndR (R2 ) ¦ A (e1 ) = η1 , A (e2 ) = η2 . ØJwÑ MB (A ) = . 0 1 lã”þw (ò5Ö¿‡ã«??), A Š^ J´r²¡S±²1uü‡‹I¶ pî†‚Œ ¤ •/ ‚, C†•²1u x ¶Ú²1u1˜!n–•é ‚ (=•§• y = x †‚) @ †‚Œ¤ ²1o>/ ‚. aq/, XJ˜‡‚5C† S ∈ End(R2 ) 3IOÄe Ý ´˜ ! 1 c ‡þn ,KS Š^ Jkaq AÛ)º. ù ²¡‚5C†¡•²¡ ˜‡ƒ 0 1 C (shear, shearing transformation). 3.2 ‚5N Ý L« 143 ~ 3.2.14. • Ä õ ‘ ª ¤ • þ ˜ m V = K[X]≤2 Ú W = K[X]≤3 . (1, X, X 2 ), W kSÄ C = (1, X, X 2 , X 3 ). •Ä‚5N T : V −→ W ; kSÄ B = V f (X) 7−→ (X − 1)f (X) . K  −1 1  MB, C (T ) =  0 0 éu˜„ f = a0 + a1 X + a2 X 2 ∈ V , ÏL T 0 −1 1 0 ½Â†  0 0   −1 1 OŽŒ• T (f ) = T (a0 + a1 X + a2 X 2 ) = −a0 + (a0 − a1 )X + (a1 − a2 )X 2 + a2 X 2 . =,    −1 a0  1    f = (1, X, X 2 ) a1  7−→ T (f ) = (1, X, X 2 , X 3 ) ·   0 a2 0 ùÒ y (3.2.9.5) ¥ ~ 3.2.15. •Äõ‘ª 0 −1 1 0   0   a0  0     a1   −1 a2 1 úª. ¤ •þ˜m V = K[X]≤3 . ½Â‚5C† A ∈ End(V ) • f (X) = a0 + a1 X + a2 X 2 + a3 X 3 7−→ A (f (X)) = a3 + a2 X + a1 X 2 + a0 X 3 . V kSÄ B = (1, X, X 2 , X 3 ). K A 3kSÄ B e  0 0  MB (A ) =  0 1 Šâþ¡ A ½Â, ÖöEŒ 0 0 1 0 0 1 0 0 Ý •  1 0   0 0 y (3.2.9.5) ª3ù‡~f¥¤á. ~ 3.2.16. •Ä V = K[X]≤n−1 þÏL¦õ‘ª /ª ê (ë„~ 3.1.15) ‚5C† A : f = a0 + a1 X + · · · + an−1 X n−1 7−→ A (f ) = a1 + 2a2 X + · · · + (n − 1)an−1 X n−2 . B •kSÄ (1, X, · · · , X n−1 ). K  0  0   0   .. MB (A ) =  . .  ..    0 0 Šâþ¡ A ½Â, ÖöEŒ 0 .. . .. . ··· 0 .. . .. . .. . ··· ··· .. . .. . .. . 0 0 0 0 ··· ··· ··· ··· 1 0 0 2 0 .. . .. . ··· ··· .. . .. . .. . .. . ··· 0 0 .. . .. . .. .                n − 1 0 y (3.2.9.5) ª3ù‡~f¥¤á. 1nÙ ‚5N 144 éu‚5N ·K 3.2.17: EÜ, ƒA Ý LyÑŸo 9ÙÝ L« 'XQ? V, W, U •k•‘•þ˜m, B, C, D ©O• V, W, U ¥ ½ kSÄ. 1. é?¿ T ∈ Hom(V, W ) Ú S ∈ Hom(W, U ), MB, D (ST ) = MC, D (S) · MB, C (T ). (Uì (3.1.23) ˜ã¤ã, ·‚rEÜN S ◦ T {P• ST .) {ü/`, ü‡N ƒEÜN Ý u 5ü‡‚5N Ý ¦È. U = V, D = B 9 S = T −1 Œ• MC, B (T −1 ) = (MB, C (T ))−1 . AO/, e T Œ_, KÏL 2. éu?¿‚5C† A ∈ End(V ) Ú?¿õ‘ª f ∈ K[X], MB (f (A )) = f (MB (A )). •Ò´`, õ‘ª f ¥“\‚5C† A ¤ ¥¤ Ý . ‚5C†ƒÝ uA Ý “\õ‘ª f AO/, éu?¿ k ∈ N, MB (A k ) = MB (A )k . y². Øä (1) ‚P y²¿Ø(J. • •B, ·‚Ú\˜‡PÒ. éu V ¥?¿•þ| α1 , · · · , αr , · T (α1 , · · · , αr ) := (T (α1 ) , · · · , T (αr )) . (3.2.17.1) éu?¿Ý (3.2.17.2) A ∈ Mr×s (K), Šâ T ‚55ŸŒ± y T (α1 , · · · , αr )A = T (α1 , · · · , αr ) · A . P B = (v1 , · · · , vn ), C = (w1 , · · · , wm ), D = (u1 , · · · , up ). K U ì (3.2.17.1) ª MB, C (T ) ½ÂŒ• T (v1 , · · · , vn ) = (w1 , · · · , wn )MB, C (T ) (3.2.17.3) òN PÒÚÝ S Šû (3.2.17.3) ªü>Œ± (ST )(v1 , · · · , vn ) = S T v1 , · · · , vn = S (w1 , · · · , wm )MB, C (T ) (|^úª (3.2.17.2)) (ŠâÝ MC, D (S) –d·‚y² Øä (1). òØä (1) (Ø † M ~ 3.2.18. ½Â) = S(w1 , · · · , wm ) · MB, C (T ) = (u1 , · · · , up ) · MC, D (S) · MB, C (T ) = (u1 , · · · , up ) · MC, D (S) · MB, C (T ) . ‚55Ÿ(ÜŒ A ∈ End(R2 ) ´ò²¡S (2). ¤k:7X : O(0, 0) ^= „~ 3.1.9). Šâ (3.1.9.2) ªŒ•, A 3IOÄ (e1 , e2 ) e ¿ n ∈ N, A n Ò´ò²¡S¤k:7X :^= Ý nα ! cos(nα) − sin(nα) Ý • . Šâ·K 3.2.17 (2), sin(nα) cos(nα) (3.2.18.1) cos α sin α − sin α cos α !n = Ýα ‚5C† (ë ! cos α − sin α Ý •A= . éu? sin α cos α ‚5C†, Ïd A n 3IOÄe ! cos(nα) − sin(nα) . sin(nα) cos(nα) 3.2 ‚5N Ý L« 145 þª ,•Œ±ÏL8B{±9n ¼ê Ú úª . w,±þAÛ•{‰Ñ Øy‡ •\{'. ¯¢þ, ±þ^=C† A w,´Œ_ . Ù_C† A −1 Ò´7X :^= Ý −α éA N , §3IOÄe Ý • (3.2.18.2) ! cos(−α) − sin(−α) = sin(−α) cos(−α) cos α − sin α sin α cos α ! = cos α sin α − sin α cos α !−1 . ù†·K 3.2.17 (1) ¥ (Ø¬Ü. 5¿, XJéu?¿ ê k ∈ N∗ , ·‚òPÒ A −k n)• (A −1 )k , ò A−k n)• (A−1 )k , @oÏL (3.2.18.2) ·‚Œ±@• (3.2.18.1) ªéu¤k ê n (ÃØ K) Ñ¤á. 3.2.3 •–"zÝ½n !¥, ·‚òÏL‚5N ÚÝ éA'Xy²e¡˜^-‡½n: ½n 3.2.19 (•–"zÝ½n rank–nullity theorem): . K T : V → W ´k•‘•þ˜mƒm ‚5N dim Ker(T ) + dim Im(T ) = dim V . ·‚8 ~P rank(T ) = dim Im(T ) ¿òÙ¡•‚5N T • (rank). dim Ker(T ) ¡• ‚5N T "zÝ (nullity). ("zÝq vkAOÚ˜ PÒ.) XJ T ´l •þ˜m K n K m ÏL†¦˜‡ ½Ý A N X 7→ AX. @o Im(T ) u A ˜m C (A), Ker(T ) K u A "z˜m N (A) (ë„~ 3.1.26), ¤± rank(T ) = rank(A), dim Ker(T ) = dim N (A). Ïd· ‚é‚5N ½Â • Vg±9"zÝ VgÑÚƒcéÝ ¤‰ ½Â¹Â˜—. (Ý " zÝVgQ3 (2.2.10) ˜ã‰Ñ.) ã 3.1: •–"zÝ½n «¿ã. ã¡5 : ‘Äz‰ 1nÙ ‚5N 146 •y²½n 3.2.19, ·‚I‡˜‡•˜„ Ún 3.2.20: 9ÙÝ L« Ún. b f U1 −−−−→ U2     ϕy yθ g U10 −−−−→ U20 ´•þ˜m … gϕ = θf . ‚5N ¤ †ãL. =, U1 , U2 , U10 , U20 þ••þ˜m, f, g, ϕ, θ þ•‚5N , 1. ±eü‡•¹'X¤á: =, ϕ ò f ØN\ g u´, ÏL•› (3.2.20.1) • ϕ(Ker(f )) ⊆ Ker(g) ; θ(Im(f )) ⊆ Im(g) . Ø, θ ò f ”. ”N\ g ½Â•Œ±l ϕ Ú θ p ϕ0 : Ker(f ) −→ Ker(g) ; v 7→ ϕ(v) 2. XJ ϕ Ú θ þ•Ó , K (3.2.20.1) ¥ ‚5N Ñ‚5N Ú θ0 : Im(f ) −→ Im(g) ; u 7→ θ(u) . ϕ0 Ú θ0 þ•Ó . y². (1) d•Ä–¢{. (½3‘,Ç‘žù).) (2) d ϕ •ü Œ±• ϕ0 •´ü . aq/, l θ •ü Œ± • θ0 ´ü . Ïd, •‡2 y ϕ0 Ú θ0 þ•÷ =Œ. • d, • Ä ? ¿ ½ ƒ w ∈ Ker(g). Ï • Ker(g) ⊆ U10 , ϕ : U1 → U10 • V , ¤ ± • 3 v ∈ U1 ¦ ϕ(v) = w. du w ∈ Ker(g), g(w) = 0. ŠâãL †5, θf (v) = gϕ(v) = g(w) = 0. ¤±, f (v) ∈ Ker(θ). θ q´ü , ¤± f (v) = 0, = v ∈ Ker(f ). ù , ϕ0 3 v ?k½Â, ¿… 0 ϕ (v) = ϕ(v) = w. Ïd w ∈ Im(ϕ0 ). ùÒ´`, ϕ0 ´÷ . 2w Im(g) ¥?¿ ½ ƒ z. Šâ”8 ½Â, •3 y ∈ U10 ¦ g(y) = z. Ï• ϕ : U1 → U10 ´÷ , ¤±q•3 x ∈ U1 ¦ ϕ(x) = y. u´, ŠâãL †5, θf (x) = gϕ(x) = g(y) = z. ù , f (x) ´ Im(f ) ¥ ƒ, θ0 3 f (x) ?Œ½Â, … θ0 f (x) = θf (x) = z. Ïd θ0 ´÷ . Ún–dy .. ±þÚn y²•, Hy›…w Ã , ù«Øy¢3´“êÆ¥š~š~-‡ Ä õ. §•w« ¦^ †ãL ¿Â: ·‚Œ±÷X †ãL¥ †ÞÏŠJ.5y²˜ •35 (Ø. ù«y²gŽkž¬ <‚o(•“ãLJl{”(diagram chase). ½n 3.2.19 (•–"zÝ½n) y². • V Ú W ˆ N T éA Ý . •Ä (3.2.9.6) ª¥ †ãL ˜|kSÄ B Ú C. - A = MB, C (T ) •‚5 T V −−−−→ W    M MB y y C K n −−−−→ K m fA Ù¥ fA L«N X 7→ AX. 3.2 ‚5N Ý L« 147 ±þãL÷vÚn 3.2.20 (2) ¥ b . Ïd·‚k•þ˜m Ó Ker(T ) ∼ = Ker(fA ) = N (A) ∼ ∼ 9 Im(T ) = Im(fA ) = C (A). (PÒ“=” ¹Â3½Â 3.1.29 ¥Qk)º.) ¤±, dim Ker(T ) + dim Im(T ) = dim N (A) + dim C (A) = dim N (A) + rank(A) . Šâ½n 2.2.12 (1), ·‚• þª•mà u n = dim V . ½nuý.. 3±þy²L§¥, ·‚¢Sþ„y² ±e(Ø: ·K 3.2.21: T : V → W • k•‘•þ˜mƒm W ?¿kSÄ C, e- A = MB, C (T ), K dim Ker(T ) = dim N (A) , •Ò´`, éu T 3?¿kSÄe ˜½ u A •. Ý A, ‚5N ‚5N . @o, éu V ?¿kSÄ B Ú rank(T ) = rank(A) . T "zÝ˜½ uA "zÝ, T •• ÏLòÝ Ú‚5N ƒéX, ¿|^•–"zÝ½n, ~~Œ±é•B/y²˜ 'uÝ • (Ø. ( ,, ·‚• yÖöÏLõg•5}Áˆ«ŒU •{.) ±e·‚Þ˜ ~f`²ù˜ :. ~ 3.2.22. é?¿Ý A, B ∈ Mm×n (K), y² rank(A + B) ≤ rank(A) + rank(B). (XJ‡ Ù¦y{, ÖöŒ±ë• [22, þþ, 1 113 •·K 4.4].) •Ä‚5N A : K n −→ K m ; X 7−→ AX )˜ Ú B : K n −→ K m ; X 7−→ BX . K (3.2.22.1) rank(A + B) = rank(A + B) = dim Im(A + B) . N´wÑ Im(A + B) ⊆ Im(A ) + Im(B). ¤±, |^'uf˜mÚ 'Ø ª (2.3.31.3), ·‚ dim Im(A + B) ≤ dim Im(A ) + Im(B) ‘ê ª (½n 2.3.30 (2)) ½ƒ ≤ dim Im(A ) + dim Im(B) (3.2.22.2) = rank(A ) + rank(B) = rank(A) + rank(B) . nÜ (3.2.22.1) Ú (3.2.22.2) = ~ 3.2.23. ¤–y. A ∈ Mm×n (K), B ∈ Mn×s (K). y² Sylvester •Ø ª: rank(AB) ≥ rank(A) + rank(B) − n . (XJ‡ )˜ Ù¦y{, ÖöŒ±ë• [22, þþ, 1 114 •·K 4.6 Ú1 133 •~ 5.5].) •Ä‚5N A : K n −→ K m ; X 7−→ AX Ú B : K s −→ K n ; Y 7−→ BY . 1nÙ ‚5N 148 ØJwÑ A p L« Ñ˜‡‚5N A 0 : Im(B) −→ Im(A B) ; ‡ 9ÙÝ v 7−→ A (v) . ù‡(Ø, Öö•Œ±òÚn 3.2.20 (1) Aûe¡ †ãL B K s −−−−→ K n     Idy yA AB K s −−−−→ K m …, Šâ½Â„Œ±• nŒ A 0 ´÷ , =, Im(A 0 ) = Im(A B). u´, éN A 0 ¦^•–"zÝ½ rank(B) = dim Im(B) = dim Ker(A 0 ) + dim Im(A 0 ) = dim Ker(A 0 ) + dim Im(A B) (3.2.23.1) = dim Ker(A 0 ) + rank(AB) . , , N´wÑ Ker(A 0 ) = {v ∈ Im(B) | A (v) = 0} ⊆ Ker(A ) = {v ∈ K n | A (v) = 0} . u´, dim Ker(A 0 ) ≤ dim Ker(A ) = n − rank(A) . (3.2.23.2) nÜ (3.2.23.1) Ú (3.2.23.2) = ¤–y. ±þ~ 3.2.22 Ú 3.2.23 ¥‰Ñ ª'X. ü‡Ý ¦È½¦Úƒ g • K 3.7. Þ~`²: ~ 3.2.22 Ú 3.2.23 ¥y² Ø Ò. Ø •† 5ü‡Ý ªQkŒU • Ò, qkŒU Ø²…Ø î‚ A g • K 3.8. (ë„ [22, þþ, 1 118 •SK 12]). A, B ∈ Mn (K) ÷v AB = BA. C= B ´ ò A, B þeü ¤ 2n × n Ý . y²: rank(A) + rank(B) ≥ rank(C) + rank(AB). (J«: ò A, B, C éA · ‚5N A , B, C , y²: Ker(C ) = Ker(A ) ∩ Ker(B) ±9 Ker(A ) + Ker(B) ⊆ Ker(A B).) ~ 3.2.24. (ë„ [22, þþ, 1 147 •·K 6.2].) Kk±eØ ª (¡• Frobenius •Ø ª) ¤á A ∈ Mm×n (K), B ∈ Mn×k (K), C ∈ Mk×s (K). rank(AB) + rank(BC) ≤ rank(ABC) + rank(B) . •y²þ¡Ø ª, •Ä3IOÄeÝ C ©O• C, B, A B n‡‚5N A K s −→ K k −→ K n −→ K m . N´wÑ±e'X¤á (3.2.24.1) Im(BC ) ⊆ Im(B) , Im(A BC ) ⊆ Im(A B) , A (Im(BC )) = Im(A BC ) , A (Im(B)) = Im(A B) . 3.2 ‚5N Ý L« ¯¢þ, (3.2.24.1) ¥ ü‡ 149 ª`²‚5N A : Kn → Km p A1 : Im(BC ) −→ Im(A BC ) 2(Ü (3.2.24.1) ª¥cü‡•¹'XŒ± (3.2.24.2) Im(BC _ ) Im(B) Ù¥ç†•• (3.2.24.2)) ü‡†ÞÑ“Lg,¹\N Ú † ‚5N A2 : Im(B) −→ Im(A B) . Xe †ãL A1 / Im(A BC ) _ A2 Ñü‡÷ / Im(A B) . dž, N´ y (½öòÚn 3.2.20 (1) AûãL Ker(A1 ) ⊆ Ker(A2 ) . u´, ÏLé A1 Ú A2 ùü‡÷ A^•–"zÝ½nŒ rank(BC) − rank(ABC) = dim Im(BC ) − dim Im(A BC ) = dim Im(BC ) − dim Im(A1 ) = dim Ker(A1 ) ≤ dim Ker(A2 ) = dim Im(B) − dim Im(A2 ) = dim Im(B) − dim Im(A B) = rank(B) − rank(AB) . -#£‘ nþª=Œ –y Ø ª. „k˜ 'uÝ • ¯KŒ±^©¬Ý E|5?n (ë„ [22, þþ, 1 145–148 •]). ù ,•´éÐ ?n•{. ØL, • •ÏÖöN¬Ý ©¬ nØXÛ†‚5N nØïáé X, ·‚„´5}Ál‚5N Ý5•Äùa¯K. ~ 3.2.25. (ë„ [22, A ∈ Mm×n (K), B ∈ Mk×l (K), C ∈ Mm×l (K). ! þþ, 1 145 •·K 6.1].) A C 2- M = . K 0 B rank(A) + rank(B) ≤ rank(M ) . •y²þ¡Ø ª, •Ä •þ˜m K n+l ¥ IOÄ ε1 , · · · , εm , εm+1 , · · · , εm+k . - IOÄ e1 , · · · , en , en+1 , · · · , en+l Ú K m+k ¥ V1 := span(e1 , · · · , en ) = {(x1 , · · · , xn , 0, · · · , 0)T ∈ K n+l | xi ∈ K} , W1 := span(ε1 , · · · , εm ) = {(y1 , · · · , ym , 0, · · · , 0)T ∈ K m+k | yi ∈ K} . •ÄXe‚5N ¤ †ãL V1 _ (3.2.25.1) τ K n+l A ι T / K m+k ρ π Kl / W1 _ B / Kk Ù¥ † éu?¿8Ü Y X Y N Ú§ ?¿f8 X, ò?¿ x ∈ X N • x , ´òN x À• Y ¡• X Y g,¹\ (natural inclusion) (N ), ë„ (A.2.6) ˜ã"—. ƒ, ù l 1nÙ ‚5N 150 • A ´3kSÄ (e1 , · · · , en ) Ú (ε1 , · · · , εm ) eÝ L«• A ‚5N 9ÙÝ L« ,= A (e1 , · · · , en ) = (ε1 , · · · , εm ) · A . • B ´3IOÄeÝ L«• B ‚5N , =,     y1 y1  .  B    .  7−→ B ·  ..  .  .   .  ym ym • T ´3IOÄeÝ • M ‚5N , =,        xn+1 x1 x1   x1     .   .  A ·  .  + C ·  ..   .  .. . .   .      .  !      xn      xn  T x x A C    n+l n   .  7−→   · =   x  xn+1   xn+1 0 B   n+1       .   .  ..    .    . B ·  ..     .  xn+l xn+l xn+l • τ Ú ι þ•g,¹\N • π : K n+l → K l .   x1  .   .     .  xn+1    xn     7−→  .. . ρ ½Â•  x   .   n+1   .  xn+l  .   .  xn+l   z1  .   .     .  zm+1    zm  ρ  .   7−→  . . ½ÂÚ π aq, =,  z   .   m+1   .  zm+k  .   .  zm+k (ãL (3.2.25.1) †5žÖög1 y !) ŠâÚn 3.2.20, •3 π p Ñ ‚5N  (3.2.25.2) π 0 : Ker(T ) −→ Ker(B) ;  x1  .   .     .  xn+1    xn      7−→  ..  x   .   n+1   .  xn+l  .   .  xn+l … τ (Ker(A )) ⊆ Ker(T ). ½=, Š• K n+l f˜m Ker(A ) ⊆ Ker(T ). e5, ØJ y Ker(A ) = Ker(π 0 ). Ïd, é‚5N π 0 ¦^•–"zÝ½nŒ (3.2.25.3) dim Ker(T ) = dim Ker(A ) + dim Im(π 0 ) ≤ dim Ker(A ) + dim Ker(B) . ,˜•¡, é T, A , B ¦^¦^•–"zÝ½n (Ú·K 3.2.21) Œ• dim Ker(T ) = n + l − rank(M ) , (3.2.25.4) dim Ker(A ) = n − rank(A) , dim Ker(B) = l − rank(B) . nÜ (3.2.25.3) Ú (3.2.25.4) =Œy²–y Ø ª. 3.2 ‚5N Ý L« 151 g • K 3.9. (é' [22, þþ, 1 147 •, íØ 2].) 3~ 3.2.25 ¥?˜Úb ª¥ N π 0 •÷ , … rank(M ) = rank(A) + rank(B). 3.2.4 A 1÷•. y²: (3.2.25.2) SK S K 3.2.1. •Ä R4 ¥ •þ α1 = (1, 1, 0, 0) , α2 = (0, 0, 1, 1) , α3 = (1, 0, 0, 4) , α4 = (0, 0, 0, 2) . e1 = (1, 0, 0, 0), · · · , e4 = (0, 0, 0, 1) • R4 (Š•¢•þ˜m) IOÄ. y² α1 , α2 , α3 , α4 ¤ R4 ˜|Ä, ¿¦Ñz‡ ei 3kSÄ (α1 , α2 , α3 , α4 ) e •þ. ‹I ( ) S K 3.2.2. y²: 3n‘E•þ˜m C3 ¥, •þ| α1 = (2i, 1, 0), α2 = (2, −1, 1), α3 = (0, 1 + i, 1 − i) ¤ ˜ | Ä. 2 © O ¦ Ñ I O Ä ¥ ˆ ‡ • þ ε1 = (1, 0, 0), ε2 = (0, 1, 0), ε3 = (0, 0, 1) 3 k S Ä (α1 , α2 , α3 ) e ‹I. S K 3.2.3. 3•þ˜m K n ¥, ¦•þ α = (a1 , a2 , · · · , an ) 3kSÄ (α1 , α2 , · · · , αn ) e ¥ éz‡ j = 1, 2, · · · , n , αj = 1, 1, · · · , 1 , 0 , 0 , · · · , 0 . | {z } ‹I, Ù j‡ S K 3.2.4. a ∈ K. y²: gê ≤ n õ‘ª ¤ •þ˜m K[X]≤n ¥, •þ| 1, X + a , (X + a)2 , · · · , (X + a)n ¤˜|Ä. 2¦Ñ•þ f (X) = a0 + a1 X + · · · + an X n 3ù|kSÄe ‹I. S K 3.2.5. é±ez‡œ¹, ¦•þ˜m K 4 ¥ kSÄ (α1 , α2 , α3 , α4 ) LÞÝ , ¿¦•þ α = (1, −1, 1, −1) 3ùü|kSÄe ‹I: 1. α1 = (1, 0, 0, 0), α2 = (0, 1, 0, 0), α3 = (0, 0, 1, 0), α4 = (0, 0, 0, 1), β1 = (1, 1, 0, 0), β2 = (1, 0, 1, 0), β3 = (1, 0, 0, 1), β4 = (1, 1, 1, 1); 2. α1 = (1, 2, −1, 0), α2 = (1, −1, 1, 1), α3 = (−1, 2, 1, 1), α4 = (−1, −1, 0, 1), β1 = (2, 1, 0, 1), β2 = (0, 1, 2, 2), β3 = (−2, 1, 1, 2), β4 = (1, 3, 1, 2). S K 3.2.6. ½ÂN  f : K 4 −→ K 3 ; 1. y² f ´‚5N    x1 −x1 + x2 + 2x3 + x4 x   2   −2x2 + x3   7−→  . x3  −x1 − x2 + 3x3 + x4 x4 . 2. ©O‰Ñ Ker(f ) Ú Im(f ) ˜|Ä. kSÄ (β1 , β2 , β3 , β4 ) 1nÙ ‚5N 152 3. •Ä K 4 kSÄ B = (ε1 , ε2 , ε3 , ε4 ) Ú K 3 9ÙÝ L« kSÄ C = (η1 , η2 , η3 ), Ù¥ ε1 = (1, 0, 1, 1)T , ε2 = (0, 1, 0, 1)T , ε3 = (0, 0, 1, 0)T , ε4 = (0, 0, 2, 1)T , η1 = (1, 1, 1)T , η2 = (1, 0, −1)T , η3 = (0, 1, 0)T . ¦Ý MB, C (f ). S K 3.2.7. ½ÂN f : K 3 −→ K 4 ; 1. y² f ´‚5N . 2. ©O‰Ñ Ker(f ) Ú Im(f ) 3. •Ä K 3     x1 + x3 x1 −x + x + x   1   2 3 . x2  7−→   2x1 + x3  x3 x2 + 2x3 ˜|Ä. kSÄ B = (η1 , η2 , η3 ) Ú K 4 kSÄ C = (ε1 , ε2 , ε3 , ε4 ), Ù¥ η1 = (1, 1, 1)T , η2 = (1, 0, −1)T , η3 = (0, 1, 0)T , ε1 = (1, 0, 1, 1)T , ε2 = (0, 1, 0, 1)T , ε3 = (0, 0, 1, 0)T , ε4 = (0, 0, 2, 1)T . ¦Ý MB, C (f ). 1 −4 S K 3.2.8. - A = −1 4 ! . ½Â‚5N fA : M2 (K) −→ M2 (K) ; 1. ©O¦Ñ Ker(fA ) Ú Im(fA ) X 7−→ AX . ‘êÚ˜|Ä. 2. ¦ fA 3kSÄ E = (ε1 , ε2 , ε3 , ε4 ) e Ý ME (fA ), Ù¥ ! ! ! 1 0 0 1 1 0 ε1 = , ε2 = , ε3 = , ε4 = 0 0 0 0 0 1 S K 3.2.9. e Ý . A ´k•‘•þ˜m V þ 1. V = K 3 , A ‚5C†. ée 0 1 1 0 ! . ˆ‡œ¹, ¦ A 3•½ ½Â• A : (x, y, z)T 7−→ (2x − y, y + z, z)T . B = (e1 , e2 , e3 ) • K3 IOÄ. 2. V = K 3 , A ´÷v±e^‡ ‚5C†: A η1 = (−5, 0, 3)T , A η2 = (0, −1, 6)T , A η3 = (−5, −1, 9)T , Ù¥ η1 = (−1, 0, 2)T , η2 = (0, 1, 1)T , η3 = (3, −1, 6)T . B = (e1 , e2 , e3 ) • K3 IOÄ. kSÄ B 3.2 ‚5N Ý L« 153 3. V = K 3 , A ´÷v±e^‡ ‚5C†: A η1 = (−5, 0, 3)T , A η2 = (0, −1, 6)T , A η3 = (−5, −1, 9)T , Ù¥ η1 = (−1, 0, 2)T , η2 = (0, 1, 1)T , η3 = (3, −1, 6)T . B = (η1 , η2 , η3 ). 4. V = K[X]≤n (Ù¥ n ≥ 1), A : V → V ½Â• A : f (X) 7→ f (X+1)−f (X), B = (ε0 , · · · , εn ), Ù¥ X(X − 1)(X − 2) · · · (X − r + 1) ε0 = 1 , ε r = , r = 1, 2, · · · , n . r! S K 3.2.10. •Ä¢•þ˜m R2 þ ü‡ÝKC† A , B, Ù¥ A ´²1u†‚ U = {(x, y) | x + y = 0} Ý•†‚ W = {(x, y) | x − y = 0} ÝK, B ´²1u x ¶Ý• y ¶ ÝK. ¦ A , B Ú A B 3 R2 IOÄe Ý . S K 3.2.11. f˜m, Ù¥ α, β • ½ ¢ê, β 6= 0, V ´d±e 6 ‡¼ê ε1 , · · · , ε6 3 Map(R, R) ¥)¤ ε1 = eαx cos(βx) , ε2 = eαx sin(βx) , ε3 = xeαx cos(βx) , 1 1 ε4 = xeαx sin(βx) , ε5 = x2 eαx cos(βx) , ε6 = x2 eαx sin(βx) . 2 2 1. y² ε1 , · · · , ε6 ¤V ˜|Ä. D : V → V ; f 7→ f 0 ´ÏL¼ê¦ ½Â ‚5C†. ¦ D 3kSÄ B = (ε1 , · · · , ε6 ) e Ý . ! a b S K 3.2.12. A= . •Ä V = M2 (K) ¥ kSÄ E = (e1 , e2 , e3 , e4 ), Ù¥ c d 2. e1 = 1 0 ! 0 , e2 = 0 0 0 ! 1 , e3 = 0 0 1 0 0 ! , e4 = 0 0 0 1 ! . ½ÂN T : V −→ V ; y² T ´ V þ X 7−→ AX − XA . ‚5C†, ¿¦Ñ T 3kSÄ E e Ý . S K 3.2.13. V ´k•‘•þ˜m… dim V ≥ 2. y²: •3 S, T ∈ EndK (V ) ¦ T S ùp, ST L«EÜN V − →V − → V . (T S ¹Âaq½Â.) S K 3.2.14. A ∈ Mn (K). y²: XJ•3 ê r þk rank(Ak ) = rank(Ak+r ). êk¦ S K 3.2.15. T : V → W •k•‘•þ˜mƒm dim T (U ) ≥ dim U − dim V + rank(T ). ST 6= T S. rank(Ak ) = rank(Ak+1 ), @oéu¤k ‚5N , U ´ V f ˜ m. y ²: S K 3.2.16. A ´k•‘•þ˜m V þ Ker(A ) ⊕ Im(A ). ‚ 5 C †. b S K 3.2.17. A ´k•‘•þ˜m V þ A B = 0 … rank(A ) + rank(B) = dim V . ‚5C†. y²: •3‚5C† B ∈ End(V ), ¦ rank(A ) = rank(A 2 ). ¦ y: V = 1nÙ ‚5N 154 S K 3.2.18. V ´ K-•þ˜m, f, g ∈ Hom(V, K). b c ∈ K ¦ g = cf . 9ÙÝ L« Ker(f ) = Ker(g). ¦y: •3š"~ê S K 3.2.19. A ∈ Mm×n (K) Ú B ∈ Mn×s (K) ÷v AB = 0. y²: rank(A) + rank(B) ≤ n. S K 3.2.20. A1 , · · · , Ar ∈ Mn (K) ÷v A1 A2 · · · Ar = 0. y²: rank(A1 ) + rank(A2 ) + · · · + rank(Ar ) ≤ (r − 1)n . 3.3 eškƒ‡ 3.3.1 ‚5N ÄC††Ý (², 3 !¥·‚E Ý ÄC†úª ƒq V Ú W Ñ´k•‘ K-•þ˜m. ·‚®²• , éu˜‡‚5N T : V → W , z ½ V ÚW kSÄ B, C ± , T k ƒA Ý L« MB, C (T ). ØL, éuÓ ‚5N T , XJ·‚òkSÄ B, C •†•Ù¦ k 0 0 0 0 SÄ B , C , @oÝ MB , C (T ) Ú MB, C (T ) ˜„5`ÒØ2ƒÓ ('X~ 3.2.12 ¥ð N 3 ØÓÄeŒ±kØÓ Ý L«). ·‚y3‡ïÄ MB0 , C 0 (T ) Ú MB, C (T ) ƒm ƒp'X. ·K 3.3.1: B, B 0 •k•‘•þ˜m V Ä. P = PB, B0 Ú Q = PC, C 0 ©O´l B ü|kSÄ, C, C 0 •k•‘•þ˜m W B0 Ú C C 0 LÞÝ . ü|kS Kéu?¿ T ∈ Hom(V, W ), (3.3.1.1) MB0 , C 0 (T ) = Q−1 · MB, C (T ) · P = PC 0 , C · MB, C (T ) · PB, B0 . 0 y². B = (v1 , · · · , vn ), B 0 = (v10 , · · · , vn0 ), C = (η1 , · · · , ηm ) 9 C 0 = (η10 , · · · , ηm ). Šâk'½Â, ˜•¡, 0 T (v10 , · · · , vn0 ) = (η10 , · · · , ηm ) · MB0 , C 0 (T ) = (η1 , · · · , ηm )Q · MB0 , C 0 (T ) = (η1 , · · · , ηm ) Q · MB0 , C 0 (T ) . ,˜•¡, T (v10 , · · · , vn0 ) = T (v1 , · · · , vn ) · P (Šâ (3.2.17.2)) = T (v1 , · · · , vn ) · P = (η1 , · · · , ηm )MB, C (T ) · P = (η1 , · · · , ηm ) MB, C (T ) · P . nÜ±þOŽ(JŒ• Q · MB0 , C 0 (T ) = MB, C (T ) · P , ù† (3.3.1.1) ¥1˜‡ ª´ d Q = PC, C 0 ´l C C 0 LÞÝ ž, Ún 3.2.5 QwŠ·‚ Q−1 Ò´l C 0 C LÞÝ ùÒ´ (3.3.1.1) ¥1 ‡ ª. . PC 0 , C . 5P 3.3.2. ·‚ØïÆÖö‚kPM •{P4 (3.3.1.1) ¥ úª, ´ïÆÖö¿©ÙGín L§ , 3I‡ž‘ž yù‡úª. ¯¢þ, (3.3.1.1) ¥ úªÚ½n 3.2.6 ¥ ‹IC†úª± 3.3 ÄC††Ý ƒq 9 (3.2.9.6) ¥ †ãLŒ±éXå5. ù«éXŒ±V)•e¡˜‡ 155 MB0 , C0 (T )• K na (3.3.2.1) / Km < MB 0 MC 0 V PB, B0 • } Kn †ãL: T /W PC, C0 • MB MC MB, C (T )• " / Km ùp, ãL¥ ü‡n / †5•â ´½n 3.2.6, ü‡F/ †5•â´ (3.2.9.6). ù †5Œ±íÑ (3.3.2.1) ù‡ãL¥ Œ••/ †5, ù‡ †5 ŸþÒ´ (3.3.1.1) ª N. ù•Ò´þ¡·‚y²·K 3.3.1 g´, •ØLy3†^ †ãL Šó£ãv . S £ ˜e·‚ùL Ý ƒ- Vg (ë„·K 1.2.33): ü‡Ý •3•3Œ_Ý Q ∈ Mm (K) Ú P ∈ Mn (K) ¦ B = Q−1 AP . A, B ∈ Mm×n (K) ƒ-´• ·K 3.3.3: B, C ©O• n ‘ K-•þ˜m V Ú m ‘•þ˜m W N , M = MB, C (T ). Kéu?¿ M 0 ∈ Mm×n (K), ±e•ã d: kSÄ, T : V → W •‚5 1. Ý M Ú M 0 ƒ-. 2. •3 V kSÄ B 0 Ú W kSÄ C 0 ¦ M 0 = MB0 , C 0 (T ). y². (2)⇒(1). † l (3.3.1.1) ª=Œ (Ø. 0 (1)⇒(2). b M † M ƒ-. K•3Œ_Ý P ∈ Mn (K) Ú Q ∈ Mm (K) ¦ M 0 = Q−1 M P . ©O- B 0 = B · P Ú C 0 = C · Q Œ± V ÚW kSÄ B 0 , C 0 (Ún 3.2.3). Šâ (3.3.1.1) ªŒ •, T 3kSÄ B 0 , C 0 e Ý Ò´ M 0 . g • K 3.10. T : V → W •k•‘•þ˜mƒm ÄBÚW kSÄ C ¦ MB, C (T ) = c¡?Ø ˜„ ±eíØ: íØ 3.3.4 (‚5C† C†. K ‚5N ‚5N Ir 0 kS ! 0 . 0 œ¹. XJ·‚•Ä˜‡•þ˜m V ‚5C†, Œ±l·K 3.3.1 B, B 0 •k•‘•þ˜m V kSÄ, A ∈ End(V ) •‚5 ÄC†úª): −1 MB0 (A ) = PB, B0 MB (A )PB, B0 . (3.3.4.1) •Ò´`, XJ P = PB, B0 ´l B ÎkSÄe Ý , @o§‚ƒm B 0 LÞÝ 'Xª• , A0 = MB0 (A ) Ú A = MB (A ) ©O´ A 3# A0 = P −1 AP . y². 3·K 3.3.1 ¥ , r = rank(T ). y²: •3 V C = B, C 0 = B 0 =Œ. ,šÍ¶˜<•íØ 3.3.4 ¥úªKü : 1nÙ ‚5N 156 9ÙÝ L« •þ˜mÀÐÄ, LÞÝ ·¶ P. ÄC†ž#Ý , P _†¦ A ¦ P. ~ 3.3.5. IOÄ, B 0 = (η1 , η2 , η3 ) •±e•þ|¤ B = (e1 , e2 , e3 ) • K 3 kSÄ η1 = (1, 1, 1)T , η2 = (1, 1, 0) , η3 = (1, 0, 0) . •Ä÷v A (e1 ) = (−1, 1, 0)T , A (e2 ) = (2, 1, 1)T , A (e3 ) = (0, −1, −1)T •˜‚5C† A : K 3 → K 3 .  K3kSÄ B e A  0  1 0 1 1 −1 Ý −1  • A = 1 0  2 0  1 −1, 1 −1 3 k S Ä B0 e A Ý ª A0 = P −1 AP (¤á. • A0 =  0  1 . −2  B0 lB LÞÝ 1  • P = 1 1 y3·‚5`²: ‚5C†  1  0. ÖöŒ± 0 1 1 0 y ÄC†úª¢SþéA• ƒm ˜«-‡ d'X. ½ Â 3.3.6. ü ‡ • A, B ∈ Mn (K) ¡ • 3 K-þ ƒ q (similar over K), X J • 3 Œ _ Ý −1 P ∈ Mn (K) ¦ B = P AP . “3 R þ (½ C þ) ƒq”Ï~ ¡•“¢ƒq (½Eƒq)”. Ï~r“3 K þƒq”{¡•“ƒq”. A † B ƒqž, k žÖö Ö (~X [22]) æ^PÒ A ∼ B 5L«. y: þãƒq'X´8Ü Mn (K) þ w,, XJ L ´ K ˜‡*•, K Ý ž) •3 L þƒq.‡ ·K 3.3.7: B • n ‘ K-•þ˜m V Kéu?¿ B ∈ Mn (K), ±e•ã 1. Ý K 3þe©¥g,² Ý ž, ·‚ d'X. A, B ∈ Mn (K) 3 K þƒqž, §‚ (À• Mn (L) ¥ kSÄ, A : V → V •‚5C†, A = MB (A ). d: A Ú B ƒq. 2. •3 V kSÄ B 0 ¦ B = MB0 (A ). y². 3‰ÖööS. ‡ ¢Sþ, XJ A, B ∈ M y²I‡^ ••q n (K) 3˜‡*• L þƒq, K§‚•˜½3 K þƒq. ù‡¯¢ Ý knIO/ (rational canonical form) nØ. k, ÖöŒ±ëw [16, 1 182 •, ½n7.20] ½ö [35, 1 290 •, 1 7 ÙSK 41]. 3.3 ÄC††Ý 3.3.2 A ƒq Š!A 157 •þÚÝ é z é Ý w,´/ª•{ü ãÝ . ÏL Ùc¡ ?Ø, ·‚• éu˜‡‚5C† A ∈ End(V ), ÏLÀ V kSÄŒ±ò A éA ˜‡• , C†Ä À ŒU¬¦Ó˜‡‚ 5C†éA Ý )Cz. ¤±˜‡g, ¯K´, éu‰½ ‚5C† A , UÄÏL· À kSÄ, ¦ A Ý L«äké /ª ? ! Ì‡8IÒ´ éù‡¯KÚ\ƒ' Vg, , ?1˜ ÐÚ ?Ø. ½ Â 3.3.8. V ´ n ‘ K-•þ˜m, A ∈ End(V ). XJ•3 V kSÄ B ¦ Ý MB (A ) ´ é Ý , ·‚Ò`‚5C† A Œ±é z (diagonalizable), {¡Œé z. éu• A ∈ Mn (K), XJ A 3 K þƒqu Mn (K) ¥,‡é Ý , ½=, •3Œ_Ý P ∈ Mn (K) ¦ P −1 AP ´é , @o·‚¡Ý A 3 K þŒ±é z, {¡Œé z. XJò n n A À•‚5C† fA : K → K ; X 7−→ AX 3IOÄe Ý , @oŠâ·K 3.3.7, Ý A Œé zÚ‚5C† fA Œé z´ d . (3.3.9) b V ´ n ‘ K-•þ˜m, A ∈ End(V ). XJ A Œé z, @oŠâ½Â, •3 V kSÄ B = (v1 , · · · , vn ) ¦ MB (A ) •é ù¿›X: •3~ê λ1 , · · · , λn ∈ K ¦   λ1   ..  = (λ1 v1 , · · · , λn vn ) . A (v1 · · · , vn ) = (v1 , · · · , vn ) ·  .   λn . •Ò´`, •þ| v1 , · · · , vn ∈ V Ú~ê| λ1 , · · · , λn ∈ K U ÷v A vi := A (vi ) = λi vi (éz‡ i ∈ [[1, n]] ¤á). ‡L5, XJ•3‚5Ã' •þ| v1 , · · · , vn ∈ V ±9~ê| λ1 , · · · , λn ∈ K ¦ éuz‡ i ∈ [[1, n]] , @o , vn )  B = (v1 , · · ·  λ1   .. .  .   λn ¤V kSÄ, þk A vi = λi vi , …Uì½ÂŒ• A 3ù|Äe Ý MB (A ) Ò´é ½ Â 3.3.10. A • (k•½Ã•‘) K-•þ˜m U þ ‚5C†. XJ v ∈ U ´š"•þ, …éu,‡ λ ∈ K k A v = λv, @o·‚` v ´ A 3 K þ ˜‡A •þ (eigenvector), λ Á •þ v éA A Š (eigenvalue). ‡L5, XJk•Ä˜‡~ê λ ∈ K, •3š"•þ v ∈ U ¦ A v = λv, @o·‚` λ ´ A ˜‡A Š, v ´áuA Š λ A •þ. ØJwÑ, v ?¿š"~ê E´ÓÃ Š λ A •þ. éuÝ A ∈ Mn (K), ‚5C† fA : K n → K n ; X 7→ AX A Š (A •þ) •¡• A 3 K ¥ A Š (3 K þ A •þ). k Öò eigenvector Ú eigenvalue ©O€È• •þÚ Š. ˜„ ó, ‚5C† A ŠÚA •þŒU´Ø•3 . ØL, A ŠÚA •þ p•6 . •Ò´`, Šâ½Â, A ŠÚA •þ´Óž•3½öÓžØ•3 . ½Â´ƒ g • K 3.11. žÖöy²: †‰½A •þéA A Š•k•˜˜‡. 2Þ~`²: éuÓ˜‡ A Š λ, ŒUk‚5Ã' ü‡•þ v1 , v2 Ñ´áu λ A •þ. 158 1nÙ ‚5N ½n 3.3.11: V • n ‘ K-•þ˜m, A ∈ End(V ). K A Œé z …= •3 n ‡‚5Ã' •þÑ´ A ¿©7‡^‡´ V k˜|Ä dA A •þ ¤. A y². ù•ØL´¦^½Â 3.3.10 ¥ 9ÙÝ •þ. †óƒ, A Œé Vg-#Lã˜e (3.3.9) ˜ã¥?ØÑ L« z (Ø. éu‰½ ‚5C† A ∈ End(V ), XJ w•,‡•þ v ∈ V ´ A A •þ, @oOŽ A v ƒ Ú v ' ˜e'~'XÒŒ± • v éA A Š. ‡L5, XJ®²• ,~ê λ ∈ K Á ˜‡A Š, @o¦Ñáu λ A •þ¿Ø(J. ½ Â 3.3.12. ½Â• A ´ K-•þ˜m U þ ‚5C†, λ ∈ K. ò A 'u λ A f˜m (eigenspace) E(λ, A ) := {v ∈ U | A v = λv} = Ker(λI − A ) , Ù¥ I L«ð C†. k aq/, éu˜‡Ý Ö (~X [22]) ò E(λ, A ) P• Uλ . A ∈ Mn (K), ¡ E(λ, A) := {x ∈ K n×1 | Ax = λx} = N (λIn − A) • A 3 K n×1 ¥'u λ A Td λ A f˜m. f˜m E(λ, A ) •¹š"•þž, Šâ½ÂŒ• λ ´ A ˜‡A ¤kA •þ†"•þ|¤. (5¿: A •þo´•š"•þ !) Ún 3.3.13: 1. λ ´ A A ´ n ‘ K-•þ˜m V þ ˜‡A ‚5C†, λ ∈ K. Ke •ã Š. dž, E(λ, A ) d: Š. 2. E(λ, A ) 6= 0. 3. rank(λI − A ) < n = dim V . 4. ‚5C† λI − A ØŒ_. 5. éu V ?¿kSÄ B = (v1 , · · · , vn ), A 3 B e 6. •3 V ˜|kSÄ B = (v1 , · · · , vn ), ¦Ý A = MB (A ) ÷v rank(λIn − A) < n. A = MB (A ) ÷v rank(λIn − A) < n. 7. éu V ?¿kSÄ B = (v1 , · · · , vn ), A 3 B e =‚5•§| (λIn − A)X = 0 kš"). 8. •3 V ˜|kSÄ B = (v1 , · · · , vn ), ¦Ý §| (λIn − A)X = 0 kš"). Ý Ý A = MB (A ) ÷v N (λIn − A) 6= 0, A = MB (A ) ÷v N (λIn − A) 6= 0, =‚5• y². (1) Ú (2) d5dƒ' ½Â=•. (3) Ú (4) d5´Šâ·K 3.1.31 ¥Øä (1) Ú (3) d5. Ù¦I‡ d5Øä´•–"zÝ½n {üíØ. (3.3.14) Ún 3.3.13 •·‚Jø OŽA ŠÚA •þ ˜«äN•Y: ÏLÀ ˜|kSÄ B = (v1 , · · · , vn ), OŽ A Ý A = MB (A ), , • Ÿo ~ê λ ∈ K U¦ rank(λIn − A) < n. dd=ŒéÑ A ¤kA Š. éuz‡A ŠŒ±2?˜ÚŽÑéA A f˜m, ùÒƒ u¦Ñ A ¤kA •þ.   1 2 2   ·‚^þã•{¦Ý A = 2 1 2 A ŠÚA f˜m. 2 2 1 3.3 ÄC††Ý ƒq 159 é λI − A ‰Ð 1C†       2(λ − 1) −4 −4 λ−1 −2 −2 λ−1 −2 −2       λ−1 −2  λ−1 −2  −→  −2 λ−1 −2  −→  −2  −2 0 −λ − 1 λ + 1 0 −λ − 1 λ + 1 −2 −2 λ−1   0 −4 + (λ − 1)2 −4 − 2(λ − 1)   −→ −2 λ−1 −2  0 −λ − 1 λ+1     −2 λ−1 −2 −2 λ−1 −2     −→  0 λ2 − 2λ − 3 −2(λ + 1) =  0 (λ + 1)(λ − 3) −2(λ + 1) 0 −(λ + 1) λ+1 0 −(λ + 1) λ+1 y3w,Œ±wÑ λ = −1 ž rank(λI − A) = 1 < 3. Ïd λ1 = −1 ´ A ˜‡A Š. b λ 6= −1. K±þÐ C†Œ±UY?1     −2 λ−1 −2 λ−1 −2 −2     λ−1 −2  −→ · · · −→  0 (λ + 1)(λ − 3) −2(λ + 1)  −2 0 −(λ + 1) λ+1 −2 −2 λ−1     −2 λ − 1 −2 −2 λ − 1 −2     −→  0 λ − 3 −2 −→  0 0 λ − 5 0 −1 1 0 −1 1   −2 λ − 1 −2   −→  0 −1 1  0 0 λ−5 ddŒ„ λ2 = 5 ´ A , e5ÖöŒ± y: ˜‡A Š, Ù¦?Û (ØÓu −1 Ú 5) E(λ1 , A) = span (−1, 1, 0)T , (−1, 0, 1)T , A A A Š. E(λ2 , A) = span (1, 1, 1)T . •þTÐ´8Ü E(λ1 , A) ∪ E(λ2 , A) \ {0} ƒ¥ g • K 3.12. y²: XJ A = (aij ) ∈ Mn (K) ´þn A ¤kA Š. 2žÖög•: þn / • ´Ä˜½Œ±é ¯K ‰Y.) ~êÑØ´ A •þ. , @o A é zQ ? (¯¢þ• ‚ ƒ a11 , · · · , ann ‰Ñ Ý =Œé ù‡ (3.3.15) U Y • (3.3.14) ˜ ã ? Ø ~ f. Ø J y, 3 E1 := E(λ1 , A) ¥ Œ ± η1 = T T T (−1, 1, 0) , η2 = (−1, 0, 1) • ˜ | Ä, 3 E2 := E(λ2 , A) ¥ Œ ± η3 = (1, 1, 1) • ˜ | Ä, 3 … C = (η1 , η2 , η3 ) ¤ K ˜|kSÄ. Šâ½n 3.3.11, Ý A ´Œé z . @o, XÛäNéÑ˜‡Œ_Ý P ¦ P −1 AP •é Q?   −1 −1 1   ¯¢þ, N´wÑlIOÄ B = (e1 , e2 , e3 ) C LÞÝ • P =  1 0 1, ‚5C 0 1 1     λ1 −1     † fA : K 3 → K 3 3 C e Ý • D :=  λ1 −1 . = λ2 5 1nÙ ‚5N 160 9ÙÝ L« ŠâÄC†úª (íØ 3.3.4), ½ö† ÏLOŽ y, Œ±• D = P −1 AP . ùÒ´`, ¦^ IOÄ (A •þ ¤ ) kSÄ C LÞÝ Œ±äN¢y˜«ò A é z •ª. ±þù«•{w,Œ±í2 ˜„œ/. =, éu?¿Ý A ∈ Mn (K), XJU é A n ‡‚5Ã' A •þ v1 , · · · , vn (§‚ •35´ A Œé z ¿©7‡^‡), z‡ vi éA A Š• λ λi Ø˜½üüpÉ), @o‚5C† fA 3kSÄ (v1 , · · · , vn ) e Ý Ò´é i (ùp  λ1   ..  . XJ P ´lIOÄ (v1 , · · · , vn ) e LÞÝ , @oÄC†úªL² D :=  .  λn P −1 AP = D. ùÄ þ´<óò˜‡Ý é z •~^öŠ•{, ÏdF"Öö˜½@ýN¬¿ ÏL·þ öS5ÙöÝºù«Ž{. Ý zkžÿ¬k Û A^. ·‚5Þü‡•~„ ~f.   2 0 0   ~ 3.3.16. A = 1 2 −1. é?¿ n ∈ N∗ , ¦ An . 1 0 1 ÏLc¡0 •{, Œ±¦Ñ A ¤kA Š: λ1 = 2, λ2 = 1. éƒA ˜|Ä: E(λ1 , A) = span(η1 , η2 ), E(λ2 , A) = span(η3 ), Ù¥ lIOÄ é A f˜m©Oé η1 = (0, 1, 0)T , η2 = (1, 0, 1)T , η3 = (0, 1, 1)T .    0 1 0 2    −1 kSÄ (η1 , η2 , η3 ) LÞÝ • P = 1 0 1, P AP = D :=  0 1 1   . 2 1 u´ An = (P DP −1 )n = (P DP −1 )(P DP −1 ) · · · (P DP −1 ) = P D(P −1 P )D(P −1 P ) · · · (P −1 P )DP −1 = P Dn P −1     2m 0 0 2n    −1  n =P 2n  P = 2 − 1 2n 1 − 2n  . 2n − 1 0 1 1 ùp, é ¤•ŒU. D ?¿g˜éNÓŽ, ~ 3.3.17. Í¶ Fibonacci ê é zL§ P −1 AP = D ¦ ´•÷vÐ©^‡ a0 = a1 = 1 ÚXe4íúª an+1 = an + an−1 vn = Avn−1 , ê an , n ≥ 0: Ù¥ A = 1 1 4íúª‰Ñ (vn )n≥0 ù‡S 1 0 ! . l (3.3.17.1) N´wÑ (3.3.17.2) ?¿gÕŽ• é?¿ n ≥ 1 . ·‚y3^‚5“ê •{¦ÑTê Ï‘úª. •d, •Ä R2 ¥ •þS vn = (an+1 , an )T , n ≥ 0. ê Ý /ª4íúª: (3.3.17.1) A an+1 a1 1 = v n = An = An . an a0 1 3.3 ÄC††Ý ƒq OŽ A ŠŒ± ÄŒ±é A Ý P = λ1 1 161 λ2 1 1 !λ1 = 2 (1 + λ1 0 λn+1 − λn+1 1 2 λn1 − λn2 1 A =√ 5 ò (3.3.17.3) ª“\ 5) Ú λ2 = 21 (1 !− P −1 AP = ¦ n (3.3.17.3) √ √ 5) ü‡A 0 . , λ2 Š. ÏL¦A Œ±Uì~ 3.3.16 ¥ λ1 λn+1 − λ2 λn+1 2 1 λ1 λn2 − λ2 λn1 f˜m •{ŽÑ ! . (3.3.17.2) =Œ¦Ñ 1 1 − λn+1 an = √ λn+1 =√ 1 2 5 5 ! √ √ (1 − 5)n+1 (1 + 5)n+1 − 2n+1 2n+1 é?¿ n ∈ N ¤á. e¡·‚20 ˜«ÏLA f˜m5 O‚5C† (½• ) ´ÄŒ±é z •{. 3· ‚•• ˜ nØþ 5Ÿ£ã! Ø• ‚5C† (½• ) äNLˆª (ÏdÃ{¢–äN êŠOŽ) žÿ, ù«•{ Ò¬NyÑAÏ dŠ. Ún 3.3.18: A •,•þ˜mþ Oáu λi A •þ. K•þ| v1 , · · · , vr ‚5Ã'. ‚5C†, λ1 , · · · , λr ´ A ØÓA Š, vi , 1 ≤ i ≤ r ´© y². é•þêþ r ‰8B{. r = 1 ž(Øw,¤á (Ï•A •þÑ´š"•þ). b ®²• ?¿ r − 1 ‡áuØÓA Š A •þ˜½‚5Ã'. XJ v1 , · · · , vr ‚5Ã ', Ø” vr Œ± v1 , · · · , vr−1 ‚5LÑ. =, •3~ê a1 , · · · , ar−1 ∈ K ¦ vr = a1 v1 + · · · + ar−1 vr−1 . (3.3.18.1) K±‚5C† λr I − A Šû (3.3.18.1) ªü>Œ± Ñ (3.3.18.2) 0 = (λr − λ1 )a1 v1 + · · · + (λr − λr−1 )ar−1 vr−1 . Ï•8Bb 0. =, ®²wŠ·‚ v1 , · · · , vr−1 ‚5Ã', ¤± (3.3.18.2) ª¥m>‚5|Ü ¤kXê• (λr − λ1 )a1 = · · · = (λr − λr−1 )ar−1 = 0 . ´ Ï • λ1 , · · · , λr ´ p É , ¤ ± d þ ª Œ a1 = · · · = ar−1 = 0. “ £ (3.3.18.1) ª Œ vr = 0. ù † vr ´ A • þ (Ï ´ š " • þ) g ñ. – d, · ‚ Ï L 8 B { y ² Ú n ( Ø. íØ 3.3.19: Š, K A Œé A ´ n ‘ K-•þ˜m V þ z. ‚5C†. XJ A (3 K S)k n ‡üüpÉ y². • A n ‡pÉA Š©OÀ ˜‡A •þ. dd qÏ•T•þ|¤¹•þê8 u dim V , ¤±T•þ| ¤ V A Œé z. · K 3.3.20: A ´ k • ‘ K-• þ ˜ m V þ Vi = E(λi , A ) •ƒA A f˜m. Kf˜m Ú V1 + · · · + Vr ´†Ú. A •þ|‚5Ã' (Ún 3.3.18). ˜|Ä. u´d½n 3.3.11 =• ‚ 5 C †, λ1 , · · · , λr ∈ K ´ ü ü p É ~ ê, 1nÙ ‚5N 162 9ÙÝ L« y². Šâ·K 2.3.34, •I y"•þ 0 ¤ 0 = v1 + · · · + vr ž, XJz‡ vi áu Vi , K7L v1 = · · · = vr = 0. ¯¢þ, XJ v1 , · · · , vr Ø •", Ø” v1 , · · · , vt þØ• 0, vt+1 = · · · = vr = 0, Ù¥ t ≥ 1. Kd ª 0 = v1 + · · · + vr Œ• v1 + · · · + vt = 0. ,˜•¡, v1 , · · · , vt ´ A áuØ ÓA Š A •þ, Ïd§‚‚5Ã' (Ún 3.3.18). ù† v1 + · · · + vt = 0 gñ. ½n 3.3.21: A ´k•‘ K-•þ˜m V þ ‚5C†, λ1 , · · · , λr ∈ K ´ A 3 K þ¤kØÓ A Š, Vi = E(λi , A ) •ƒA A f˜m. Pr K A Œé z …= V = V1 ⊕ · · · ⊕ Vr , (Šâ·K 2.3.34) • …= i=1 dim Vi = dim V . d^‡¤áž, ? Vi ˜|Ä vi1 , · · · , vini , Ü¿ V ˜|kSÄ (ë„íØ 2.3.36) B = v11 , · · · , v1n1 ; v21 , · · · , v2n2 ; · · · ; vr1 , · · · , vrnr , K A 3±þkSÄe Ý •é . y². e V = V1 ⊕ · · · ⊕ Vr , Kéu½n•ã¥À kSÄ B, †   λ1 In1   ..  MB (A ) =  .   λr Inr yŒ• Ïd A Œé z. ‡ƒ, e A Œé z, KŠâ½n 3.3.11 Œ••3 V Ä ε1 , · · · , εn ¦ z‡ εj , 1 ≤ j ≤ n Ñ Á A •þ. Šâb , V1 , · · · , Vr ´ A ¤kA f˜m, Ïdz‡ εj Ñáu V1 , · · · , Vr ¥ ,˜‡. Ïd, z‡ εj Ñáu V1 + · · · + Vr . ¤± V = span(ε1 , · · · , εn ) •¹u V1 + · · · + Vr . ·K 3.3.20 ®²`² V1 + · · · + Vr ´†Ú. ¤±k V = V1 ⊕ · · · ⊕ Vr . ~ 3.3.22. A ´k•‘•þ˜m V þ ‚5C†, … A 2 = I (ùp I L«ð C†). K A ˜ ½Œ±é z. •y²ù‡Øä, 5¿ A 2 = I du (A − I)(A + I) = 0. XJ A + I Œ_, KddíÑ A − I = 0, A = I w,Œ±é z. aq/, XJ A − I Œ_, Kd (A − I)(A + I) = 0 Œ• A + I = 0, l A = −I ´Œé z . ±e·‚b A + I Ú A − I þØŒ_. u´ rank(A + I) < dim V , rank(A − I) < dim V (ž Öög••Ÿo). ¤±, Šâ 3.3.13 Œ• λ1 = 1 Ú λ2 = −1 Ñ´ A A Š. ¯¢þ, ù´ A = k ü‡A Š. Ï•XJ λ ´ A A Š, v ´áu λ ˜‡A •þ, K v = Iv = A (A v) = A (λv) = λA v = λ(λv) = λ2 v . λ2 = 1. e- V1 = E(λ1 , A ) Ú V2 = E(λ2 , A ), KÏ• V1 + V2 ´†Ú, dim V1 + dim V2 = dim(V1 + V2 ) ≤ dim V . (3.3.22.1) ,˜•¡, Sylvester •Ø ª (~ 3.2.23) 3 cœ¹eíÑ rank(A − I) + rank(A + I) ≤ dim V + rank((A − I)(A + I)) = dim V + rank(0) = dim V . ÏL•–"zÝ½nqŒ±òþ¡Ø ª=z• dim V − dim Ker(A − I) + dim V − dim Ker(A + I) ≤ dim V . ùÒ`² (3.3.22.2) dim V1 + dim V2 = dim Ker(A − I) + dim Ker(A + I) ≥ n . (Ü (3.3.22.1) Ú (3.3.22.2) =• dim V = dim V1 + dim V2 . ¤±, V = V1 ⊕ V2 = E(λ1 , A ) ⊕ E(λ2 , A ). Šâ½n 3.3.21, A Œé z. 3.3 ÄC††Ý 3.3.3 ƒq 163 ØCf˜m!Oé zÚþn z ½n 3.3.21 wŠ·‚, XJk•‘•þ˜m V þk˜‡Œé z ‚5C† A , @o V Œ± ©)• A A f˜mƒ†Ú, ¦^ù †Ú©)Œ±¦‚5C† A C éN´ï Ä. @o, éu˜„ ‚5C† A ∈ End(V ), XJ§ØUé z, kvkŒU2ò˜m V ©)•, f˜m †Ú, ±•BïÄ‚5C† A Q ? ‡£‰ù‡¯K·‚I‡Ú\Xe½Â. c¡ ½ Â 3.3.23. A ´ (k•½Ã•‘) •þ˜m U þ ‚5C†. XJ M ´ U f˜m, ¿… A (M ) ⊆ M (=, éu?¿ v ∈ M þk A v ∈ M ), ·‚¡ M ´ A ˜‡ØCf˜m (invariant subspace). džÏLò A •›3f˜m M þ½Â, ¿…@•”8•¹3 M ƒS, @oŒ±@• A p Ñf˜m M g ˜‡‚5C† A |M : M −→ M , ¡• A 3 M þ v 7−→ A v , •› (restriction)§ . ~ 3.3.24. U • K-•þ˜m, A ∈ End(U ). (1) "•þ˜m 0 Ú U w,´ A ØCf˜m. ¡§‚• A ²… (trivial) ØCf˜m. (2) žÖö y, Ker(A ) Ú Im(A ) •´ A ØCf˜m. (3) XJ W ⊆ U ´ A ˜‡ 1 ‘ØCf˜m, K?Û)¤ W •þ v ´ A A •þ (ž Öög••Ÿo). ‡L5, XJ v ∈ U ´ A ˜‡A •þ, K W = span(v) ´ A ˜‡ 1 ‘Ø Cf˜m. òù‡¯¢†½n 3.3.11 (Üå5, ·‚Ò• : ˜‡k•‘•þ˜m V þ ‚5C† A Œ é z …= V Œ±©)• A 1 ‘ØCf˜m †Ú. (4) e λ ´ A ˜‡A Š, E = E(λ, A ) ´éA A f˜m, K E ´ A ØCf˜m. XJæ^ØCf˜m *:, kžŒ±é•B/lAÛ Ýn)A ŠÚA •þ. ~ 3.3.25. •Ä²¡ R2 S7 :^= θ ^=C† A : R2 → R2 (ë„~ 3.1.9). XJ θ ∈ / πZ := {πk | k ∈ Z}, K A vk 1 ‘ØCf˜m, Ïd A (3 R ¥) vkA Š. ~ 3.3.26. ½²¡ R2 ˜‡ 1 ‘f˜m L. é?¿˜: P ∈ R2 , P 0 ´Ó P 'u†‚ L é¡ 0 2 @‡:. ÏL P 7→ P ù˜éA'XŒ± R g ˜‡C† A . ØJ y, A ´‡‚5C†, ¡•'u L ‡ (reflection) ½º”é¡ (mirror symmetry). dž, L ÚL :† L R† @^† ‚ L0 ‰Ñ A ü‡ØCf˜m. Ïd A kü‡‚5Ã' A •þ. (žÖög•: §‚éA A Š©O´Ÿo ?) g • K 3.13. 3~ 3.3.26 ¥b †‚ L •§• y = ax. ¦þã‡ A. 2¦Œ_Ý P ∈ M2 (R) ¦ P −1 AP •é . ïÄØCf˜m ·K¤«. ·K 3.3.27: Ke •ã 1. •3 V Ì‡¿Â3uBun)‚5C† V ´k•‘ K-•þ˜m, A ∈ End(V ). d: ˜|kSÄ¦ A 3ù|Äe Ý Oé C† A 3IOÄe z (quasi-diagonalization), X±e n1 , · · · , nr ∈ N ÷v dim V = n1 +· · ·+nr .  A1   ´/X     A2 ..      . Ar ¥z‡f• § d?½Â •› Ai • ni . ,´ (A.2.8) ˜ã¥˜„œ¹ ˜‡A~. Ý Oé ,Ù 1nÙ ‚5N 164 2. V Œ±©)• A Mi ‘ê´ ni . y². b ˜ ε1 , · · · , εn ´ V ØCf˜m M1 , · · · , Mr ˜|kSÄ¦ 9ÙÝ L« †Ú: V = M1 ⊕ · · · ⊕ Mr , Ù¥z‡f˜m A 3ù|Äe Ý •Oé  A1   /    A2 ..   .   . Ar ·‚Œ±ò ε1 , · · · , εn ù|kSÄ¥ •þ©¤ r |: ε11 , · · · , ε1n1 ; ε21 , · · · , ε2n2 ; · · · ; εr1 , · · · , εrnr . z˜|S •þ)¤f˜m Mi = span(εi1 , · · · , εini ). K A (εi1 , · · · , εini ) = (εi1 , · · · , εini )Ai . Ïd Mi Ñ´ A ØCf˜m, …N´ y V = M1 ⊕ · · · ⊕ Mr (ë„íØ 2.3.36). ‡L5, XJ®• V k†Ú©) V = M1 ⊕ · · · ⊕ Mr , Ù¥z‡ Mi Ñ´ A ØCf˜m. @o z‡ Mi Ä εi1 , · · · , εini , 2ò§‚•gü ¤ V ˜|kSÄ. K† yŒ•, A 3ù|Ä e Ý Óé . ¯¢þ ué ‚þ fÝ ´ A 3z‡ Mi þ •› A |Mi 3¤À Ä εi1 , · · · , εini ƒe Ý . ò·K 3.3.27 †~ 3.3.24 (3) ¥ (Ø(Ü, Œ±wÑ: XJ V Œ±©)¤ A ØCf˜m† Ú V = M1 ⊕ · · · ⊕ Mr , K z‡•› A |Mi ÑŒ±é zž, A •Œ±é z. ‡L5, ·‚kXe ·K: ·K 3.3.28: V ´k•‘ K-•þ˜m, A ∈ End(V ), U ⊆ V ´ A XJ A Œ±é z, K A |U ∈ End(U ) •Œ±é z. ØCf˜m. y². λ1 , · · · , λm • A (3 K ¥) ¤k pÉA Š, Ei := E(λi , A ) ´ A ƒAA f˜m. P A 0 = A |U ∈ End(U ). Šâ½ÂŒ± y: A 0 A ŠÑ´ A A Š (Ïd´ λ1 , · · · , λm ¥ ˜‡), …éz‡ i Ñk E(λi , A 0 ) = Ei ∩ U . qdÚn 3.3.18 Œ• E(λ1 , A 0 ) + · · · + E(λm , A 0 ) ´†Ú. y3·‚•I‡ y U ⊆ E(λ1 , A 0 ) + · · · + E(λm , A 0 ) = (E1 ∩ U ) + · · · + (Em ∩ U ) , (3.3.28.1) Ï•ù‡(ØÚc¡ ?Ø(Üå5L²: U = E(λ1 , A 0 ) ⊕ · · · ⊕ E(λm , A 0 ). l Šâ½n 3.3.21 Œ• A 0 = A |U Œé z. • y ² (3.3.28.1) ª, • Ä U ¥ ? ¿ À ½ • þ u. Ï • A ∈ End(V ) Œ é z, ¤ ± V = E1 + · · · + Em (½n 3.3.21). ÏdŒ±ò u ∈ U ⊆ V ¤ u = w1 + · · · + wm •Iy²z‡ wi ∈ U . b Ø,. KØ” Ù¥z‡ wi ∈ Ei = E(λi , A ) . wm ∈ / U . OŽŒ• (A − λ1 I)u = A u − λ1 u = A w1 + · · · + A wm − λ1 (w1 + · · · + wm ) = (λ2 − λ1 )w2 + · · · + (λm − λ1 )wm . 3.3 ÄC††Ý ƒq 165 2±‚5C† A − λ2 I Šûþªü>, OŽz{Œ (A − λ2 I)(A − λ1 I)u = (A − λ2 I) (λ2 − λ1 )w2 + · · · + (λm − λ1 )wm = (λ3 − λ2 )(λ3 − λ1 )w3 + · · · + (λm − λ2 )(λm − λ1 )wm . ‡Eù öŠ, •ªŒ (3.3.28.2) (A − λm−1 I) · · · (A − λ2 I)(A − λ1 I)u = (λm − λm−1 ) · · · (λm − λ2 )(λm − λ1 )wm . Ï• U ´ A Ø C f ˜ m, Œ ± y (3.3.28.2) ª † > • þ á u U . u ´, a := (λm − λm−1 ) · · · (λm − λ2 )(λm − λ1 ) ´ š " ~ ê, (3.3.28.2) L ² awm ∈ U . q Ï • λi ü ü p É, a 6= 0. ù , awm ∈ U íÑ wm ∈ U . ùÚ·‚ƒcb wm ∈ / U gñ. Ïd·‚y² (3.3.28.1) ¤á. ·K–dy.. ·‚®²?Ø ‚5C† (½• ) †Œ±Oé z ¹Â. • Œ±Oé “þn z”. é zÚOé z. (3·K 3.3.27 ¥·‚½Â ‚5C z ¹ÂžÖög1‰Ñ.) „k˜‡aq öŠ´¤¢ ½ Â 3.3.29. V ´ n ‘ K-•þ˜m, A ∈ End(V ). XJ•3 V kSÄ B ¦ Ý MB (A ) ´þn , ·‚Ò`‚5C† A Œ±þn z ((upper) triangularizable), {¡Œþn z. éu• A ∈ Mn (K), XJ A 3 K þƒqu Mn (K) ¥,‡þn , ½=, •3Œ_Ý P ∈ Mn (K) ¦ P −1 AP ´þn , @o·‚¡Ý A 3 K þŒ±þn z, {¡Œþn z. XJò A À•‚5C† fA : K n → K n ; X 7−→ AX 3IOÄe Ý , @oŠâ·K 3.3.7, Ý A Œþn zÚ‚5C† fA Œþn z´ d . þn ·K 3.3.30: d: zÚØCf˜mƒm éXd±e(J‰Ñ: V ´ n ‘ K-•þ˜m, (v1 , · · · , vn ) ´ V 1. A 3kSÄ (v1 , · · · , vn ) e Ý ´þn ˜|kSÄ, A ∈ End(V ). K±e^‡ . 2. éuz‡ j = 1, · · · , n, þk A vj ∈ span(v1 , · · · , vj ). 3. éuz‡ j = 1, · · · , n, span(v1 , · · · , vj ) Ñ´ A ØCf˜m. y². 3‰ÖööS. ½3‘,Ç‘žùÇ. ·‚ò^·K 3.3.30 5y²š" ‚I‡ky²: ·K 3.3.31: e V ´ C þ A 3 V ¥kA •þ). k•‘E•þ˜mþ?Û‚5C†ÑŒ±þn n ‘•þ˜m, n ≥ 1, K V þ z. •d· ?Û‚5C† A 3 C ¥kA Š (Ï y². d·K y²‡^ “êÄ ½n (ë„½n A.4.3). T½n ˜‡íØ (ë„íØ A.4.9) Œ ±`²: éu?ÛÄ‘Xê• 1 EXêõ‘ª f ∈ C[X], eÙgê m = deg(f ) ≥ 1, K•3Eê λ1 , · · · , λm ¦ (3.3.31.1) f (X) = (X − λ1 ) · · · (X − λm ) . •y²·K (Ø, ? š"•þ v ∈ V . Ï• V ´ n ‘ ‚5ƒ'. u´•3Ø •" ~ê a0 , a1 , · · · , an ∈ C ¦ , ¤±•þ| v, A v, · · · , A n v ˜½ 0 = a0 v + a1 A v + · · · an A n v . 1nÙ ‚5N 166 9ÙÝ L« •Äõ‘ª f (X) = a0 +a1 X +· · ·+an X n . Ùgê• m ≤ n (Ïd am 6= 0, am+1 = · · · = an = 0). −1 Ø” am = 1 (3þª¥†m¦± am •†PÒ -#½Â f =Œ), l f Ä‘Xê• 1. þ¡ ªL² f (A )v = 0. Šâc¡¤ã“êÄ ½n íØ, •3/X (3.3.31.1) ª ©)ª. dž 0 = f (A )v = (A − λ1 I) · · · (A − λm I)v . Ï• v 6= 0, ¤±ùL²EÜN (A − λ1 I) · · · (A − λm I) Ø´ü . Ï Ø´ü , Ï ØŒ_. ŠâÚn 3.3.13, ƒA λi ´ A ˜‡A Š. ·K 3.3.32: †¤Ý Ù¥– k˜‡ A − λi I e V ´ C þ n ‘•þ˜m, n ≥ 1, K V þ ?Û‚5C† A Œ±þn ŠóLã, ?Û A ∈ Mn (C) 3 C þÑŒ±þn z. z. y². ·‚é V ‘ê¦^8B{‡ . e n = 1, (Øw,¤á (1 • ,@•´þn ). ±eb n = dim V > 1. Uì8B{ n, ·‚b éu?¿‘ê u n E•þ˜m U , Ù þ ?Û‚5C†ÑŒ±þn z. dž·‚•I‡y²: n ‘˜m V þ ‚5C† A •˜½Œ ±þn z. Šâ·K 3.3.31, Œ±é λ ∈ C • A ˜‡A Š. uÁ f˜m E(λ, A ) = Ker(λI − A ) = Ker(A − λI) š". ¤±•–"zÝ½n`² U := Im(A − λI) ´' V î‚ f˜m, = dim U < dim V = n. d , † yŒ• U ´ A ØCf˜m. ÏdŒ±½Â A 0 = A |U ∈ End(U ). Šâ8Bb , A 0 Œ±þn z. Šâ·K 3.3.30, •3 U ˜|Ä v1 , · · · , vm ¦ éz‡ j ∈ [[1, m]] , y3ò v1 , · · · , vm *¿• V A vj = A 0 vj ∈ span(v1 , · · · , vj ) . ˜|Ä v1 , · · · , vm , vm+1 , · · · , vn . j ∈ [[m + 1, n]] ž, A vj = (A − λI)vj + λvj ∈ Im(A − λI) + λvj = U + λvj ⊆ span(vj ) + U = span(vj ) + span(v1 , · · · , vm ) ⊆ span(v1 , · · · , vm , · · · , vj ) . ¤±y3, A vj ∈ span(v1 , · · · , vj ) éu¤k z. 3.3.4 SK S K 3.3.1. •Ä K 3 ¥ b j ∈ [[1, n]] ¤á. Šâ·K 3.3.30, ùL² A Œ±þn kSÄ B = (η1 , η2 , η3 ), Ù¥ η1 = (−1, 1, 1)T , η2 = (1, 0, −1)T , η3 = (0, 1, 1)T .   1 0 1   ‚5C† A ∈ End(K 3 ) 3kSÄ B e Ý ´  1 1 0. −1 2 1 ¦ A 3 K 3 IOÄe Ý . S K 3.3.2. (ε1 , ε2 , ε3 ) e (ε1 , ε2 , ε3 ) ´  3 ‘•þ˜mV a11 a12 a13   Ý • A = a21 a22 a23 . a31 a32 a33 ‡ O(5`, ·‚¦^ ˜|kSÄ. ‚5C† A ∈ End(V ) 3kSÄ ´¤¢ “1 êÆ8B{” (½n A.1.4). =, e–y²,‡'u ê n Øä A(n) é¤k ê n ¤á, Œ±©¤±eü‡Ú½. 1˜Ú•I‡ y n = 1 žØä A(1) ¤á. 1 Ú·‚Œ±b éu,‡ n > 1, Øä A(m) éu¤k u n ê m Ñ®²¤á, , ·‚|^ù‡b 2y² A(n) ¤á=Œ. 3.3 ÄC††Ý ƒq 167 1. ¦ A 3kSÄ (ε3 , ε2 , ε1 ) e Ý 2. ¦ A 3kSÄ (ε1 , 3ε2 , ε3 ) e Ý . 3. ¦ A 3kSÄ (ε1 + ε2 , ε2 , ε3 ) e Ý . . S K 3.3.3. E = (ε1 , ε2 , ε3 , ε4 ) ´•þ˜m V   1 0 2 1 −1 2 1 3    ´ . 1 2 5 5 2 −2 1 −2 ˜|kSÄ, ‚5C† A ∈ End(V ) 3 E e Ý η1 = ε1 − 2ε2 + ε4 , η2 = 3ε2 − ε3 − ε4 , η3 = ε3 + ε4 , η4 = 2ε4 . 1. y² η1 , η2 , η3 , η4 ´ V ˜|Ä. 2. ¦ A 3kSÄ (η1 , η2 , η3 , η4 ) e Ý . E = (ε1 , ε2 , ε3 , ε4 ) Ú B = (η1 , η2 , η3 , η4 ) ´ K 4 ü|kSÄ, A ∈ End(K 4 ) 3 E  1 −1 0 1 0 0 1 1   • . ée z‡œ¹, ¦ A 3 B e Ý . 1 0 0 −1 S K 3.3.4.  e Ý −1 1 0 1 1. ε1 = (1, 2, −1, 0)T , ε2 = (1, −1, 1, 1)T , ε3 = (−1, 2, 1, 1)T , ε4 = (−1, −1, 0, 1)T ; η1 = (2, 1, 0, 1)T , η2 = (0, 1, 2, 2)T , η3 = (−2, 1, 1, 2)T , η4 = (1, 3, 1, 2)T . 2. ε1 = (1, 1, 1, 1)T , ε2 = (1, 1, −1, −1)T , ε3 = (1, −1, 1, −1)T , ε4 = (1, −1, −1, 1)T ; η1 = (1, 1, 0, 1)T , η2 = (2, 1, 3, 1)T , η3 = (1, 1, 0, 0)T , η4 = (0, 1, −1, −1)T . S K 3.3.5. •Ä K 3 ü|kSÄ E = (ε1 , ε2 , ε3 ) Ú B = (η1 , η2 , η3 ), Ù¥ ε1 = (1, 0, 1)T , ε2 = (2, 1, 0)T , ε3 = (1, 1, 1)T ; η1 = (1, 2, −1)T , η2 = (2, 2, −1)T , η3 = (2, −1, −1)T . A ∈ End(K 3 ) ÷v A εi = ηi , i = 1, 2, 3. ¦ ME (A ) Ú MB (A ). S K 3.3.6. A, B ∈ Mn (K). y²: 1. e A Œ_, K AB Ú BA ƒq. 2. e A Ú B ƒq, Kéu?¿õ‘ª f (X) ∈ K[X], f (A) Ú f (B) •ƒq. S K 3.3.7. V ´k•‘•þ˜m. 1. A 2 = 0 (ùp 0 L«"N A ∈ End(V ), r = rank(A ). y²e ^‡ d: ). 2. Im(A ) ⊆ Ker(A ). 3. •3 V , ˜|kSÄ B ¦ ü‡ 0 •L«· Œ MB (A ) äk "Ý . ) 0r 0 ∗ 0 ! /ª. (ùp 0r L« r × r "Ý , 1nÙ ‚5N 168 4. •3 V ˜|kSÄ B 0 ¦    A=  S K 3.3.8. y²é  0  0 0  0 Ir  0 MB (A ) äk 0 0 0 0  1 2 3  S K 3.3.9. P =  4      Ú A0 =    e1 = ½ÂN ! 0 , e2 = 0 0 0    3knê• Q þƒq.  2 1 ! −1 . •Ä V = M2 (K) ¥ 1 1 0 L« /ª. 3 4 −1 2 9ÙÝ ! 1 , e3 = 0 kSÄ E = (e1 , e2 , e3 , e4 ), Ù¥ 0 1 ! 0 , e4 = 0 0 0 ! 0 . 1 T : V → V ; X 7→ P −1 XP . 1. y² T ´ V þ ‚5C†, ¿¦Ñ T 3kSÄ E e Ý . 2. ´Ä•3š"Ý X ∈ M2 (K) Ú λ ∈ K ¦ T (X) = λX ? e´, ž‰Ñù ~ê λ. eÄ, ž)º•Ÿoù X Ú λ Ø•3. ˜‡Ý X Ú S K 3.3.10. éu˜‡ n E• A, ^ fA L«‚5C† Cn×1 → Cn×1 ; X 7→ AX. ée œ /, ¦Ñ fA ¤kA Š±9ƒAA f˜m ˜|Ä.     ! ! 5 6 −3 0 0 1 0 a 3 4     ; (3) A = −1 0 1  ; (4) A = 0 1 0 ; ; (2) A = (1) A = −a 0 5 2 1 2 −1 1 0 0       1 1 1 1 0 2 1 3 1 0 1 1 −1 −1       (5) A = −2 0 3 ; (6) A = −4 −1 0  ; (7) A =   ; 1 −1 1 −1 −1 −3 0 4 −8 −2 1 −1 −1 1 S K 3.3.11. ½Â A ∈ EndR (R[X]≤4 ) • A (f (X)) := Xf 0 (X). ¦ A ¤kA ŠÚA •þ. S K 3.3.12. é u ˜ „ K-• þ ˜ m U , = ¦ Ø b ½ U ´ k • ‘ ˜ m, E Œ ½ Â ? ¿ ‚ 5 C † A ∈ End(U ) A ŠÚA •þ (ë„½Â 3.3.10). 1. •Ä¢Xêõ‘ª˜m U = R[X]. ½Â A : U → U •‚5C† f 7→ f 0 . ¦ A ÚA •þ. 2. •Ä¢ê‘ê ¤ ¤kA Š ˜m U = RN . ½Â A : U → U •‚5C† (a0 , a1 , · · · ) 7−→ (0, a0 , a1 , · · · ) (ë„~ 3.1.20.) y² A vkA Š. S K 3.3.13. A ´•þ˜m V þ ‚5C†, r ∈ N∗ . b r A α = 0. y²: •þ| α, A α, A 2 α, · · · , A r−1 α ‚5Ã'. •3•þ α ∈ V ÷v A r−1 α 6= 0, 3.3 ÄC††Ý ƒq 169 A ´ n ‘•þ˜m V þ ‚5C†. b   0 1   0 1     .. ..   . . SÄ B ¦ MB (A ) =  .     .. . 1  0  λ0     S K 3.3.15. λ0 • ½ ~ê, •Ä n • J =     S K 3.3.14. A n−1 6= 0, A n = 0. ¦y: V k˜|k 1 λ0  1 .. . .. . .. .     . ¦y J Ú§   1 =˜ λ0 J T ƒq. S K 3.3.16. V ´˜‡ C-•þ˜m. ÏLò V † C ƒm ê¦•›u V Ú R ¥ ± V † R ƒm ˜‡ê¦$Ž. ± V ¥ 5 \{•\{, ëÓ V † R ƒm ˜‡ R-•þ˜m. òT•þ˜mP• VR . 1. y²: XJ A ´ V ·‚F"rN A g C-‚5C†, @o A •´ VR À•¢•þ˜mƒm 2. y²: XJ ε1 , · · · , εn ´ C-•þ˜m V g ƒ?1, Œ ê¦, Œ± R-‚5C†. ¢‚5C†ž, ·‚U^ AR 5L«Ó˜‡N . ˜|Ä, K ε1 , iε1 , · · · , εn , iεn ¤ R-•þ˜m VR 3. ˜|Ä.. E = (ε1 , · · · , εn ) ´ C-•þ˜m V ˜|Ä, - ER = (ε1 , iε1 , · · · , εn , iεn ) . A = (aij ) ∈ Mn (C) ´ A 3 V kSÄ E e Ý . ¦ AR 3 VR kSÄ ER e Ý . ˜|Ä, ‚5C† A ∈ End(V ) 3 E e n ) ´•þ˜m V E = (ε1 , · · · , ε 0 1     0 ...   ME (A ) = A :=  . ..   . 1  0 y² 0 ´ A •˜ A Š. 2¦Ñ A 'uA Š 0 A f˜m E(0, A ). Ý • 4. y²: é?¿ A ∈ EndC (V ), det(AR ) = | det(A )|2 . S K 3.3.17. E = (ε1 , · · · ,! εn ) ´•þ˜m V ˜|Ä, ‚5C† A ∈ End(V ) 3 E e Ý J1 A := ME (A ) ´ A = /ª Oé , Ù¥ J1 , J2 ©O´ r Ú n − r • , ¿…Ñ/ J2 X   0 1    0 ...     . ..   . 1  0 S K 3.3.18. 1nÙ ‚5N 170 y² 0 ´ A •˜ S K 3.3.19. Š. 2¦Ñ A 'uA A A ´š" K-•þ˜m U þ 1. y²: e v ∈ U ´ A áuA áuA Š f (λ) A •þ. Šλ A B ∈ End(U ) † A Œ L« f˜m E(0, A ). A ‚5C†, λ ∈ K ´ A ˜‡A Š. •þ, Kéu?¿õ‘ª f ∈ K[X], v ´ f (A ) 2. y²: XJ A Œ_, K λ 6= 0, λ−1 ´ A −1 3. b Š0 9ÙÝ Š, … E(λ, A ) = E(λ−1 , A −1 ). A †. y² E(λ, A ) ´ B ØCf˜m. S K 3.3.20. éSK 3.3.10 ¥ Ñ ˆ‡E• A, ?ØÙ´ÄŒ±é z. 3Œ±é z œ¹ e, éÑ˜‡Œ_Ý P ¦ P −1 AP •é .   2 2 −2   S K 3.3.21. y²E• A =  2 5 4  Œ±é z, ¿éÑ˜‡Œ_Ý P ¦ P −1 AP −2 −4 5 •é . S K 3.3.22. •Ä 3 Ý  2  B = −2 −2 b B Ú C äkƒÓ 1. ¦ x, y 2. A  0  −2 , 0 −2 x −2  2  C = 0 0 0 2 0  0  0 . y õ‘ª. Š. ä B Ú C ´Äƒq. S K 3.3.23. U ´ K-•þ˜m, λ1 , λ2 ´‚5C† A ∈ End(U ) Á áu λ1 , λ2 A •þ. y²: v1 + v2 Ø´ A A •þ. S K 3.3.24. U ´š" K-•þ˜m, A ∈ End(U ). y²e 1. A ´XþC†, =, •3~ê c ∈ K ¦ 2. U ¥¤kš"•þÑ´ A A •ã ü‡ØÓA d: A = cI. •þ. S K 3.3.25. V ´š" k•‘ K-•þ˜m, A ∈ End(V ). y² A Œ±é ØƒÓ ~ê λ1 , · · · , λr ∈ K ¦ (λ1 I − A )(λ2 I − A ) · · · (λr I − A ) = 0 . (J«: éu¿©5, Œ±•Ä|^SK 3.2.20.) S K 3.3.26. n ≥ 2, •Ä n •  0  1   0  A = . . .   .. . 0 ¦A ¤kEA Š, ¿ Š, v1 , v2 ©O −1 0 0 ··· 0 −1 0 1 . 0 .. . . .. . −1 .. . .. . ··· .. . .. . .. .  0 ..  .   ..  .   .  0   −1 ··· ··· 0 1 0 .. .. ä A 3 C þ´ÄŒ±é z. z …= •3p 3.3 ÄC††Ý ƒq 171 S K 3.3.27. n ≥ 2, V ´ n ‘ K-•þ˜m, E = (ε1 , · · · , εn ) ´ V End(V ) 3 E e Ý ©O´ A, A? (= A N‘Ý ). 1. y² A , B Œ 2. b 0Á ˜|kSÄ. b A, B ∈ †. ˜‡A S K 3.3.28. •Ä n • Š. ¦ Ker(B)  1  ..  . P =   1 ‘êÚ˜|Ä.    =  1 In−1 ! 1 . 1. é?¿ k ∈ N, OŽ P k . 2. y²: P 3 C þŒ±é 3. y²: •3Œ_Ý z. Q ∈ Mn (C) ¦  a1  a  n  −1 a Q  n−1  ..  . a2 S K 3.3.29. A ´•þ˜m U þ M + N •´ A ØCf˜m. a2 a1 an .. . a2 é?¿Eê a1 , · · · , an , a3 a2 a1 .. . a4 ··· ··· ··· .. .  an an−1    an−2  Q  ..  .  ··· a1 ‚5C†, M, N ⊆ U ´ A S K 3.3.30. n ≥ 2, V ´ n ‘ K-•þ˜m, A ∈ End(V ). b   λ 1   λ 1     .. ..   . . • .     .. . 1  λ y²: Ø•3˜éš²… A Ñ´é ØCf˜m M, N ¦ . ØCf˜m. y²: M ∩ N Ú A 3V ,˜|kSÄe Ý V = M ⊕ N. S K 3.3.31. é?¿ a ∈ K, ½Â K 2 þ ‚5C† x ay Ta : 7−→ . y (1 − a)x y²: XJ K 2 ,‡f˜m M ´z˜‡ Ta ØCf˜m, K M = 0 ½ M = K 2 . S K 3.3.32. V ´š" k•‘ K-•þ˜m, A , B ∈ End(V ) Œ †. y²: XJ A , B ÑŒ± é z, @o˜½•3 V ˜|kSÄ E ¦ ME (A ) Ú ME (B) Óž•é . S K 3.3.33. V ´š" k•‘ K-•þ˜m, A ∈ End(V ) Œ±é z. y²: éu A ?ÛØCf˜m M , ˜½•3 A ØCf˜m N ¦ V = M ⊕ N. S K 3.3.34. V ´š" k•‘E•þ˜m, A ∈ End(V ). y²: XJéu A ?¿ØCf˜m M , þ•3 A ØCf˜m N ¦ A Œ±é z. (J«: é dim V ‰8B.) V = M ⊕ N , @o 1nÙ ‚5N 172 9ÙÝ L« S K 3.3.35. V ´ n ‘ K-•þ˜m, A ∈ End(V ) 3 K ¥k n ‡ØÓ A Š λ1 , · · · , λn . y² A 3 V ¥ ØCf˜m‡ê• 2n , ¿ Ñ A ¤kù ØCf˜m. S K 3.3.36. V ´ n ‘ K-•þ˜m, A ∈ End(V ) 3,˜|kSÄ E = (ε1 , · · · , εn ) e     A=    a1 a2 .. . an−1 Ý •     .    an y²: A Œ±é z ¿©7‡^‡´éz‡ i ∈ [[1, n]], span(εi , εn+1−i ) Ñ´ A m. S K 3.3.37. ‰½ 2m ‡• A1 , · · · , A2m ∈ Mn (K).     M =    A1 A2 .. . A2m−1     .    A2m y²: M 3 K þŒé Ñ3 K þŒé z. z ¿©7‡^‡é ! Ai Mi := A2m+i−i • i = 1, 2, · · · , m ØCf˜ 1oÙ 1 ª9ÙA^ 1 ª VgŒU´•U“L‚5“ê‘§A ˜‡êÆVg. ˜•¡, 1 ª´• ½ö ‚5C†±êŠ/ªLyÑ5 ˜«S3A . §éuÝ Ú‚5C† nØïÄäk-‡¿Â, 3NõAÛ½ÔnA^¥, ù‡êŠVgåX'…Š^. ,˜•¡, 1 ª•Œ±@•´'u˜| 1•þ½ö •þ ¼ê, ù‡¼ê3ä “õ-‚5”5Ÿ ¼ê+N¥Œ±`´˜‡•~„•; . ~f. ÙòXÚ/0 1 ª Vg, § Ä 5Ÿ!OŽ•{ÚnØA^. • ¦e© Øã• •{'² , ·‚3 Ù¥òæ^˜ A½ PÒ. Äk, K 3 Ù¥o´L« C ˜‡f•. e A • K þ n • , α1 , · · · , αn • A 1• þ|, β1 , · · · , βn • A •þ|, ·‚¬Šâ1©I‡²~r A ¤   α1    α2   A=  ..  ½ö A = (β1 , β2 , · · · , βn ) .  .  αn XJ f ´l Mn (K) kžÿ·‚'X A ˜‡¼ê, ·‚¬ò f (A) ¤   α1    α2   f (A) = f  ½ö f (A) = f (β1 , β2 , · · · , βn ) .  ..   .  αn K 1 i 1½1 j , @oŒ±ò A Ú f (A) ©O{     .. .. . .     A = αi  , f (A) = f αi  . . .. .. • ½ö A = (· · · , βj , · · · ) , f (A) = f (· · · , βj , · · · ) . • A 1 ªÏ~æ^ det(A) ½ |A| ü«PÒƒ˜. ( ˜«PÒ3Ø¬ÚåÜÂžƒ ^.) XJò A Ý ƒäN Ñ5, ¤   a11 a12 · · · a1n    a21 a22 · · · a2n   A= . .. ..  ..  .  .. . .  an1 an2 · · · ann 173 ~ 1oÙ 174 1 ª9ÙA^ @o det(A) ½ |A| •~^±ePÒO“  a11   a21 det   ..  . an1 ··· ··· .. . a12 a22 .. . an2 ···  a1n  a2n  ..   .  ann 1 4.1.1 ½ $ 4.1 a11 a21 .. . an1 1 ··· ··· .. . a12 a22 .. . an2 ··· a1n a2n .. . ann ª ª†‹IAÛ ! a b (4.1.1) · ‚ Q ² é • ½ Â L 1 ª (ë „ (1.2.21) ˜ ã): é u ? ¿ Ý ∈ c d M2 (K), § 1 ª (determinant) ½Â• ad − bc ù‡êŠ. Uì ÙÚóÜ©0 PÒ, d½Â Œ±^ÎÒLã• " # a b a b := ad − bc . (4.1.1.1) det = c d c d Šâ (1.2.21) ˜ã ‚q• a c , `Ý ?Ø, 1 ! b d Œ_ a ª c b d a c Šš"´Ý ! b Œ_ d du`1•þ| α = (a, b) , β = (c, d) ¿©7‡^‡. · •• 2, ½ö`, 1•þ −→ −−→ | α, β ‚5Ã'. XJb ²¡ K 2 S AÛ•þ OA Ú OB ©O u α Ú β, K α, β ‚5Ã' −→ −−→ −→ −−→ ¿›XAÛ•þ OA Ú OB Ø ‚. †‡ Ý`, •þ α = OA Ú β = OB ‚ ¿©7‡^‡´ a b 1 ª u 0. c d 1 ª a c b Ø d u 0 ž, T1 y3·‚ ¯K´: ¯¢þ, * ½Âª (4.1.1.1) Œ±uy: 1 ª a c b d −−→ −−→ SÈ. XJ OB 0 L«•þ OB 7 : O ^ž ^= 90◦ ¤ −−→ Œ±wÑ OB 0 ‹I ´ (d, −c). ¤±, XJ K ⊆ R, K " a det c ª ŠkŸoAÛ¹Âí ? −→ u•þ OA = (a, b) †•þ (d, −c) −−→ AÛ•þ, @ol OB ‹I• (c, d) −→ OA b  −→ −−→0  = det   = OA · OB = |OA| · |OB| · cos (θ − π/2) = |OA| · |OB| · sin(θ) , d −−→ OB # −→ −−→ −→ −−→ Ù¥ θ ∈ [−π, π] L« OA = α OB = β k• (oriented angle), =, XJl OA = OB ´_ ž •• ^=, K" θ # Š, ƒ‡K θ KŠ. −→ −−→ a b ddŒ„, det ýéŠÒ´± OA = (a, b) Ú OB = (c, d) •˜| > ²1o>/ c d " # " # a b a b OAP B ¡È (ã« ??). θ ∈ (0, π) ž, det > 0, θ ∈ (−π, 0) ž, det < 0. ·‚Œ c d c d " # a b ±`1 ª det L«²1o>/ OAP B k•¡È (oriented area) —— ùp, ·‚¦^“k c d 4.1 $ 1 ª 175 •”ù‡•½c==´• rN: d??Ø “¡È”Œ±•â•þ (a, b) (c, d) =•5«© K Ò. ·‚þ¡ ?Øwþ ´±•þ| α = (a, b), β = (c, d) ‚5Ã'•cJ , ć[ŽŽŒ −→ −−→ ±uy: α, β ‚5ƒ'ž, OA † OB ‚, dž·‚Œ±@•±ù‡ü‡•þ• > ²1o " # a b >/´“òz ”, =¡È• 0. ù• u1 ª det Š. Ïd, 3?Ûœ¹e, oŒ±ò1 c d " # a b ª det AÛ)º•˜‡²1o>/ “k•”¡È. c d e5·‚ïÄ 1 (4.1.2) ¯¢þ, 3c˜ã û½Ñ ª ˜ Ä 5Ÿ. ?Ø¥·‚´r1 " a ª det c # b À•p•ü d ˜‡êŠ. •Ò´`, XJP V = K 1×2 , ¿ò V × V ¥ ª, @o 1 ªŒ±À•l V × V ƒ ˜‡¼ê: K det : V × V = K 1×2 × K 1×2 −→ K " # " # " # " α a b α a = det = 7−→ det β c d β c (4.1.2.1) ÃØlAÛ n^5Ÿ: Ý„´l“ê Ý, ÑØJ • V‚5 (bilinearity): det 'uz˜1 " # α kS1•þ| β " # α ¤p•ü kSé / β C # b := ad − bc . d y (äNL§3‰Öö) (4.1.2.1) ¥ ¼ê÷v±e Ñäk‚55Ÿ. =, éu?¿ λ1 , λ2 ∈ K Ú α1 , α2 , β ∈ V = K 1×2 , þk±e ª¤á: " # " # " # λ1 α1 + λ2 α2 α1 α2 det = λ1 det + λ2 det , β β β " # " # " # β β β det = λ1 det + λ2 det . λ1 α1 + λ2 α2 α1 α2 • †5 (alternating property). =, é?¿ α ∈ V = K 1×2 " # α , det = 0. α " # e1 • 5‰5 (normalization). =, det = 1, Ù¥ (e1 , e2 ) ´ K 1×2 e2 8 Öö¬w g • K 4.1. , éup 1 ª ó, ±þn^5Ÿ ƒA‡ (kS) IOÄ. •´•Ä 5Ÿ. α1 , α2 , β1 , β2 ∈ K 1×2 . ª " # " # " # α1 + α2 α1 α2 det = det + det β1 + β2 β1 β2 ´Ä¤á? y²½ÞÑ‡~. " \@•ATXÛ^· 1 ª5Lˆ1 # α1 + α2 ª det ? y²\ β1 + β2 (Ø. 176 c¡ (4.1.1) ˜ã·‚`² ²n‘AÛ¥•¬g,/Ñy 1 ª3²¡‹IAÛ¥k²w 1 ª ^Éƒ/. 1oÙ 1 ª9ÙA^ ^?. e5·‚25` (4.1.3) · ‚ • , 3 Ô n Æ ½ ˜ m A Û Æ ¥, • þ È (cross product, • ¡ • þ È ½ ¥ þ È vector product∗ ) ´˜«š~k^ $Ž. y3·‚- V = K 1×3 . •þ ÈŒ±@•´ V ¥ ƒ üüƒm ˜«$Ž, $Ž(JE,3 V ¥. • •B, d?·‚6žæ^ÔnÆ¥ ~^PÒ, ò V = K 1×3 ¥IOÄP• i = (1, 0, 0) , j = (0, 1, 0) , k = (0, 0, 1) . l‹IAÛ Ý`, i, j, k ©O´÷X Ox ¶, Oy ¶Ú Oz ¶ •• ü •Ý•þ. 3ÔnÆ ¥, <‚~` i, j, k ¤˜‡mÃX (right-handed system), =, XJ^mÃo• -ž /G [ •þ i ±Ø‡L 180◦ =• j ž =Ä••, @omÃ-• ••Ò´•þ k ••. p éu V ¥•þ v, ·‚± |v| L« v •Ý. =, XJ v = xi + yj + zk, K |v| = x2 + y 2 + z 2 . α, β ∈ V . UìÔn½AÛ •ª£ã, •þ α † β È α × β ½Â•÷v±e^‡ • þ: • e α = 0 ½ β = 0, K5½ α × β = 0. (ùp 0 L«"•þ !) e¡b α, β þØ•", ¿± ∠αβ L« α † β Ïdò ∠αβ Š‰Œ ½•«m [0, π]. • Äk5½ α × β ƒm Y . ùp·‚Ø•Ä Ý ••, •Ý• |α × β| = |α| · |β| · sin ∠αβ. e ∠αβ ∈ {0, π} (= α, β ‚), d^‡L² α × β = 0. α, β Ø ‚ž, † öþR† •k…=k˜‡. dž·‚5½ α × β † α, β ÑR†, ¿…U¦kS•þ| (α, β , α × β) mÃX. • ¤ Šâ±þ½Âá=Œ• i × j = k, j × k = i, k × i = j. (4.1.3.1) , , N´ y Èäk±e5Ÿ: • ‡é¡5 (anti-symmetry): α × β = −β × α. • àg5 (homogeneity): é?¿~ê λ ∈ K, (λα) × β = λ(α × β) = α × (λβ). , ˜^-‡ ƒév@ow, 5Ÿ´: • © Æ (distributive law): é?¿ α, β1 , β2 ∈ V , ok α × (β1 + β2 ) = α × β1 + α × β2 , (β1 + β2 ) × α = β1 × α + β2 × α . 'u±þ© Æ y², k, ÖöŒ±Š•˜‡öS, ½ög1 k' ë•Ö (~X [33, þ þ, 1 100 •]). ò à g 5 Ú © Æ ( Ü 3 ˜ å, Œ ± w Ñ: é u é ? ¿ α, β1 , β2 ∈ V 9 ? ¿ λ1 , λ2 ∈ K, (4.1.3.2) ∗, α × (λ1 β1 + λ2 β2 ) = λ1 α × β1 + λ2 α × β2 , (λ1 β1 + λ2 β2 ) × α = λ1 β1 × α + λ2 β2 × α . ˜‡~^ ¶¡´ È (outer product). ØL, ò5ÆS “ênØžÖöŒU„¬‘ ,˜«“ È” Vg. @ ‡VgÚdg Èk «O, Ÿþ•kƒÏƒ?. • •ÏÐÆö~ (¾, ·‚3d¦þØ¦^“ È”ù«Lã. 4.1 $ 1 ª •Ò´`, •þ 177 È÷vV‚5. Šâ‡é¡5!V‚5±9úª (4.1.3.1), ·‚Œ± OŽÑ α × β ‹I. ·K 4.1.4: α = (a1 , a2 , a3 ), β = (b1 , b2 , b3 ). K a2 b2 y². ŠâK a1 a3 , − b1 b3 Ñ˜„œ¹eXÛÏL•þ α, β È α×β a1 a3 , b1 b3 a2 b2 ‹I ‹I• ! , α = a1 i + a2 j + a3 k, β = b1 i + b2 j + b3 k. |^ (4.1.3.1) Ú (4.1.3.2) OŽŒ• α × β = (a2 b3 − a3 b2 )i + (a3 b1 − a1 b3 )j + (a1 b2 − a2 b1 )k . ùÒ´–y 4.1.2 n (J. 1 ª†˜mAÛ !¥·‚ V = K 1×3 . (4.1.5) c¡·‚é V ¥•þ½Â üüƒm È. ,˜•¡ V ¥ •þƒmqŒ±üü‰ :È (=SÈ). XJ‰ n‡•þ α, β, γ ∈ V , ·‚Œ±½Â§‚ ·ÜÈ (mixed product) ½nIþÈ!Iþn-È (triple scalar product, scalar triple product) • α · β × γ := α · (β × γ) . ùp, ·‚k± ü‡•þ β, γ U‰½ †müS‰ È, 2±1˜‡•þ α Ú β × γ ‰:È, • ª (J´˜‡Iþ (=êŠ). Ïd, ·‚¢Sþ½Â ˜‡¼ê (4.1.5.1) V × V × V −→ K ; (α, β, γ) 7−→ α · β × γ := α · (β × γ) . ·‚‘ ò 1 ªš~ƒq 5Ÿ. Ïd, XJò V × V × V ¥  yù‡¼ê† α   •ü β  /ª, Œ±ò (4.1.5.1) ¥ ¼êU •˜‡¼ê γ     α α     (4.1.5.2) det : V × V × V −→ K ; β  7−→ det β  := α · (β × γ) . γ γ ¿…òd¼ê¡•n ·‚Äky² (4.1.5.3) 1 ƒ ¤p ª¼ê. •þ| α, β , γ ‚5ƒ' …=   α   det β  = 0 . γ ¯¢þ, XJ β, γ ‚5ƒ', K β × γ = 0 (žÖög••Ÿo). XJ β, γ ‚5Ã' ‚5ƒ', K•3~ê b, c ∈ K ¦ α = bβ + cγ. u´Šâ:È V‚55Ÿ,   α   det β  = α · (β × γ) = (bβ + cγ) · (β × γ) = bβ · (β × γ) + cγ · (β × γ) . γ α, β, γ 1oÙ 178 Ï• β × γ † β, γ ÑR†, 1 ª9ÙA^   α   β · (β × γ) = γ · (β × γ) = 0. ÏdþªL² β  = 0. γ   α   ‡L5, b det β  = α · (β × γ) γ (ÄKg,k α, β, γ ‚5ƒ'). u 0. ·‚Žy² α, β , γ ‚5ƒ'. Ø” Äk·‚äó: dž β, γ, β × γ 7½‚5Ã', Ï u˜ ~ê a, b, c ∈ K k ª ¤ V = K 1×3 β, γ ‚5Ã' ˜|Ä. ù´Ï•, eé a(β × γ) + bβ + cγ = 0 (4.1.5.4) ¤á, Kü>† β × γ ‰:ÈŒ a.(β × γ) · (β × γ) = 0. Ï• β, γ ‚5Ã', ¤± β × γ ´š"• þ, l § •Ý (β × γ) · (β × γ) ´š"~ê. u´d a.(β × γ) · (β × γ) = 0 Œ• a = 0. 2£ (4.1.5.4) ª, ¿|^ β, γ ‚5Ã'5Œ• b = c = 0. Q,·‚þ¡ äó¤á, Œ±ò α L«• β, γ, β × γ ‚5|Ü. =, •3 a, b, c ∈ K ¦ a(β × γ) + bβ + cγ = α. “\ α · (β × γ) = 0 ù‡b ¥, 2Šâ β · (β × γ) = γ · (β × γ) = 0 Œ• a.(β × γ) · (β × γ) = 0, l a = 0. u´ α = a(β × γ) + bβ + cγ = bβ + cγ, α, β, γ ‚5ƒ'. –d·‚y² (4.1.5.3). −→ −−→ −−→ ‘ œ¹aq, y3·‚Œ±y²: XJ˜m¥ AÛ•þ OA, OB, OC ©O“L α, β , γ,  α −→ −−→ −−→ −→ −−→ −−→   K1 ª det β  Š u± OA, OB, OC •c ²18¡N (parallelepiped) P(OA, OB, OC) γ −→ −−→ −−→ k•NÈ. ùp, (3NÈš" œ¹e) XJkS•þ| OA, OB, OC ¤mÃX, ·‚@•k•N È• Š, ÄK•KŠ. −−→ −−→ −−→ −→ −−→ ¯¢þ, XJP δ = OD • β × γ = OB × OC, θ ∈ [0, π] • OA † OD ƒm Y (Ø•Ä −−→ −−→ −→ −−→ −−→ ••). K |OD| = |OB × OC| ´²18¡N P(OA, OB, OC) 3²¡ OBC þ .¡È, |OA| · | cos θ| −→ −−→ −−→ K´T²18¡N3²¡ OBC þ p. Ïd, P(OA, OB, OC) NÈ• † −−→ −−→ |OA| · |OD| · | cos θ| = |OA| · |OB × OC| · | cos θ| . ù ´   α −−→ −−→   −→ −−→ −−→ det β  = OA · (OB × OC) = |OA| · |OB × OC| · cos θ γ −→ −−→ −−→ ýéŠ. ùp, XJkS•þ| OA, OB, OC   α   dž det β  Œu", ÄKT1 ª u". γ ¤mÃX (·‚b½§‚‚5Ã'), K θ ∈ [0, π2 ], (4.1.6) UYþ˜ã ?Ø. Šâ:ÈÚ È ƒ'5Ÿ, ÖöØJ y: (4.1.5.2) ¥ 1 ª÷vƒq n^5Ÿ: õ-‚5 (multilinearity), †5!5‰5. =, ¼êÚ 4.1 $ 1 ª 179 • õ-‚5: éu?¿ λ1 , λ2 ∈ K Ú α1 , α2 , β, γ ∈ V = K 1×3 , þk±e ª¤á:       α2 α1 λ1 α1 + λ2 α2       det  β  = λ1 det  β  + λ2 det  β  , γ γ γ       β β β       det λ1 α1 + λ2 α2  = λ1 det α1  + λ2 det α2  , γ γ γ       β β β       det  γ  = λ1 det  γ  + λ2 det  γ  . λ1 α1 + λ2 α2 α1 α2   α   • †5: e α, β, γ nöƒ¥kü‡ (½•õ‡) ƒÓ, K det β  γ   e1   1×3 • 5‰5: e e1 , e2 , e3 T• K kSIOÄ, K det e2  = 1. e3 d , Šâ:È OŽúªÚ·K 4.1.4, ·‚Œ± n 1 u 0. ª ˜‡OŽúª: e α = (a1 , a2 , a3 ) , β = (b1 , b2 , b3 ) , γ = (c1 , c2 , c3 ) K    α a1    det β  = det  b1 γ c1 a2 b2 c2  a3  b3  c33 = (a1 , a2 , a3 ) · b2 c2 = a1 b2 c2 b3 b1 − a2 c3 c1 b1 b3 , − c1 c3 b1 b3 , c1 c3 b3 b1 + a3 c3 c1 b2 c2 b2 c2 ! = a1 b2 c3 + a2 b3 c1 + a3 b1 c2 − a1 b3 c2 − a2 b1 c3 − a3 b2 c1 . Uì±þúª, ·‚Œ±rn 1 ª ŠÝ ½Â•   a11 a12 a13 a11 a12 a13   a21 a22 a23 = det a21 a22 a23  a31 a32 a33 a31 a32 a33 (4.1.6.1) a22 a23 a21 : = a11 − a12 a32 a33 a31 ˜m M3 (K) þ a23 a21 + a13 a33 a31 ¼ê det : M3 (K) → K, Ù a22 a32 = a11 a22 a33 + a12 a23 a31 + a13 a21 a32 − a13 a22 a31 − a11 a23 a32 − a12 a21 a33 . • BuPÁn 1 ª úª, ÖöØ”æ^ã 4.1 ¤« é ‚{K (•¡ Sarrus {K† ). äN5`, OŽn 1 ªž, Œ±kò1˜Ú1 U^Sˆ 31n m>, , òÌé ‚ † Pierre Frédéric Sarrus (1798–1861) ´{IêÆ[. 1oÙ 180 1 ª9ÙA^ (main diagonal, principal diagonal, •l†þ•ë me• é ‚) •• n^ ‚0B ê© O‰¦È¿D± ÎÒ, ògé ‚ (minor diagonal, •¡‡é ‚ anti-diagonal, •lmþ•ë †e• é ‚) •• n^ ‚0B ê©O‰¦{¿D±K ÎÒ, • ò¤ 8‘U ì‰½ ÎÒ\å5Ò ‡¦ 1 ªŠ. ù«•{•Œ±æ^é1öŠ •ª?1. =, òn 1 ª cü1ˆ 31n1e•, , aq/òÌé ‚•• n‡¦ÈD± ÎÒ, gé ‚•• n‡¦ÈD±K Î Ò, §‚ƒ\ (J• u1 ª Š. ã 4.1: n XJÖö 1 ¿, •Œ±æ^n ª Sarrus {K (é 1 ª ‚{K). ã¡5 : ‘Äz‰ ½Â5“/ª/”PÁ·K 4.1.4 ¥ úª: e α = (a1 , a2 , a3 ) = a1 i + a2 j + a3 k , β = (b1 , b2 , b3 ) = b1 i + b2 j + b3 k K a2 α×β = b2 (4.1.6.2) ùp 1 ‡ a3 a1 i− b3 b1 ÒÙ¢Ò´ (4.1.6.1) ª¥ e¡·‚?Ø˜ a3 a1 j+ b3 b1 1 i a2 k = a1 b2 b1 j a2 b2 1. k a3 b3 AÛA^. (4.1.7) 3 (2.1.18) ˜ã·‚Q²?ØL, ˜m¥o‡:´Ä ¡ ¯KŒ±=z•n‡•þ´Ä ‚5ƒ' ¯K. (4.1.5.3) fÐwŠ·‚XÛÏL1 ª5 ä‚5ƒ'. ¤±·‚Œ±^1 ª5ïÄ˜m¥: ¡¯K. ~X, e A(a1 , a2 , a3 ), B(b1 , b2 , b3 ) Ú C(c1 , c2 , c3 ) ´˜m¥Ø ‚ n‡:, @o?¿˜: 4.1 $ 1 P (x, y, z) ª 181 u²¡ ABC S ¿©7‡^‡´Ù‹I÷v x − a1 b1 − a1 c1 − a1 (4.1.7.1) y − a2 b2 − a 2 c2 − a2 z − a3 b3 − a3 = 0 , c3 − a3 −→ −−→ −→ Ï• (4.1.7.1) ª AÛ¿Â ´ AP , AB, AC n‡•þ ¡ (q½ö`´§‚•c ²18¡N −→ −−→ −→ P(AP , AB, AC) NÈ• 0). ·‚Œ±ò (4.1.7.1) ¡•²¡ ABC n:ª•§. −−−→ XJ ½˜: P0 (x0 , y0 , z0 ), ±9l P0 Ñu ü‡‚5Ã'•þ α1 = P0 P1 = (x1 , y1 , z1 ) Ú −−−→ −−−→ −−−→ α2 = P0 P2 = (x2 , y2 , z2 ), @o²L P0 P1 , P0 P2 ùü‡AÛ•þ (±9: P0 ) •˜²¡ Π ÷v± e•x: éu˜m¥?¿˜: P (x, y, z), −−→ −−−→ −−−→ −−→ P ∈ Π ⇐⇒ P0 P ´ P0 P1 , P0 P2 ‚5|Ü ⇐⇒ •þ| P0 P , α1 , α2 ‚5ƒ'. ¤± Π Œ±d±e•§(½: x − x0 x1 x2 (4.1.7.2) y − y0 y1 y2 z − z0 = 0. z1 z2 ·‚ò (4.1.7.2) ª¡•²¡ Π ˜:ü•þª•§. 5¿ùp ²¡ Π Œ±)º•²L: P0 …† •þ α1 , α2 Ñ²1 @‡²¡. XJ®•²¡ Π ²L‰½ ü‡: P0 (x0 , y0 , z0 ) Ú P00 (x00 , y00 , z00 ), ¿… Π ²1u,˜‡† −−−→0 P0 P0 Ø ‚ •þ β = (a, b, c), @o|^þ¡ ˜:ü•þª•§Œ± Ñ Π ü:˜•þª• §: x − x0 x00 − x0 a (4.1.7.3) , ÚL ˜ , ·‚ØïÆÖö ÙöAûäN¯K=Œ. kPM |^•þ ÈÚ·ÜÈ (½öéA ¯K. e¡ g•KÒ´˜‡~f. y − y0 y00 − y0 b z − z0 z00 − z0 = 0 . c ±þ úª, Ún g • K 4.2. •Ä˜m¥ü^†‚ :•ª•§: x − x1 y − y1 z − z1 L1 : = = ; A1 B1 C1 1 L2 : Ù¥: Pi (xi , yi , zi ) 9•þ αi = (Ai , Bi , Ci ), i = 1, 2 ž)º±e¯¢: ´F"ÖöU ª) „Œ±ïÄ˜m¥ n, ˜ Ù¦AÛ y2 − y1 B1 B2 z2 − z1 C1 . C2 x − x2 y − y2 z − zz = = A2 B2 C2 ‹I@•´®• 1. †‚ L1 † L2 -Ü −−−→ ¿©7‡^‡´ α1 × α2 = α1 × P1 P2 = 0. 2. †‚ L1 † L2 ²1 Ø-Ü 3. †‚ L1 † L2 ƒ (u•˜˜:) 4. †‚ L1 † L2 É¡ +¬n)Ù¥ . −−−→ ¿©7‡^‡´ α1 × α2 = 0 6= α1 × P1 P2 . ¿©7‡^‡´ α1 × α2 6= 0 = x2 − x1 ¿©7‡^‡´ A1 A2 y2 − y1 B1 B2 x2 − x1 A1 A2 z2 − z1 6= 0. C1 C2 1oÙ 182 ˜mAÛ¥ 4.1.3 1 ª9ÙA^ ål¯K (4.1.8) 4·‚k5?Ø˜m¥: ²¡ ål. ˜m¥²¡ Π ˜„•§• ax + by + cz + d = 0 . K n = (a, b, c) ´ Π ˜‡{•þ, §†¤k²1u Π •þR†. e M (x1 , y1 , z1 ) •˜m¥?¿ −−→ ˜:, d M :•²¡ Π ‰R‚ (perpendicular), b Rv (foot of perpendicular) • Q. K M Q ´ n −−→ ~ê . • M Q = λn. e ²¡ Π S,˜: P (x0 , y0 , z0 ), K (ÏLã«Œ±wÑ) −−→ −−→ |M P · M Q| = |M Q|2 . ¤± |M Q| = OŽŒ• −−→ −−→ −−→ −−→ |M P · λn| |M P · n| |M P · M Q| = = . |M Q| |λn| |n| −−→ M P · n = a(x1 − x0 ) + b(y1 − y0 ) + c(z1 − z0 ) = ax1 + by1 + cz1 + d . ùp, Ï• P (x0 , y0 , z0 ) ‹I÷v²¡ Π •§, ¤± −ax0 − by0 − cz0 = d. ¤±, : M (x1 , y1 , z1 ) ²¡ Π : ax + by + cz + d = 0 ålOŽúª• |ax1 + by1 + cz1 + d| √ . a2 + b2 + c2 (4.1.8.1) 5¿, ù‡úª¢Sþ3 M ∈ Π ž•¤á. g • K 4.3. OŽü‡²1²¡ Π1 : 11x − 2y − 10z + 15 = 0 , ƒm Π2 : 11x − 2y − 10z + 45 = 0 ål. (4.1.9) M •˜m¥˜:, L •˜m¥ †‚. •¦ M L ål, Œ±3 L þ ˜: A ¿• −−→ −−→ −−→ Ä L ˜‡•••þ v = AB. K M L ålŒ±@•´± AM , v = AB • > ²1o>/ 3 AB ù^>þ p. ÏdTålŒ±ÏLeªOŽ −−→ |AM × v| . |v| (4.1.9.1) ~X, e L :•ª•§• K: M (3, 0, 2) Ù¥: A ∈ L Œ± y−1 z x−2 = = , 2 1 1 L ål• −−→ 3 |AM × v| =√ |v| 2 • (2, 1, 0), •••þ v Œ± • v = (2, 1, 1). (4.1.10) L1 , L2 •˜m¥ É¡†‚, ©O± v1 , v2 ••••þ. K n := v1 × v2 ´† L1 Ú L2 ÑR† ••. : P 3 L1 þ£Äž, ²L: P …± n ••••þ †‚ LP o´† L1 Ú L2 Ñ R†. P £Ä ,‡· ˜ž, LP ¬† L2 •ƒ . ù«œ¹•¬é•˜ ˜: P â¬Ñy. džeP Q • LP † L2 :, ·‚¡‚ã P Q • L1 Ú L2 úR‚ (common perpendicular). ú R‚ •ÝÒ´ L1 † L2 ƒm ål. 4.1 $ 1 ª 183 ·‚NoOŽù‡ålQ ? −−−→ −−−→ ·‚Œ±3†‚ L1 þ ½˜‡: P1 , 3 L2 þ ½˜‡: P2 . 2 AÛ•þ P1 A1 = P2 A2 = −−−→ −−−→ −−−→ −−−→ −−−→ v1 , P1 B1 = P2 B2 = v2 . K± P1 P2 , P1 A1 , P1 B1 •c ²18¡Nkü‡.¡3²¡ P1 A1 B1 Ú P2 A2 B2 þ. ùü‡²¡ÑÚ P Q R† (Ï• P Q † v1 , v2 ÑR†) …ƒ (~X P ∈ L1 , L1 •¹ u²¡ P1 A1 B1 ), :=• P, Q. (Ö¿ã« ???) −−−→ −−−→ −−−→ ¤±, ‚ã P Q •ÝÒ u²18¡N P(P1 P2 , P1 A1 , P1 B1 ) 3.¡ P1 A1 B1 þ p (height). ddŒ„, L1 , L2 ål• −−−→ −−−→ −−−→ −−−→ |P1 P2 · P1 A1 × P1 B1 | | P1 P2 · v 1 × v 2 | (4.1.10.1) |P Q| = = . −−−→ −−−→ |v1 × v2 | |P1 A1 × P1 B1 | ±þúª3 L1 , L2 ²1ž¿Ø·^. Ñ). ´Ð3§‚²1žål•\N´¦Ñ (žÖög•XÛ¦ g • K 4.4. ¦y: ±eü^†‚´É¡†‚ x y z L1 : = = , L2 : x − 1 = y + 1 = z − 2 . 1 2 3 2¦ L1 Ú L2 ƒm 4.1.4 ål. SK S K 4.1.1. OŽ±e1 ª: −3 2 1 S K 4.1.2. a, b, c ´ x3 − 1 1 3 −2 2 −1 , 3 n‡ØÓ a c b S K 4.1.3. OŽ1 −1 2 1 2 1 4 Š. y² b a c c b =0. a ª a a a S K 4.1.4. ®•˜m† 1. 1 3 0 b c a+b a+b+c ; 2a + b 3a + 2b + c ‹IX¥o: x y x+y y x+y x x+y . x y ‹IXe A(1, −1 , −3) , B(2, 0, −2) , C(0, −5, 12) , D(2, 1, 0) . −−→ −→ −−→ äkS•þ| AB, AC, AD ´Ä ¤mÃX. 2. ¦²¡ ABC S K 4.1.5. ˜„•§. aij (x), 1 ≤ i, j ≤ 3 Ñ´'uCþ x a11 (x) a12 (x) a13 (x) Œ‡¼ê, f (x) = a21 (x) a22 (x) a23 (x) . a31 (x) a23 (x) a33 (x) ¦y: a011 (x) a012 (x) a013 (x) a11 (x) a12 (x) a13 (x) a11 (x) a12 (x) a13 (x) 0 0 0 0 f (x) = a21 (x) a22 (x) a23 (x) + a21 (x) a22 (x) a23 (x) + a21 (x) a22 (x) a23 (x) a31 (x) a23 (x) a33 (x) a31 (x) a23 (x) a33 (x) a031 (x) a023 (x) a033 (x) 1oÙ 184 S K 4.1.6. b 1 ª9ÙA^ α = (2, −1, 3), β = (1, −3, 2), γ = (3, 2, −4). ¦÷v v · α = −5 , v · β = −11 , v · γ = 20 •þ v ∈ R1×3 . S K 4.1.7. y²: α, β , γ, δ ∈ R1×3 . 1. e α + β + γ = 0, K α × β = β × γ = γ × α. 2. e α × β + β × γ + γ × α = 0, K α, β, γ ‚5ƒ'. 3. e α × β = γ × δ, α × γ = β × δ, K•þ| α − δ , β − γ ‚5ƒ'. S K 4.1.8. ¦e ˆ²¡ ëê•§Ú˜„•§: 1. ²L: A(1, 2, 3) …²1u•þ α = (1, −2, 1) Ú β = (0, 1, 2). 2. ²L: A(1, 1, 2), B(3, −2, 0) Ú C(0, 5, −5). 3. ²L: A(1, 2, −1) Ú z ¶. 4. ²L: A(4, 0, −2), B(5, 1, 7) …²1u z ¶. S K 4.1.9. ¦²¡ Π1 Ú Π2 ƒm ål, Ù¥ Π1 : x − 2y − 2z − 12 = 0 ; p 4.2 !¥, é?¿ Π2 : x − 2y − 2z − 6 = 0 . 1 ª ê m Ú?¿ K-•þ˜m W , ·‚æ^PÒ m‘ W , , ·‚o (m) V = K 1×n ´• K þ }| { z := W × · · · × W . n ‘1•þ˜m.   α1  .  .  W = V = K 1×n ž, W (m) ¥ ƒÏ~Uìp•ü •ªP¤  W •n  .  /ª. αm ‘ •þ˜m K n×1 ž, W (m) ¥ ƒEUÏ~î•ü¤1 •ª5L«.   α1  .  (n) (n)  m = n ž, ·‚•¬ŠâI‡r V ¥ ƒ  ..  †8  ¤ n • , l ò8Ü V αn n×1 (n) Ü Mn (K) Àƒ. aq/, W =K ž, 8Ü W •Œ±†8Ü Mn (K) À• Ó. 4.2.1 1 ª¼ê9Ù5Ÿ þ˜!·‚?Ø$ ½Â. 1 ªžQw §‚Ñäkõ-‚5 5Ÿ. y3·‚Œ±‰˜‡˜„ 4.2 p 1 ª 185 ½ Â 4.2.1. m ∈ N∗ , W • K-• þ ˜ m. W þ ˜ ‡ m - õ ‚ 5 ¼ ê (m-fold multilinear function) ´•?Û÷v±e^‡ ¼ê f : W (m) → K: éu?¿ α1 , · · · , αm ∈ W Ú?¿ i ∈ [[1, m]], N fα1 ,··· ,αi−1 , αi+1 , ··· , αm : W −→ K ; v 7−→ f (α1 , · · · , αi−1 , v , αi+1 , · · · , αm ) ´ K-‚5N . m = 1 ž, W þ ˜-õ‚5¼êÒ´l W K ‚5N . ·‚•ò§‚{¡• W þ ‚5¼ê (linear function). -õ‚5¼êÏ~{¡•V‚5¼ê (bilinear function). m ≥ 2 ž, ˜‡ m--õ‚5¼ê f : W (m) → K ¡•´ †. (alternating), ½ö` f äk †5, XJé?¿•þ| α1 , · · · , αm ∈ W , •‡3 αi ¥k,ü‡ (½•õ‡) ƒÓ, Òk f (α1 , · · · , αm ) = 0. m = 1 ž, ½¤k‚5¼ê f : W → K Ñ´ †. . ½ Â 4.2.2. f • Mn (K) K ¼ê. 1. XJ3ò Mn (K) Óu V (n) (Ù¥ V = K 1×n ) ƒ , f ´l V (n) @o·‚` f ´ Mn (K) þ 1‚5¼ê (row linear function). K n -õ‚5¼ê, aq/, XJ3ò Mn (K) Óu (K n×1 )(n) ƒ , f ´l (K n×1 )(n) @o·‚` f ´ Mn (K) þ ‚5¼ê (column linear function). K n -õ‚5¼ê, 2. XJ f ´1‚5¼ê, ¿…Š• V (n) f ´ †. . aq/, éu Mn (K) þ 3. XJ f 3ü Ý K ¼êžäk ‚5¼êŒ±½Â In ? †5, ·‚Ò` f äk †5. Š• 1, ·‚¡ f ´5‰ (normalized), ½ö f äk5‰5. 4. XJ f : Mn (K) → K ´5‰ †.1‚5¼ê, ·‚¡ f ´ Mn (K) þ (determinant function), ½ö f ´ K þ n 1 ª¼ê. Ï~·‚¬r“1 g • K 4.5. K þ Ún 4.2.3: e5Ÿ: ˜ ª¼ê”{¡•“1 1 ˜‡1 ª (determinant)”. ª¼ê ª¼ê´Ÿo¼ê ? m ≥ 2, f : W (m) → K ´ K-•þ˜m W þ 1. f ´ ‡ é ¡ [[1, m]], þk †5½ö †. m--õ‚5¼ê. K f äk± (anti-symmetric). =, é u ? ¿ α1 , · · · , αm ∈ W ± 9 ? ¿ Ø Ó • I i, j ∈ f (α1 , · · · , αi−1 , αi , αi+1 , · · · , αj−1 , αj , αj+1 , · · · , αm ) = −f (α1 , · · · , αi−1 , αj , αi+1 , · · · , αj−1 , αi , αj+1 , · · · , αm ) (3þª¥·‚b i < j. ùw,ØK•?Ø •Ò´`, XJòkS•þ| (α1 , · · · , αm ) ¥ Kf ŠUCÎÒ. 2. éu?¿ α1 , · · · , αm ∈ W , ?¿ØÓ ˜„5.) ü‡•þé† ˜, Ù¦•þ ˜ØC, •I i, j ∈ [[1, m]] ±9?¿ c ∈ K, þk f (α1 , · · · , αi , · · · , αj , · · · , αm ) = f (α1 , · · · , αi , · · · , cαi + αj , · · · , αm ) . •Ò´`, ·‚r•þ| α1 , · · · , αm ¥ þ αj þ, ¼ê f Š¿ØUC. ,˜‡•þ αi ¦±?¿~ê λ \ ,˜‡• 1oÙ 186 1 ª9ÙA^ 3. éu?¿ α1 , · · · , αm ∈ W , XJÙ¥,‡ αi = 0, K f (α1 , · · · , αm ) = 0. 4. éu W ¥?¿‚5ƒ' y². (1) • •þ| α1 , · · · , αm , þk f (α1 , · · · , αm ) = 0. PÒ{², Ø” i = 1, j = 2. Šâ f õ-‚5Ú †5, 0 = f (α1 + α2 , α1 + α2 , α3 , · · · ) = f (α1 , α1 , α3 , · · · ) + f (α1 , α2 , α3 , · · · ) + f (α2 , α1 , α3 , · · · ) + f (α2 , α2 , α3 , · · · ) = 0 + f (α1 , α2 , α3 , · · · ) + f (α2 , α1 , α3 , · · · ) + 0 . n= ¤–y. (2) EŠâ f õ-‚5Ú †5Œ•, f (· · · , αi , · · · , cαi + αj , · · · ) =cf (· · · , αi , · · · , αi , · · · ) + f (· · · , αi , · · · , αj , · · · ) =0 + f (· · · , αi , · · · , αj , · · · ) . ¤±äó¤á. (3) αi = 0. Ï• αi + αi = αi Šâõ-‚5Œ•, f (· · · , αi , · · · ) = f (· · · , αi + αi , · · · ) = f (· · · , αi , · · · ) + f (· · · , αi , · · · ) . ±þ´'u K ¥~ê ª, k f (· · · , αi , · · · ) = 0. (4) Ø” αm ´ α1 , · · · , αm−1 ‚5|Ü, =, •3 c1 , · · · , cm−1 ¦ αm = c1 α1 + · · · + cm−1 αm−1 . ÏL‡E¦^5Ÿ (2) Œ•, f (α1 , · · · , αm−1 , αm ) = f (α1 , · · · , αm−1 , c1 α1 + · · · cm−1 αm−1 ) = f (α1 , · · · , αm−1 , c1 α1 + · · · cm−1 αm−1 + (−cm−1 )αm−1 ) = f (α1 , · · · , αm−1 , c1 α1 + · · · cm−2 αm−2 ) = ············ = f (α1 , · · · , αm−1 , c1 α1 ) = f (α1 , · · · , αm−1 , c1 α1 + (−c1 )α1 ) = f (α1 , · · · , αm−1 , 0) . Šâ5Ÿ (3) Œ•þª• íØ 4.2.4: ˜‘ u 0. (Ø–d n ≥ 2, f • Mn (K) þ y. †.1‚5¼ê. 1. e A ∈ Mn (K) ØŒ_ (~X,˜1• 0), K f (A) = 0. 2. e P ´1ãÐ Ý O/, f (P ) = −f (In ). (= P ´ü Ý In 3. e P ´1 aÐ Ý Pn (λ · i) (= P ´òü K f (P A) = λf (A). AO/, f (P ) = λf (In ). †,ü1 Ý 4. e P ´1naÐ Ý Pn (c · i, j) (= P ´òü Ý Ý ), K f (P A) = f (A). AO/, f (P ) = f (In ). Ý ) K f (P A) = −f (A). A In 1 i 1¦±š"~ê λ Ý In 1 i 1¦±~ê c \ 1j1 ), 4.2 p 1 ª 5. e?˜Úb f ´1 f (AB) = f (A)f (B). 187 ª¼ê (=?˜Úb f (In ) = 1), Kéu?¿ A, B ∈ Mn (K), þk y². (1) e A ØŒ_, K A 1•þ|‚5ƒ'. Ïd(ØdÚn 4.2.3 (4) Œ . (2) Ý P A † A O=3u P A ´ A ¥,ü1é†¤ , ¤±ùžÚn 4.2.3 (1) ¥(Ø A~. (3) P A ´ A 1 i 1¦± λ ¤ Ý . Ïdù´1‚55Ÿ † íØ. (4) P A ´ A 1 i 1¦±~ê c \ 1 j 1 Ý , ¤±ù´Ún 4.2.3 (2) ¥(Ø A ~. (5) e A ØŒ_, K AB •ØŒ_. džŠâ (1) Œ•, –y ª ü>Ñ u 0. ÏdØ” A Œ_. u´, (Šâín 1.2.29) A Œ± ¤˜ Ð Ý P1 , · · · , P1 ¦È A = Pr · · · P1 . ·‚F" y²éu?¿Ý B ∈ Mn (K), f (Pr · · · P1 B) = f (Pr ) · · · f (P1 )f (B) (4.2.4.1) AO/, B = In ž (4.2.4.1) L² f (A) = f (Pr · · · P1 ) = f (Pr ) · · · f (P1 ), ù‡ªf“£ (4.2.4.1) = f (AB) = f (A)f (B). •y² (4.2.4.1), |^é r 8B{Œ±† z{ r = 1 œ/. •Ò´‡y²: P •Ð Ý ž, f (P B) = f (P )f (B). •d•I‡©n«œ¹?Ø, , ©O¦^ (2), (3), (4) ¥ (Ø= Œ. g • K 4.6. f : Mn (K) → K •1 ¦y: f (A) = a11 · · · ann , = f (A) ª¼ê, A = (aij ) ∈ Mn (K) •þn u A é ‚ ƒƒÈ. íØ 4.2.5: éu?¿ n ∈ N∗ , K þ n 1 . ª¼ê•õ•k˜‡. y². n = 1 œ¹3‰Öög• (ë„g•K 4.5). b n ≥ 2, f, g : Mn (K) → K Ñ´1 ª¼ê. K f (In ) = g(In ) = 1. u´, ŠâíØ 4.2.4 (Ø (2)–(4), f Ú g 3?ÛÐ Ý ? ŠƒÓ. 2ŠâíØ 4.2.4 (5), f Ú g 3?ÛŒ_Ý ? Š7,•ƒÓ (Ï•Œ_Ý o´Ð Ý ¦È). é?ÛØŒ_ Ý A, íØ 4.2.4 (1) `² f (A) = 0 = g(A). nþŒ• f = g. íØ 4.2.6: e f ´ K þ Ú§ =˜1 ªƒ . n 1 ª¼ê, Kéu?¿ A ∈ Mn (K), f (A) = f (AT ). =, ?Û• y². e A ØŒ_, K f (A) = 0 = f (AT ). e A Œ_, K•3Ð ŠâíØ 4.2.4 (5) f (A) = f (Pr ) · · · f (P1 ) = f (P1 ) · · · f (Pr ) , Ý P1 , · · · , Pr ¦ A = Pr · · · P1 . f (AT ) = f (P1T · · · PrT ) = f (P1T ) · · · f (PrT ) Ïd, •I‡ yéu?¿Ð Ý P , f (P ) = f (P T ). XJ P •1ã½1 aÐ Ý , K P ´é¡Ý , Ïd f (P ) = f (P T ). XJ P ´1naÐ Ý , K P T •´1naÐ Ý . Šâ T íØ 4.2.4 (4) Œ• f (P ) = f (In ) = 1 = f (P ). –d(Øy.. íØ 4.2.7: e f ´ K þ n y². ÏLíØ 4.2.6 ?11Ú 1 ª¼ê, K f ´ ƒm =†=Œ. †. ‚5¼ê. ØJ²x, 3±þ˜X (Ø (íØ 4.2.4 — 4.2.7) ¥, XJl˜‡ ‚5¼êÑu (±§“Oƒ c 1‚5¼ê f ), ·‚•Œ aq (Ø. AO/, ?Û5‰ †. ‚5¼ê•´ †.1‚5¼ê. ¤±, 1 ª¼ê•Œ±½Â•“5‰ †. ‚5¼ê”. 1oÙ 188 1 ª9ÙA^ éu n ≤ 3 œ¹, ·‚3 § 4.1 !¢Sþ®²`² n 1 ª (¼ê) •35, …‰Ñ wªLˆª (ë„ (4.1.1.1) ªÚ (4.1.6.1) ª). ·‚ò3 § 4.2.4 ! (½n 4.2.28) ¥y²?¿ 1 ªÑ•3 (Ù•˜5K®3íØ 4.2.5 ¥Øy). y34·‚#NgCJc¦^1 ª • 35, 3Ø• 1 ª¼ê˜„Lˆª œ¹e, |^§ 5Ÿ Ñ˜ -‡ Ä (Ø. X Ù Úó¥¤ã, • A ∈ Mn (K) 1 ªò± det(A) ½ |A| 5L«. ·‚k|^1 ª (•3Ú)•˜5y²'u©¬Ý 1 ª ˜‡Ä (Ø. §‚ r, s ∈ N∗ . Kéu?¿ A ∈ Mr (K), Ún 4.2.8: A Is y². ·‚•y1˜‡ •Ä¼ê ª. ,˜‡ ª = |A| = y² Is . A aq. f : Mr (K) −→ K ; A 7−→ A Is . |^ r + s 1 ª¼ê 1‚5! †5Ú5‰5, N´ y f ´5‰ †.1‚5¼ê. Ïd f Ò´•˜ @‡ r 1 ª¼ê. =, f (A) = |A| é¤k A ∈ Mr (K) ¤á. r, s ∈ N∗ . Kéu?¿ A ∈ Mr (K), B ∈ Ms (K), C ∈ Ms×r (K) Ú D ∈ Mr×s (K), Ún 4.2.9: A C 0 A = |A| · |B| = B 0 D . B y². Ï•Ý =˜öŠØUC1 ª (íØ 4.2.6), ·‚•Iy²1˜‡ ª. Ø”b Œ_ (ë„~ 1.2.44). K ! ! ! ! ! Ir 0 A 0 A 0 A 0 Ir 0 P := ÷v P = = −CA−1 Is C B 0 B 0 Is 0 B Ïd, Šâ1 ª ±¦{ 5ŸÚÚn 4.2.8, |P | · P ´é ª. 4.2.2 ‚ ƒ N‘Ý ù˜ •1 A 0 = 0 B A C en , Ir 0 · 0 Is 0 = |A| · |B| . B dg•K 4.6 Œ• |P | = |P T | = 1. ¤±þª‰Ñ·‚–y † Laplace Ðm !¥·‚UYb½?¿ ~. (4.2.10) ·‚k5òƒc'u1 1. 1 ª'uÝ c ∈ K, A, B þ 1 (½  1 ª®² ª •35®•, 5|^ƒ' (½ö1 ª½Â¥‰Ñ 5Ÿ5‰˜ OŽ ¢ ) Ì‡5Ÿo(˜e: ) äkõ-‚5. =, éu?¿ i ∈ [[1, n]], αi , αi0 ∈ K 1×n Ú?¿      .. .. ..  .  . .       det αi + αi0  = det αi  + det αi0  ;  .  . . .. .. ..     .. ..  .  .     det cαi  = c det αi  .  .  . .. .. 4.2 p 1 ª 189 (ùp3z‡úª¥, ØÓ1 ª ƒÓ ˜?p•ŽÑÒÑ@•éAƒÓ ƒ.) aq/, éu?¿ i ∈ [[1, n]], βi , βi0 ∈ K n×1 Ú?¿ c ∈ K, det(· · · , βi + βi0 , · · · ) = det(· · · , βi , · · · ) + det(· · · , βi0 , · · · ) , det(· · · , cβi , · · · ) = c det(· · · , βi , · · · ) . 2. e A ∈ Mn (K) Ø Œ _ (' X A |A| = 0. ,˜1½,˜ • 0, 2 ' X A k ü 1 ½ ü 3. e A0 ∈ Mn (K) ´d A ∈ Mn (K) é†,ü1 (½,ü †óƒ, ²L˜g1ãÐ 1 (½ 4. e A0 ∈ Mn (K) ´ò A ∈ Mn (K) †óƒ, 1 aÐ 1 (½ 6. 1 ª¼ê 7. ?Û• 1 (½ †Ù=˜ n ª • ü‡šÛÉ ´šÛÉ ~ 4.2.11. OŽ1 KÒ˜g. Š¦±ƒA ª , K |A0 | = c|A|. Ý ~ê. ) ¦±˜‡~êƒ \ ,˜1 (½ )¤ Š. e5·‚0 . uÙé ‚ ƒƒÈ. Ï~¡•ÛÉ (singular matrix), 1 ¦ÈE´šÛÉ . …= §´Œ_ 1 ª5Ÿ, OŽ1 ªš" • K¡•šÛÉ . ª 1˜«Ä •{Ò´ÏLÐ C†z•þ ª OŽ(J•c˜‡1 (4.2.12.1) ªƒ 1 Šâ (4.2.10) ˜ã Þ ½en Ý œ¹. ½ Â 4.2.12. ŠUC ) ¦±˜‡~ê c ¤ ,˜1 (½ ) C†Ø¬UC1 1 g • K 4.7. 1 ª•" (nonsingular matrix). ¦y: 2. ˜‡• ª ª , K |A0 | = −|A|. Ý ±¦{. =, éu?¿ A, B ∈ Mn (K), |AB| = |A| · |B|. 8. þ (½e) n 1. ƒÓŒ ,˜1 (½ ) C†¦1 5. e A0 ∈ Mn (K) ´ò A ∈ Mn (K) Ý , K |A0 | = |A|. =, 1naÐ ) C†¬¦1 )¤ ƒ Ó), K 1 1 1 1 1 1 −1 −1 1 −1 1 −1 ª u −16, ˜«š~-‡ 1 −1 −1 1 Ú ˜‡ u 312. 4íª1 −2 1 3 2 5 −9 −1 8 −1 3 13 7 5 −5 −7 −10 ªOŽ•{. •d·‚kÚ\˜ n ≥ 2, A = (aij ) ∈ Mn (K). é?¿•I i, j ∈ [[1, n]], ·‚P i A = l A ¥í 1 i 1Ú1 j n−1 • j . ½Â. 1oÙ 190 ÏL n − 1 1 1 ª9ÙA^ ª5½Â i Mij := A j (4.2.12.2) 9 Cij := (−1) i+j Mij = (−1) i+j i . A j ·‚r Mij ¡•Ý A (i, j)-fª (minor) ½öÝ ƒ aij {fª, r Cij • A ‡ Ïª (cofactor) ½öÝ ƒ aij “ê{fª. ±{Ïª Cij , 1 ≤ i, j ≤ n • ƒ Ý i (4.2.12.3) C = Cij 1≤i, j≤n = (−1)i+j A j 1≤i, j≤n ¡• A {ÏªÝ (4.2.12.4) (cofactor matrix). § =˜Ý C T P• A? := A?ij 1≤i, j≤n Ù¥ A?ij := (−1)i+j A (i, j)-{ j . i N‘ (Ý ) (adjugate (matrix), adjunct (matrix))§ .    1 −2 −1 0 1    ~ 4.2.13. A=2 1 0. K A {ÏªÝ • C = −3 −1 −1 2 0 −3 1   1 −3 −1   ? A = −2 −1 2 . −6 −3 −1 ·‚ò A? ¡• A  −6  −3, A −1 N‘Ý • Ú n 4.2.14: A = (aij ) ∈ Mn (K). é u ? ¿ • I i,  ∈ [[1, n]], P Cij • A (i, j)-{ Ï ª (• ¡ aij “ ê { f ª), 2 Bij ´ ò A 1 i 1 † ¤ K 1×n IOÄ¥1 j ‡•þ 0 ej = (0, · · · , 0, 1, 0, · · · , 0) (Ù¥ 1 ´1 j ‡‹I) ¤ Ý , Bij ´ò A 1 j †¤ K n×1 IOÄ¥1 i ‡•þ εi = (0, · · · , 0, 1, 0, · · · , 0)T (Ù¥ 1 ´1 i ‡‹I) ¤ Ý . =, 1j  (4.2.14.1) a11   ∗    ai−1, 1  1 i 1  0   ai+1, 1    ∗ an, 1  ∗ .. . a1, j−1 a1, j a1, j+1 ∗ a1n ∗ ∗ ∗ ∗ ··· ∗ .. . ai−1, j−1 0 ai+1, j−1 ai−1, j 1 ai+1, j ai−1, j+1 0 ai+1, j+1 ∗ an, j−1 ∗ an, j ∗ an, j+1 ∗ ∗ ··· ∗ .. .     ai−1, n   0   =: Bij  ai+1, n    ∗  ann ∗ ‡ =©©z¥ minor Ú cofactor ùü‡üc ∗ ∗ ^{3ØÓ©z¥´' Ú˜ , …•Ú·‚ùpæ^ ½ÂƒÓ. ¥©Ö¥Ï~Ñ´rd? Mij ¡• aij “{fª”, Cij (kéõÖPƒ• Aij , ~X [22]) K¡• aij ““ê{fª”. ùü‡¥©c®¥¦ ^“{”ù‡cŒU´• rN§éA fÝ Ú ƒ aij ƒm ˜'XkƒpÖ{ ¿g. ØL, Šö‡<@•, r“{fª”Ú““ê{fª”©O Š minor Ú cofactor ùü‡c ¥©Ècq k Øþ. ÏkA‡• ¡. 1˜, ¥©Ö¥„¬k |Ü ('X?Ø•˜„/ª Laplace Ðmªž, ½ö¦^Ìfª!^SÌfª?ØÝ ½ 5ž) ¦^“fª”ù‡c, ù |Üe“fª”éA =©ücE,´ minor. …, üli¡þ5` minor ´˜‡üc, “{fª”Kw ´‡Ü¤c. d , cofactor ù‡üc¥¿vk²w †““ê”ù‡cƒA ¹Â, Ïdò cofactor €È• ““ê{fª” •w Ø g,. ØL, “{fª”Ú““ê{fª”ùü‡¥©`{¤/®È, ¤±·‚•Ø‹Ž2•†§‚ . ·‚•´Ø2r§‚ Š minor Ú cofactor ¥©Èc, ´Àƒ•¥©Õk âŠ, Ø‰=©€È. 3¦^ minor Ú cofactor ž, ·‚–•uæ^“fª”Ú“{Ïª”ùü‡cŠ•éA ¥©È{. § •k ÖòÙ¡• A ;Š‘ (Ý ) (classical adjoint (matrix)) (~X [35]) ½öŠ‘Ý (adjoint matrix) (~X [22]). ´3†Ý nØ²1 ‚5N nØ¥, y“©z•–•ur“Š‘”˜cû“Š‘C†” (½Š‘N ) ù˜âŠ (ë„ Öeþ 7.2.2 !). ·‚•ÄŠ‘C†éA Ý ž, ¬uy§†ùp¤` N‘Ý Ø´˜£¯. • ; •ÜÂ, Ö¬¦ŒU/;•¦^“Š‘Ý ”ù‡c. Ïd, ·‚Ø2rd? A? ¡• A Š‘Ý , Ú^“N‘Ý ”ù«`{. 4.2 p 1 ª 191 1j  (4.2.14.2) a11   ∗    ai−1, 1  1 i 1  ai1   ai+1, 1    ∗ an, 1 ∗ .. . a1, j−1 ∗ ··· ∗ .. . ai−1, j−1 ai, j−1 ai+1, j−1 ∗  0 .. . a1, j+1 ∗ a1n ∗ ai−1, j+1 ai, j+1 ai+1, j+1 ∗ 0 1 0 .. . ∗ ∗ ∗ ··· ∗ .. . an, j−1 0 an, j+1 ∗     ai−1, n   0 ain   =: Bij  ai+1, n    ∗  ann ∗ ∗ 0 K Cij = |Bij | = |Bij |. y². ¯¢þ, (4.2.14.3) Pn (1, 2) · · · Pn (i−2, i−1)Pn (i−1, i)Bij Pn (j−1, j)Pn (j−2, j−1) · · · Pn (1, 2) = 1 ∗ ! 0 A ji ùp, é?¿•I k, l ∈ [[1, n]], k 6= l, Pn (k, l) L« †ü Ý In 1 k 1Ú1 l 1¤ 1˜ aÐ Ý (ë„½Â 1.2.24). ŠâÚn 4.2.9 ±91ãÐ Ý 1 ª• −1 ¯¢ (žÖög ••Ÿo), ÏLò (4.2.14.3) ª†mü> 1 ªŒ i i−1 j−1 (−1) · (−1) · |Bij | = A . j n= Cij = |Bij |. 0 ª Cij = |Bij | y²•{´ aq . ½n 4.2.15 (Laplace Ðmª Laplace¶ expansion): 1. (÷X1 i 1 A = (aij ) ∈ Mn (K). Ðm expansion along the i-th row) é?¿•I i ∈ [[1, n]], n X n X i i+j |A| = aij Cij = (−1) aij A . j j=1 j=1 2. (÷X1 j Ðm expansion along the j-th column) é?¿•I j ∈ [[1, n]], |A| = n X aij Cij = i=1 n X (−1)i+j aij A i=1 i . j y². ·‚•y²1˜‡úª, 1 ‡úªŒ± aq/y². Pn 1×n α∈K • A 1 i 1. K α = j=1 aij ej , Ù¥ e1 , · · · , en • K 1×n IOÄ. Šâ1 Pn ª 1‚55Ÿ, |A| = j=1 aij |Bij |, Ù¥ Bij ½ÂX (4.2.14.1) ª. u´, –y(ØlÚn 4.2.14 Œ . ~ 4.2.16. ÏL Laplace ÐmOŽÑ1 −1 2 ª 0 0 1 ¶ Pierre-Simon Laplace (1749–1827), {IêÆ[. 1 1 −1 0 0 1 0 0 0 0 0 0 −1 0 0 1 −1 0 1 2 Š• 5. 1oÙ 192 ! • ·‚5y²˜‡• Ú§ N‘Ý ƒm 1 ª9ÙA^ ˜‡-‡'X. ·K 4.2.17: A = (aij ) ∈ Mn (K), A? • A N‘Ý . K AA? = A? A = |A| · In . •Ò´`, éu?¿•I i, j ∈ [[1, n]], n X (4.2.17.1) aik Cjk = δij |A| = k=1 Ù¥ Cij • A n X aki Ckj k=1 (i, j)-{Ïª, δij ´ Kronecker delta ÎÒ (ë„ (1.2.6.9)). y². e i = j K (4.2.17.1) ª ´½n 4.2.15 ¥¤‰úª. y3·‚ i 6= j. Bjk X (4.2.14.1) ª½Â. KÚn 4.2.14.1 L² Cjk = |Bjk |. Šâ1 ª Pn Pn 1‚5, k=1 aik Cjk uÝ A0 := k=1 aik Bjk 1 ª. A0 ¢Sþ´rÝ A 1 j 1O 0 †•Ù1 i 1¤ Ý . Ïd A 1 i 1Ú1 j 1ƒÓ. |A0 | = 0. ùÒy² (4.2.17.1) ª¥ 1˜‡ Ò. ,˜‡ Ò y²3‰Öö. Ý c¡g•K 4.7 ®²wŠ·‚, ˜‡• ‰Ñ A−1 ˜‡nØúª. íØ 4.2.18: A Œ_ A = (aij ) ∈ Mn (K) ´šÛÉ duÙ1 ªš". y3·‚Œ±ÏLN‘ 1 A? . , K A Œ_, ¿… A−1 = |A| y². A^ (4.2.17.1) ª=Œ. ÖöXJP 4.2.3 Ý ¦_úª (ë„ (1.2.21)), ØJuyíØ 4.2.18 ´@‡úª í2. OŽÞ~ !¥, ·‚Eb½?¿ 1 ª •35, 5|^1 ª ˆ«5ŸÐ«1 ª ˜ O Ž•{ÚE|. ˜„‘ 1 ªŒ±©•ü«. ˜«´Ù¥Ý ƒ ´‰½êi 1 ª, ·‚ …¡ƒ • êi1 ª. ,˜«1 ª Ý ƒ•¹˜ Œ±“L?¿~ê i1, ùa1 ª·‚ … ¡• ©i1 ª. éu $ êi1 ª, Ï~ØI‡AOõ E|ÒŒ±Uìþ˜ !0 •{OŽÑ 5. ˜„Œ±•Ä •{•)Ð C†{ (z•n Ý ) ½öÏL Laplace Ðmª=z••$ 1 ª. éup 1 ª, AO´©i1 ª, ˜„5ÒŽå5' (J, •vk ½k •{, ‡‘ÅAC, Šâ¯K A:äN©Û, ÏL}ÁÏé· )û•{. ØL, du1 ªnØ{¤aÈ, c ‚'u1 ª OŽo( Nõ B ² . Ïd ! ·‚Œ±{‡0 A«' ;. ~^ 1 ªOŽE|. ~ 4.2.19. OŽ n ‰ (Vandermondek ) 1 ª 1 x1 2 Vn (x1 , · · · , xn ) := x1 .. . x1n−1 1 x2 x22 .. . xn−1 2 „ ··· ··· ··· .. . ··· ··· ··· ··· .. . ··· 1 xn x2n .. . xn−1 n k Alexandre-Théophile Vandermonde (1735–1796), {IêÆ[!zÆ[!ÑW[. ¦u 1770 cc ¦3 1772 cuL ;Í Mémoire sur l’élimination (5ž {ïÄ w6) C½ y“1 ªnØ âm©X•uêÆ. Ä:. 4.2 p 1 ). l1 n 1 1 n−1 1 ª 193 þ–1 1, òz˜1~ þ˜1 x1 (5¿: ·‚ öŠ^S´k±1 n 1~ x1 , 2±1 n − 1 1~ 1 n − 2 x1 , Xd ), Œ± 1 0 Vn (x1 , · · · , xn ) = 0 .. . 0 (U1˜ Š Laplace Ðm) (z JÑúÏf) ··· ··· ··· .. . ··· 1 x2 − x1 x22 − x2 x1 .. . xn−1 − xn−2 x1 2 2 x2 − x1 x22 − x2 x1 = .. . n−1 x2 − xn−2 x1 2 ··· ··· .. . ··· ··· ··· ··· .. . ··· ··· ··· .. . ··· 1 xn − x1 x2n − xn x1 .. . xn−1 − xn−2 x1 n n xn − x1 x2n − xn x1 .. . n−1 xn − xn−2 x1 n 1 x2 = (xn − x1 ) · · · (x2 − x1 ) . .. n−2 x2 ··· ··· .. . ··· ··· ··· .. . ··· 1 xn .. . xn−2 n = (xn − x1 ) · · · (x2 − x1 )Vn−1 (x2 , · · · , xn ) . d±þ4íúªØJ Y Vn (x1 , · · · , xn ) = (xj − xi ) . 1≤i<j≤n Q ùp ë¦ÎÒ 1≤i<j≤n L«: éz‡÷v 1 ≤ i < j ≤ n •ISé (i, j) k˜‡†ƒéA ¦ Q ÈÏf, ÎÒ 1≤i<j≤n ^5L«òù Ïf Ü¦å5¤ (J. (ïÆÖö Ñ n = 4 ½ 5 ž Q þã Vandermonde 1 ª Lˆª, ±B\ é 1≤i<j≤n ù‡ÎÒ n).) ±þ·‚OŽ Vandermonde 1 •{ÑØ”˜Á. ~ 4.2.20. OŽ n 1 ªž^ 4í gŽ. éuNõ/Gk5Æ ª An := ùp·‚@• A1 = x + y, A2 = x+y 1 xy x+y 0 xy ··· 0 0 .. . .. . 0 1 .. . .. . x+y .. . .. . xy .. . .. . ··· ··· 0 x+y 1 xy . x+y ··· ··· .. . .. . 0 0 .. . x+y 1 xy x+y 0 1 ª, ù« 1oÙ 194 ). ò1 ªUì1˜ (ò1 ‡1 Ïd·‚Œ± ,† ª9ÙA^ ÐmŒ xy 0 0 1 x+y An = An (x, y) : = (x + y)An−1 − 0 .. . 0 1 .. . xy .. . .. . ··· ··· ªU1˜1Ðm) ··· .. . .. . 0 x+y 1 xy x+y 0 .. . = (x + y)An−1 − xyAn−2 . 4íúª An = (x + y)An−1 − xyAn−2 . (4.2.20.1) , 1 Œ±wÑ A1 = x + y , A2 = (x + y)2 − xy = x2 + xy + y 2 . (4.2.20.2) é?¿ n ≥ 2, - Bn = An − xAn−1 . K (4.2.20.1) ªÚ (4.2.20.2) ªL² Bn = yBn−1 , B2 = A2 − xA1 = y 2 . u´ Bn = y n . l Šâ Bn ½Â An = xAn−1 + y n . (4.2.20.3) y3·‚– kü«•{Œ±¦Ñ An . 1˜«•{: Œ±† ÏL‡E¦^ (4.2.20.3) ªS“Œ±wÑ An = xAn−1 + y n = x(xAn−2 + y n−1 ) + y n = x2 An−2 + xy n−1 + y n = x2 (xAn−2 + y n−2 ) + xy n−1 + y n = x3 An−3 + x2 y n−2 + xy n−1 + y n = ············ = xn−1 A1 + xn−2 y 2 + · · · + x2 y n−2 + xy n−1 + y n = xn−1 (x + y) + xn−2 y 2 + · · · + x2 y n−2 + xy n−1 + y n = xn + xn−1 y + xn−2 y 2 + · · · + x2 y n−2 + xy n−1 + y n . 1 «•{: ù«•{Ì‡Äu1 ª An = An (x, y) Lˆ'u x Ú y Cn = An − yAn−1 Œ Cn = xn ±9 (4.2.20.4) é¡5. ÏLÚ\9Ïê An = yAn−1 + xn . n+1 n+1 −y XJ x 6= y, ÏLéá (4.2.20.3) Ú (4.2.20.4) )•§Œ± An = x x−y . XJ x = y, Ø” x = y 6= 0. dž·‚Œ±Ú\˜‡š"ëê t, ò 51 ª An = An (x, y) ¥ x O†• y(1 + t). Šâfâ (JŒ•, é¤k t 6= 0, An (y(1 + t), y) = y n (1 + t)n+1 − 1 . t 4.2 p 1 ª ØJ• ±, 1 ª An (y(1 + t), y) Š• t 195 ¼ê´ëY¼ê (k, An (y, y) = lim y n t→0 nþŒ•, (ÃØ x † y ´Äƒ y). ¤ (1 + t)n+1 − 1 = (n + 1)y n . t ) An (x, y) = xn + xn−1 y + · · · + xy n−1 + y n = (4.2.20.5) ÖöŒ±ÏL8B{ n X xn−i y i . i=0 ±þ·‚¦) x = y žLˆª •{Ù¢´©ÛÆ¥~^ ˜«•{. <‚¡ƒ• (ëê) ((parameter) perturbation method) ½‡ 6Ä{ (small perturbation method). , , ÖöŒ±5¿ ±þ·‚ Sþ•Úƒc·‚¦) Fibonacci ê ~ 4.2.21. OŽ n 1 Ä{ •{•Ny 4íªê Ï‘¦)•{. ù«•{¢ Ï‘ •{ (ë„~ 3.3.17) kVÜƒ?. ª Dn (λ, a1 , · · · , an ) := |λIn − An | = λ −1 0 λ ··· 0 0 .. . .. . 0 −1 .. . .. . λ .. . .. . ··· ··· .. . .. . .. . ··· ··· 0 0 0 .. . 0 an an−1 .. . .. . λ −1 a2 λ + a1 Ù¥ (4.2.21.1)  0  1   0  An =  .  ..  .  ..  0 0 0 ··· 0 1 .. . .. . 0 .. . .. . ··· ··· .. . .. . .. . ··· ··· 0 ùp, n = 1 ž·‚@• A1 = −a1 , n = 2 ž A2 = 0 1  −an  −an−1   ..  .   ..  0 .    0 −a2   1 −a1 0 0 .. . ! −a2 . −a1 ){˜: e λ = 0, KÏLé1˜1Š Laplace ÐmŒ•¤¦1 ª |λIn − An | u an . e λ 6= 0, Œ ±ò¤¦1 ª1˜1¦± λ−1 \ 1 1, 2ò# 1 1¦± λ−1 \ 1n1, XdöŠ•ª 1oÙ 196 1 ª9ÙA^ Œ • λ 0 0 λ ··· 0 0 Dn (λ, a1 , · · · , an ) = . .. .. . 0 −1 .. . .. . λ .. . .. . ··· ··· .. . .. . .. . ··· ··· 0 λ 0 0 λ ··· 0 0 = . .. .. . 0 0 .. . . λ .. . .. . ··· ··· .. . .. . .. . ··· ··· 0 ¤ þn /1 ƒÑ´ λ. ¤±•ª (4.2.21.2) ªé .. ‚þ• ˜‡ 0 0 .. . an an−1 + an λ−1 0 an−2 .. . λ −1 a2 λ + a1 0 0 .. . = ········· an an−1 + an λ−1 .. . .. . .. . λ + a1 + a2 λ−1 + · · · + an λ−(n−1) 0 λ 0 ƒ´ λ + a1 + a2 λ−1 + · · · + an λ−(n−1) , Ù{é ‚ Dn (λ, a1 , · · · , an ) = |λIn − An | = λn + a1 λn−1 + · · · + an−1 λ + an . w,, ù‡úªé λ = 0 œ¹•¤á. Uì8 ò0 âŠ, éu˜„ n Ý A, 1 ª |λIn − A| ´˜‡'u λ n gÄ˜ õ‘ª, §¡• A A õ‘ª (characteristic polynomial). A /X (4.2.21.1) ž, ·‚` A ´ õ‘ª λn + a1 λn−1 + · · · + an−1 λ + an l . (companion matrix). ±þOŽL², ˜‡Ä˜õ‘ª f (λ) l A õ‘ª u f (λ) ){ : 5¿ 1 ª Dn (λ, a1 , · · · , an ) ¥ an {fª´˜‡þn / 1 ª, λ {fª´ K´•$ /Gaq 1 ª Dn−1 (λ, a1 , · · · , an−1 ). ¤±, † ò1 ª Dn (λ, a1 , · · · , an ) U ì1˜1ÐmŒ Dn (λ, a1 , · · · , an ) = an + λDn−1 (λ, a1 , · · · , an−1 ) . |^±þ4íúª±9 D1 (λ, a1 ) = λ + a1 , =Œ ~ 4.2.22. k 1 ª 1½ ¥y' k5Æ 5ò 1 ª© •õ‡1 ª?1OŽ. (4.2.21.2) ª¥(J. ¦ÚªA:. džŒ±}Á¦^1 ~X, e a1 , · · · , an •š"~ê, ·‚Œ±U±e•{OŽ n Dn = 1 + a1 1 1 1 + a2 ··· 1 1 .. . 1 1 .. . 1 1 + a3 .. . ··· ··· ··· .. . .. . 1 1 1 1 .. . 1 1 + an ª . ª õ-‚ 4.2 p 1 ª 197 Äk, ò• ˜ Dn = ¤ (1, · · · , 1)T + (0, · · · , 0, an )T , , 1 + a1 .. . .. . ··· .. . 1 1 ··· ··· 1 + an−1 1 1 a1 1 .. 0 . .. + an Dn−1 = .. . . 0 1 |^ ‚5Œ 0 . ··· .. . ··· ··· an−1 0 .. 1 .. . .. + an Dn−1 . . 1 = a1 a2 · · · an−1 + an Dn−1 . ÏLù˜4íúª‡ES“¿* o(Œ Dn = a1 a2 · · · an−1 + an Dn−1 = a1 a2 · · · an−1 + an (a1 a2 · · · an−2 + an−1 Dn−2 ) = a1 a2 · · · an−1 + a1 a2 · · · an−2 an + an an−1 Dn−2 = a1 a2 · · · an−1 + a1 a2 · · · an−2 an + an an−1 (a1 a2 · · · an−3 + an−2 Dn−3 ) -A= n Y ! ai = i=1 A A A + + + an an−1 an−2 Dn−3 an an−1 an−2 A A A + + ··· + + an an−1 · · · a2 D1 an an−1 a2 A A A + + ··· + + an an−1 · · · a2 (a1 + 1) = an an−1 a2 ! ! n n X X 1 1 =A 1+ = a1 · · · an 1 + . a a i=1 i i=1 i = ········· = (¯¢þ, , ai •"ž, 1 g • K 4.8. OŽ±e n + 1 1 1 1 1 .. . 1 ~ 4.2.23. OŽ n 1 ª Dn ŠŒ±=z ±þœ¹5OŽ.) ª a1 a1 + b1 a1 .. . a1 a2 a2 a2 + b2 .. . a2 a c b a b b ª Dn = c .. . c c .. . ··· a .. . ··· ··· .. . .. . ··· c ··· ··· ··· .. . ··· an an an .. . an + bn b b .. . . (n = 1 ž@• D1 = a.) b a 1oÙ 198 ). ò1˜1 ¤ (b, · · · , b) + (a − b, 0, · · · , 0) 1 ª9ÙA^ /ª, |^1‚5Œ± 1 c 1 a 1 b ··· ··· Dn = (a − b)Dn−1 + b c .. . c c .. . ··· a .. . ··· .. . ··· c b a ··· ··· 1 b .. . 1 0 1 a−c 1 b−c = (a − b)Dn−1 + b 0 .. . 0 0 .. . ··· a − c ··· .. .. . . ··· n−1 = (a − b)Dn−1 + b(a − c) 0 1 b−c .. . b−c a−c . |^·‚ƒc3~ 4.2.20 ¥0 4íªê Ï‘¦)•{, Œ±¦Ñ   b(a−c)n −c(a−b)n b 6= c , b−c Dn = (a − b)n + nb(a − b)n−1 b=c. ùp ˜ OŽ[!·‚3‰ÖööS. ~ 4.2.24. OŽ n + 1 ù‡1 1˜ 1 ª x a1 a1 x a2 a2 Dn+1 = Dn+1 (x, a1 , · · · , an ) := a1 .. . a1 a2 .. . a2 x .. . ··· ª A:´z˜1 ƒƒÚÑƒÓ. Ïd, ·‚Œ±kò1 , , JÑúÏf. u´ Dn+1 = x+ n X ai (ò1˜ ¦± −ai \ 1 i+1 ) = x+ n X ai = x+ n X i=1 an x ª a2 1 x a2 · 1 .. . 1 a2 .. . a2 x .. . a3 1 ··· .. . .. . .. . –1 n + 1 an .. . an x ··· 0 0 1 x − a1 0 · 1 .. . 1 a2 − a1 .. . a2 − a1 x − a2 .. . a3 − a2 · n Y (x − ai ) . i=1 Ü\ an 1 ! ai x an a1 ! i=1 an an .. . . 1 ! i=1 ··· ··· .. . ··· .. . .. . .. . ··· 0 0 .. . 0 x − an 4.2 p 1 ª 199 ~ 4.2.25. k 1 ª Œ ± ¤ Ù ¦ • N ´ O Ž Cij = (ai + bj )n−1 ‰Ñ (n ≥ 1). K Cij = n−1 X l=0 1 ª ¦ È. ~ X, e n Ý ƒd C n n − 1 l n−1−l X n − 1 k−1 n−k ai bj = a bj . l k−1 i k=1 Ïd, e n−1 0  n−1  0 n − 1 k−1 A = Aij 1≤i, k≤n = ai =  .. k  . 1≤i, k≤n n−1 n−1 1 a1 n−1 1 a2 .. . n−1 1 an 0  n−1 b1  .  .. = B = Bkj 1≤k, j≤n = bn−k j  1≤k, j≤n  b1 1 b2n−1 .. . b2 1 n−1 n−1 n−1a1 n−1 n−1   n−1 a2 ··· ··· .. . ···   ..   . n−1 n−1 a n−1 n ··· .. . ··· ···  bn−1 n ..  .    bn  1 Y (aj − ai ) . K C = AB. |A| = n−1 Y n − 1 n−1 n−1 ··· Vn (a1 , · · · , an ) = · 0 n−1 k k=0 |B| = (−1) (n−1)n 2 Vn (b1 , · · · bn ) = (−1) (n−1)n 2 · 1≤i<j≤n Y (bj − bi ) . 1≤i<j≤n ¤±, (a1 + b1 )n−1 (a2 + b1 )n−1 .. . (an + b1 )n−1 (a1 + b2 )n−1 (a2 + b2 )n−1 .. . (an + b2 )n−1 ··· ··· .. . ··· (a1 + bn )n−1 n−1 Y n − 1 Y (a2 + bn )n−1 (n−1)n 2 = (−1) · · (aj −ai )(bj −bi ) . .. k . 1≤i<j≤n k=0 (an + bn )n−1 (ù¢SþÒ´|^þ¡ ª C = AB ŽÑ |C| = |A| · |B|.) ~ 4.2.26. éz‡ k ∈ N, sk = λk1 + λk2 + · · · + λkn . K s0 s1 .. . sn−1 ¯¢þ, þª†à1 ªéA Ý s1 s2 .. . sn ··· ··· .. . ··· sn−1 Y sn = (λj − λi )2 . .. . 1≤i<j≤n s2n−2 A 1 (i, j) aij = si+j−2 = n X l=1 ƒ• λi+j−2 = l n X λli−1 · λj−1 . l l=1 Ïd, e± bil = λi−1 • ƒ ¤Ý B = (bil )1≤i, l≤n , K A = BB T . u´ |A| = |B|2 . ùp |B| ´ l ‡ Vandermonde 1 ª, § Š®²3~ 4.2.19 ¥ŽÑ. 1oÙ 200 ·‚• 25w˜‡š~kA: ~ 4.2.27. •Ä˜‡Xe/ª 1 ª. a1 a2 a3   a  n  A= an−1  .  .  . a2 a1 a2 an .. . a3 a1 .. . a4 (4.2.27.1) ·‚¡•Ì‚Ý ª9ÙA^ Ý  ù«Ý 1 (cyclic matrix). ù«Ý ··· .. . .. . .. . ···  an  an−1     an−2  ..   .  a1 ˜‡A:´Œ± ¤Xe‚5|Ü /ª A = a1 I + a2 W1 + a3 W2 + · · · + an Wn−1 , Ù¥  0 0    W1 =  0   .. . 1 c[* þ¡ Ý 1 0 0 1 0 .. . 0 0 .. . ··· ··· ··· .. . .. . ···   0 0 1 0    ..    .  , · · · · · · , Wn−1 = 0    ..  . 1 0 0 0 .. . ··· ··· .. . .. .  1 0  ..   . .   0 ··· 1 0 0 0 0 0 1 .. . 0 Wi Œ±uy, e- W = W1 , K Wn = W0 = I , éu i = 1, 2, · · · , n − 1 , ·‚k W i = Wi . u´ (4.2.27.2) A = a1 I + a2 W + · · · + an W n−1 = f (W ) Ù¥ f ´õ‘ª f (X) = a1 + a2 X + · · · + an X n−1 . l (4.2.27.2) ªŒ±wÑ, XJ λ ´Ý W ˜‡A Š, v ´áu λ ˜‡A •þ, K f (λ) ´ A = f (W ) ˜‡A Š, v ´ A áu f (λ) A •þ. W A ŠŃ´¦Ñ . ¯¢ n n þ, Ï• W = I, ¤±3õ‘ª g(X) = X − 1 ¥± X = W “\ g(W ) = 0. ¯¢þ, l (4.2.27.2) ª„Œ±wÑ?Ûgê ≤ n − 1 š"õ‘ª¥“\ W Ø¬ u 0. g(X) 3Eê•¥ Tk n ‡ØÓ Š (§‚ ¡• C ¥ n gü Š (n-th roots of unity)), §‚´ 2π 1, ω = exp i , ω 2 , · · · , ω n−1 . n |^8 'u• õ‘ª •£Œ±• , g(X) ŠT• W 3 C ¥ A Š. ¤±, W 3 C ¥ k n ‡ØÓ A Š. ŠâíØ 3.3.19, (eò W Óu†¦ W ‰Ñ ‚5C† Cn → C n , v 7→ W v,) W 3 Cn ¥k n ‡‚5Ã' A •þ. b §‚´ v1 , · · · , vn . @où •þ•´ A = f (W ) A •þ, –ØLéu A ó, ù A •þéA A ŠC¤ f (1), f (ω), · · · , f (wn−1 ). u´, •3Œ_Ý P ∈ Mn (C) ¦ P −1 AP ´é ‚ ƒ• f (1), f (ω), · · · , f (ω n−1 ) é . ¤±, ·‚Œ±¦Ñ A 1 ª• |A| = |P −1 AP | = f (1)f (ω) · · · f (ω n−1 ) . (žÖög••Ÿo |A| = |P −1 AP |.) g • K 4.9. éu~ 4.2.27 ¥ Ý W , • W 3 C ¥z‡A ŠéÑ˜‡A •þ. 4.2 p 1 ª 4.2.4 1 ª 201 •35 ! Ì‡8Iý²?¿ 1 ª •35. éu n ≤ 3 œ¹, 4.1 !¥·‚®² ƒA 1 ª. Ïd, éup 1 ª·‚Œ±æ^8Bª E. ·‚Žy²e¡ ½n. k, ÖöŒ± y, ·‚e¡é½n 4.2.28 y²Úƒc vkÌ‚Øy ¯K. EÑ ?Ø ½n 4.2.28: éu?¿ n ≥ 2, •3 K þ•˜ n 1 ª¼ê Mn (K) → K ; A 7→ det(A) = |A|, …§Œ±d n − 1 1 ª¼êÏL±eLˆª48/Lˆ: é?¿ A ∈ Mn (K), det(A) = |A| := (4.2.28.1) n X ai1 Ci1 = i=1 n X (−1)i+1 ai1 A i=1 i . 1 y². 'u•˜5 Øä®²3íØ 4.2.5 ¥y²L . y3·‚I‡‰ ¯œ´: b½ n − 1 ª®²•3, P• Mn−1 (K) → K ; M 7→ |M |. ·‚I‡ y (4.2.28.1) ª‰Ñ ¼ê det : A ∈ Mn (K) 7−→ n X i+1 (−1) i=1 1 i ai1 A 1 ÷v1‚5! †5Ú5‰5. 5‰5 yŃ´ . Ï•XJ A = In , @o a21 = · · · = an1 = 0, A 11 = In−1 . ¤±, Šâ n − 1 1 ª 5‰5Œ n X (−1)i+1 ai1 A i=1 y3·‚ 1 i + 0 + · · · + 0 = a11 |In−1 | = a11 = 1 . = a11 A 1 1 y1‚5. •d, b  A = (aij ), A0 = (a0ij ), A00 = (a00ij ) ∈ Mn (K) /X kn‡Ý ···   a12 .. . ak2 + a0k2 .. . an2 a11  .   .. · · ·     · · · , A = ak1    ..   . · · · an1 ··· =, A, A0 , A00 • k 1 k 1 Œ U Ø Ó, aij , a0ij , a00ij ©OL« A, A0 , A00 Ý A00 1 k 1 ƒ, K a11  ..  .   A00 = ak1 + a0k1  ..   . an1 • i 6= k ž, a00ij = aij = a00ij é¤k A, A0 • 'X Ïd, ÃØ i, k ´Äƒ 1 ··· ª¼ê   a11  .   .. · · ·     0 · · · , A = a0k1    ..   . · · · ··· an1 u A j ∈ [[1, n]] ¤á, aq. u´, Šâ n−1 i = k ž, a00ij = aij + a00ij é¤k a12 .. . ak2 .. . an2 ··· a12 .. . a0k2 .. . an2 1 k 1 \ þ A0  · · ·   · · · .   · · · ··· 1 k 1. e ± A00 1i Ó A 1i , A0 1i 'XÚ A00 Ó 1‚5Œ• A00 1i = A 1i + A0 1i . j ∈ [[1, n]] ¤á, ¿… A00 1i = A 1i = A0 1i . , ok a00i1 A00 1i = ai1 A 1i + a0i1 A0 1i . ¤±, lLˆª (4.2.28.1) Œ± det(A00 ) = det(A) + det(A0 ) aq/Œ±y², XJ B ´ò A ,˜1¦±~ê c, K det(B) = c det(A). ù (4.2.28.1) ª½Â ¼ê det ÷v1‚5. ž,  Ò y • 5 y †5. •d, b Ý A = (aij ) 1 k 1Ú1 l 1ƒÓ (ùpØ” k < l). d i∈ / {k, l} ž, A 1i ¥kü1ƒÓ, Ïd A 1i = 0. ,˜•¡, A 1l ´ò A k1 1 l − 1 1£ 202 Ä 1k1 â n−1 , Ïdl A k1 z¤ A 1l •I‡?1 l − k − 1 g 5ŸŒ• A 1l = (−1)l−k A k1 . u´ Ý 1 ª 1oÙ 1 ª9ÙA^ † (ƒ ) ü1 öŠ. Š n X l k i l+1 k+1 i+1 + (−1) al1 A = (−1) ak1 A (−1) ai1 A 1 1 1 i=1 k (−1)k+1 + (−1)l+1 (−1)l−k−1 = 0 = ak1 A 1 –d·‚y² 4.2.5 (4.2.28.1) ª½Â ¼ê´˜‡1 ª¼ê. ½ny.. SK S K 4.2.1. ½ÂN f, g, h : Mn (K) → K ©O•   a11 a12 · · · a1n    a21 a22 · · · a2n  f :  .. ..  ..  ..  7−→ a11 a12 · · · a1n , .  . . .  an1 an2 · · · ann   a11 a12 · · · a1n    a21 a22 · · · a2n   g :  . .. ..  ..  7−→ a11 a21 · · · an1 , .  .. . .  an1 an2 · · · ann   a11 a12 · · · a1n    a21 a22 · · · a2n  h :  ..  .. ..  ..  7−→ a11 a22 · · · ann . .  . .  . an1 an2 · · · ann Á¯ f, g, h ùn‡¼ê= S K 4.2.2. (i) f ´ ´1‚5 ,= ´ ‚5 ,= n ≥ 2, f : Mn (K) → K ´1‚5¼ê. y²e † Q´1‚5 •ã q´ ‚5 ? d: . (ii) éu?¿ A ∈ Mn (K) Ú?¿ i, j ∈ [[1, n]], i 6= j, eP A0 • ØC) ¤ Ý , K f (A0 ) = −f (A). †A 1 i 1Ú1 j 1 (Ù¦1 (iii) éu?¿ A ∈ Mn (K) Ú?¿ i, j ∈ [[1, n]], i 6= j, eP Ã ´ò A 1 j 1 (Ù¦1ØC) ¤ Ý , K f (Ã) = f (A). 1i1 ,‡~ê \ S K 4.2.3. n ≥ 2, f : Mn (K) → K ´ †. ‚5¼ê. y²: XJ•3Œ_Ý M ∈ Mn (K) ¦ f (M ) = 0, Ké?¿ A ∈ Mn (K) þk f (A) = 0. S K 4.2.4. n ≥ 2, f : Mn (K) → K ´ †.1‚5¼ê. y²: •3~ê c ∈ K ¦ Ù¥ detn ´ n 1 ª¼ê (·‚b detn •35®•). S K 4.2.5. OŽ±e1 1 2 3 4 2 3 4 1 (1) ; 3 4 1 2 4 1 2 3 ª: 1 3 (2) 3 2 3 − 71 − 25 −12 − 29 2 7 2 5 21 5 4 5 − 17 3 2 15 5 2 3 7 ; f = c·detn , 4.2 p (3) 1 ª 203 0 a b −a 0 d −b −d 0 −c −e −f S K 4.2.6. OŽe c e f 0 1 ; (3) −2 3 −1 2 S K 4.2.7. OŽe (3) 1 2 3 4 2 3 4 1 (4) ª (1) 1 0 4 1 a b c d −b a d −c . −c −d a b −d c −b a 1 −2 0 −6 1 0 0 0 2 −1 0 0 1 4 2 1 ; (2) 2 1 0 3 0 −1 ; (4) −3 3 1 3 1 1 2 0 −1 0 −1 1 2 3 0 2 −1 3 2 1 2 −1 0 1 2 3 5 1 3 1 2 1 0 3 1 2 3 1 2 0 2 −1 1 0 −1 1 0 3 3 1 1 1 1 3 1 1 1 1 3 1 1 1 ; 1 3 ª (1) 1+x 1 1 1 1 1−x 1 1 1 1 1+y 1 4 1 ; (4) 2 3 0 1 1 0 1 1 a b 1 1 0 c a b ; (5) c d a2 b2 c2 d2 f (x) = 5x 1 x x 1 2 x 1 2 3 1 2 x 3 2 2x 1 1 ; (2) 1 1−y (a + 1)2 (b + 1)2 (c + 1)2 (d + 1)2 S K 4.2.8. ¦õ‘ª f (x) ¥ x3 † x4 ü‘ Xê. S K 4.2.9. f (x) = ¦õ‘ª f (x) ¥ x3 † x4 ü‘ S K 4.2.10. OŽ1 2x 1 3 1 x 1 x 1 2 x 1 1 2 −1 1 x a12 ··· . .. ··· a2,n−1 a1n Xê. ª a11 a21 .. . an1 1 2 1 2 ; (5) 1 −2 1 3 4 1 2 4 1 2 ; 1 5 (a + 2)2 (b + 2)2 (c + 2)2 (d + 2)2 1 1 1 2 1 0 −1 2 0 . 2 1 2 (a + 3)2 (b + 3)2 . (c + 3)2 (d + 3)2 1oÙ 204 1 ª9ÙA^ S K 4.2.11. A = (aij ) ´‰½ n • , b ´š"~ê. é?¿ 1 ≤ i, j ≤ n, - cij = aij bi−j , dd n • C = (cij ). y²: |A| = |C|. S K 4.2.12. |^ Laplace ÐmOŽe 1 2 2 1 0 1 0 2 ; (1) 2 0 1 1 0 2 0 1 1 x1 a1 (3) a2 a3 x21 S K 4.2.13. 1 x2 b1 b2 b3 x22 0 0 1 x1 x21 0 0 0 1 x2 x22 0 0 0 1 x3 x23 0 1 x3 c1 c2 c3 x23 S K 4.2.15. OŽ1 2 1 0 0 (2) (4) 1 2 1 0 0 1 2 1 0 0 1 2 λ x1 0 c 0 b x2 .. . xn a b .. . b 0 c .. ; . ··· ··· .. . .. . 0 ··· .. . .. . ··· ··· ··· 0 b 0 ‡é¡Ý . y² |A| = 0. a1 b1 c1 d1 e1 a4 b4 0 0 0 0 b .. . . b c 0 ª a2 b2 c2 d2 e2 a3 b3 0 0 0 a5 b5 0 0 0 ª 1 −1 0 ··· ··· 0 1 .. . .. . .. . −1 0 1 .. . −1 .. . ··· .. . .. . 0 ··· 0 0 1 0 0 .. . 0 −1 S K 4.2.16. ª: ; n ≥ 1 •Ûê, A • n S K 4.2.14. OŽ1 1  −1 0  A= −3 −1 2 0 1 −1 1 0 1 0 ..  1 1  . ¦ A 2 . 0 .. . .. . 0 −1 1 {Ïª M13 , M32 Ú M24 . 1   −3 1 0   S K 4.2.17. A =  2 1 −1. ¦ A z‡ ƒ aij 1 0 1 u1 2 1Ú1 3 Laplace Ðmª.   −1 2 0   S K 4.2.18. A = −3 1 1. ¦ A N‘Ý A? . 0 2 0 “ê{fª Cij , ¿ Ñ1 ª |A| ' 4.2 p 1 ª 205 S K 4.2.19. |^N‘Ý  1  2 1 S K 4.2.20. ‰½Ý ¤ n−1 • . ¦e Ý _Ý   2 −1   0 ; 1 −1 0 1 1 −1 :  3  0 ; 1 2 −1 2  1 0   0 0  0 0   . 1 −1 −1 0 −1 0 0 1 0 0 A = (aij ) ∈ M(n−1)×n (K). éz‡ i ∈ [[1, n]], ^ Mi L«ò A 1i 1. y²: X0 := M1 , −M2 , · · · , (−1)n−1 Mn ´‚5•§| AX = 0 2. b ˜‡). rank(A) = n − 1. y²: AX = 0 ¤k)Ñ´ X0 S K 4.2.21. n ≥ 2, A ∈ Mn (K). y²: |A? | = |A|n−1 . S K 4.2.22. n ≥ 2, A ∈ Mn (K). y²:    n ? rank(A ) = 1    0 (J«: •Ä N (A? ) Ú A S K 4.2.23. Ñ1 ˆ 4 0 ª 7 1 0 T ƒ ~ê . e rank(A) = n , e rank(A) = n − 1 , e rank(A) < n − 1 . “ê{fª.) −3 2 1 −5 3 0 −1 0 2 0 1 −1 7 3 2 −1 1 −1 é1 3 1 −1 0 Laplace Ðmª. S K 4.2.24. P D= a1 x1 a2 x1 a1 y1 a2 y1 a1 z1 a2 z1 b1 x1 b2 x1 b1 y 1 b2 y 1 b1 z1 b2 z1 a1 x2 a2 x2 a1 y2 a2 y2 a1 z2 a2 z2 b1 x 2 b2 x 2 b1 y2 b2 y2 b1 z2 b2 z2 a1 x3 a2 x3 a1 y3 a2 y3 a1 z3 a2 z3 b1 x 3 b2 x 3 b1 y 3 b2 y 3 b1 z3 b2 z3 x1 y1 z1 x2 y2 z2 a1 a2 b1 b2 , ∆= ~ê. ÏL±e n + 1 1 ª½Âõ‘ª , δ= y²: D = δ 3 ∆2 . S K 4.2.25. a1 , · · · , an ∈ K •‰½ 1 1 P (x) = 1 .. . 1 x a1 a2 .. . an x2 a21 a22 .. . ··· ··· ··· .. . a2n ··· xn an1 an2 .. . ann x3 y3 z3 . í 1oÙ 206 1. y² P (x) ´ n gõ‘ª. 2. b a1 , · · · , an üüØÓ. y² P (x) k n ‡ØÓ S K 4.2.26. OŽ1 S K 4.2.27. OŽ1 a1 b1 a2 b1 a3 b1 .. . an b1 S K 4.2.28. OŽ1 S K 4.2.29. OŽ1 S K 4.2.30. OŽ1 Š. ª .. . .. . ··· ··· .. . .. . .. . .. . ··· 0 7 2 5 7 0 5 0 0 0 .. . .. . .. . 0 2 .. . .. . .. . 7 .. . .. . .. . 5 .. . ··· ··· ··· ··· ··· .. . a1 bn a2 bn a3 bn ; .. . an bn ··· ··· .. . .. . .. . .. . 2 0 0 .. . .. . 0 5 7 ª a1 b2 a2 b2 a3 b2 .. . an b2 a1 b3 a2 b3 a3 b3 .. . an b3 ··· a 1 + b1 a 2 + b1 .. . an + b1 ··· ··· .. . a1 + b2 a2 + b2 .. . an + b2 ··· ª 1 + a1 b1 1 + a2 b1 1 + a3 b1 .. . 1 + an b1 1 + a 1 b2 1 + a 2 b2 1 + a3 b2 .. . 1 + an b2 1 + a1 b3 1 + a2 b3 1 + a3 b3 .. . 1 + an b3 1 2 3 .. . n−2 n−1 n 2 3 3 4 4 5 .. .. . . n−1 n n n n n ··· ··· ··· .. . ··· ··· ··· .. . ··· 1 + a1 bn 1 + a2 bn 1 + a3 bn .. . 1 + an bn ª ··· ··· ··· n−2 n−1 n .. . n n n n−1 n n .. . n n n an 0 0 .. . ª a1 −x1 0 .. . .. . 0 a2 x2 −x2 .. . .. . a3 0 x3 .. . .. . ··· ··· 0 .. . .. . ··· ··· ··· .. . .. . ··· ··· 0 −xn−1 0 xn n n n .. . n n n a1 + bn a2 + bn .. . an + bn 1 ª9ÙA^ 4.2 p 1 ª 207 S K 4.2.31. OŽe (1) (3) 1 n ª: a1 x x a2 x x x .. . x x .. . ··· a3 .. . ··· ··· .. . .. . ··· x x −a a x a a a a a a −a .. . −a −a −a x .. .. . . −a −a −a −a ··· ··· .. . .. . a .. . ··· ··· x −a a .. . a x S K 4.2.32. OŽ 2n 1 x x .. . (2) x an a0 −1 0 a1 x −1 a2 .. . 0 .. . 0 0 x 0 y x 0 .. . 0 y 0 .. . 0 0 an−2 an−1 (4) x .. . ··· .. . .. . .. . 0 .. . 0 0 ··· ··· x 0 0 y .. . .. . ··· 0 .. . .. . 0 ··· .. . .. . ··· 0 0 ··· x 0 0 .. . 0 .. . .. . 0 −1 x 0 0 .. . 0 y x ª a1 b2n a2 b2n−1 .. . an bn ∆2n (a1 , · · · , a2n ; b1 , · · · , b2n ) = .. .. . .. . bn+1 an+1 . b2 a2n−1 b1 Ù¥™ Ñ ƒÑ´". S K 4.2.33. OŽ 2n 1 a11 ª a12 a22 ··· ··· .. . ∆2n = .. cn,1 Ù¥™ Ñ a2n cn−1,2 cn,2 . ··· ··· a1n a2n .. . an,n c1,n .. . b11 b21 .. . bn,1 d11 .. . ··· ··· . .. cn−1,n cn,n dn−1,1 dn,1 ··· ··· .. b1,n−1 b2, n−1 b1,n . dn−1,n−1 dn,n−1 dn,n ƒÑ´". S K 4.2.34. éuz‡ i ∈ [[0, n − 1]], b1 , b2 , · · · , bn ∈ K. OŽ1 ª fi ∈ K[X] ´‰½ f0 (b1 ) f0 (b2 ) ··· f1 (b1 ) f1 (b2 ) ··· .. .. .. . . . fn−1 (b1 ) fn−1 (b2 ) · · · i gõ‘ª, ÙÄ‘XêP• ai . f0 (bn ) f1 (bn ) .. . fn−1 (bn ) 1oÙ 1 ª9ÙA^ 1 ··· x ··· 0 ··· ············ x ··· x ··· 1 x x 1 x x 208 S K 4.2.35. OŽe a1 0 0 −a2 a2 0 0 1 0 1 1 + a1 + x1 a2 + x1 an + x1 1 1 cos θ1 cos θ2 1 cos θn 1 a1 + b1 1 a2 + b1 1 an + b1 1 n 0 ··· −a3 · · · a3 ··· ············ 0 ··· 1 ··· 0 0 0 0 0 0 an−1 1 −an 1 + an 1 x11 x21 a 1 x22 xn−2,1 xn−1,1 xn−2,2 xn−1,2 2 cos θ 1 0 1 2 cos θ 1 0 0 0 0 a1 + x2 ··· 1 + a2 + x2 · · · ············ an + x2 ··· cos 2θ1 ··· cos 2θ2 ··· ············ cos 2θn ··· 1 ··· a1 + b2 1 ··· a2 + b2 ············ 1 ··· an + b2 S K 4.2.36. ª: ; a2 ··· a ··· 1 ··· ············ xn−2,3 · · · xn−1,3 · · · 0 ··· 1 ··· 2 cos θ · · · ············ 0 ··· 0 ··· a1 + xn a2 + xn ; 1 + an + xn cos(n − 1)θ1 cos(n − 1)θ2 x −n 1 0 x 1 1 x x an−2 an−3 an−4 an−1 an−2 an−3 1 xn−1,n−1 a 1 0 0 0 2 cos θ 1 1 2 cos θ 1 + x1 1 + x2 1 + x21 ··· 1 + x22 ··· ············ ··· 1 + x2n sin nθ1 sin nθ2 ; sin nθn 1 x−2 M ∈ Mn (K) Œ©¬• M = . .. . ! B , Ù¥ A •Œ_• D A ∈ Mn×m (K), B ∈ Mm×n (K). y²: Im A .. 1 + xn1 1 + xn2 ; 1 + xnn sin θ1 sin θ2 ; sin θn . n−1 x − 2n + 2 −1 −2 y²: |M | = |A| · |D − CA−1 B|. S K 4.2.37. ; 2 1 a n + bn A C 0 x x 0 sin(n − 1)θ1 ··· sin(n − 1)θ2 ··· ············ sin(n − 1)θn ··· −(n − 1) x − 4 .. . ; ; ; 0 0 0 1 + xn cos(n − 1)θn 1 a1 + bn 1 a2 + bn 0 1 1 B = |In − AB| = |Im − BA| . In . n x − 2n 4.3 1 ª S K 4.2.38. nØA^ 209 A ∈ Mm (K), B ∈ Mn (K), C ∈ Mm×n (K). - M = ! A . 0 C B y²: |M | = (−1)mn |A| · |B|. S K 4.2.39. a1 , a2 , · · · , an ∈ K üüpÉ. y²: é?‰ f ∈ K[X]≤n−1 ¦ éz‡ i ∈ [[1, n]] þk f (ai ) = bi . 1 4.3 4.3.1 ª b1 , b2 , · · · , bn ∈ K, •3•˜ õ‘ª nØA^ Cramer {K†)‚5•§| XJ A ∈ Mn (K) ´˜‡Œ_• , @oéu?¿ •þ b ∈ K n×1 , ‚5•§| AX = b k…= k•˜) X = A−1 b. ·‚3g•K 4.7 ¥w , A Œ_ ˜‡¿©7‡^‡´1 ª |A| Šš 1 A? ²( ". 3íØ 4.2.18 ¥·‚qw , |A| = 6 0 ž, A _Ý A−1 Œ±ÏLLˆª A−1 = |A| ? ‰Ñ. ùp A ´ A N‘Ý (ë„½Â 4.2.12). lù ¯¢¥o(Jõ, ·‚Œ± Xe½n: ½n 4.3.1: A ∈ Mn (K). 1. àg‚5•§| AX = 0 kš") ¿©7‡^‡´ |A| = 0. 2. Cramer∗∗ {K (Cramer’s rule): e |A| = 6 0, K A Œ_, ¿…éu?¿ b ∈ K n×1 , ‚5•§| AX = b k…=k•˜), Ù)•þ X = (x1 , · · · , xn )T ‹IŒ±deª‰Ñ: x1 = Ù¥ Ai ´ò A 1i †¤ |A1 | |A2 | |An | , x2 = , · · · , xn = , |A| |A| |A| •þ b ( Ù¦ ØC) ¤ Ý . y². (1) AX = 0 kš") du rank(A) < n (½n 2.2.14 (2) (a)), ˜^‡q du A ØŒ_ (g•K 2.6). þ¡·‚J |A| = 6 0 ´• A Œ_ ¿©7‡^‡. ¤±, A ØŒ_ ¿©7‡^‡ ´ |A| = 0. 1 A? . ŠâN‘Ý (2) ŠâíØ 4.2.18, |A| = 6 0 ž A−1 = |A| ½ÂŒ     x1 b1  .  . 1 1 ? ? . .  X=  .  = |A| A b = |A| A  .  xn bn    Pn  C11 C21 · · · Cn1     b1 i=1 Ci1 bi C12 C22 · · · Cn2   .    1  ..  .  .  = 1  . = . . . . .       .. .. ..  |A|  .. |A| P n C b bn i=1 in i C1n · · · · · · Cnn y3•‡`², éuz‡ j ∈ [[1, n]], |Aj | = Pn i=1 Cij bi . 0 0 ŠâÚn 4.2.14, Cij = |Bij |, Ù¥ Bij ´ò A 1 j 1 u1 i ‡ ˜) ¤ Ý . Šâ1 ª 1 j †¤ •þ b1 e1 + · · · + bn en ¤ Ý ∗∗ Gabriel Cramer (1704–1752), a¬êÆ[. †¤ ei = (0, · · · 0, 1, 0, · · · , 0)T (Ù¥ Pn 0 ‚5, i=1 Cij bi = i=1 bi |Bij | Š uò A 1 ª, •Ò´Ý Aj 1 ª. (Øy.. Pn 1oÙ 210 1 ª9ÙA^ 5P 4.3.2. Š•XêÝ •Œ_• ž‚5•§| ¦)úª, Cramer {KlnØþw´‡{' ²¯ úª. ´3¢SöŠ¥, ù«OŽ•{ OŽþ´éŒ , § Ç ØX·‚lc0 pdž {. ØL, Cramer {K3nØþ„´k§ ¿Â. ¯¢þ, Cramer {KŒ±ò) • ˜5 O8(•˜‡Ú½ (=•Ä1 ª Š´Äš"), …ù‡Ú½•I‡^ ©XêÝ A g Ñy ƒ ( ž {K¬3¥mÚ½¥ÑyÙ¦I‡?1$Ž êŠ). ù‡A5`², XJ XêÝ A ƒ Ü5gu˜‡ †‚†† R, @o^ Cramer {K O)´Ä•˜ž, ¤kI‡ $ŽÑ3 R SÜ?1, OŽ (JE3 R ¥. ù«y–¢Sþé˜„ †‚þ Ý ±9‚5• §|•¤á. ¤± Cramer {KŒ±3•2• œ¹ew«§ %å. d , XJ •þ b ‹I•Ñ3 R S, …1 ª |A| ê•áu R, @o Cramer {K‰ Ñ úªL²)•þ X ‹I• Üáu R. ~X, XJ A Xê Ü´ ê, b ‹I•Ñ´ ê, @o 1 ª |A| Š• ±1 ž, •§| AX = b )•þ X ‹I˜½• ´ ê. (ù˜: ^ž {q ÒØ •B`² .) (4.3.3) Cramer {KŒ±k˜«AÛ éu ‚5•§| x1 A x2 =b, ·‚Œ±n)•¦Xê x1 , x2 ¦ (4.3.3.1) )º. ·‚k± Ù¥ A = a11 a21 Ý a12 a22 ! œ¹Š•éuª ~f. b1 , b= , b2 Xe‚5|Ü'Xª¤á x 1 β1 + x 2 β2 = b , Ù¥ β1 , β2 • A . 5¿ •þ α1 = (a22 , −a12 )T Ú α2 = (a21 , −a11 )T ©O†•þ β2 Ú β1 α2 · β1 u"). ò (4.3.3.1) ªüà† α1 ŠSÈŒ (=SÈ α1 · β2 Ú x1 (α1 · β1 ) = α1 · b . OŽŒ• b1 b2 α1 · β1 = a11 a22 − a12 a21 = |A| , α1 · b = b1 a22 − b2 a12 = ¤±, X Cramer {Käó 2| x2 = |A |A| . éu˜„ @ 1| , x1 = |A |A| . aq/, ÏLò (4.3.3.1) ªüàÓ α2 ŠSÈŒ n, g,/Œ±òþã?ØŠí2. dž AX = b ƒ (4.3.3.2) a12 = |A1 | . a22 x 1 β1 + · · · + x n βn , u‚5|Ü'X Ù¥ β1 , · · · , βn • A . XJU é •þ α1 † β2 , · · · , βn Ñ , @oÏLò (4.3.3.2) ªüàÓ α1 ŠSÈ, BŒ±”ƒ c n = 2 A~˜ ¦Ñ x1 . XJ α1 qU¦ α1 · β1 = |A|, @o·‚ål Cramer {K úªÒ •\ C . ¤±, ·‚F"é •þ α1 = (α11 , · · · , αn1 )T ¦ α1 · β1 = |A| , α1 · β2 = · · · = α1 · βn = 0 , ½=, n X k=1 †† ù´Ä–“êÆ¥ ak1 αk1 = |A| , n X k=1 ak2 αk1 = · · · = n X akn αk1 = 0 . k=1 ˜‡-‡Vg. ÖödžŒ±òÙn)• C ¥•¹ 0, 1 …'u\~{Ú¦{µ4 ˜‡f8. 4.3 1 ª nØA^ 211 * (4.2.17.1) ª¥ 1 žÓ •{Œ±¦Ñ ‡ ªŒ±uy, α1 = (C11 , · · · , Cn1 )T fÐ÷v‡¦. u´, æ^ n = 2 n α1 · b 1 X x1 = = bk Ck1 . α1 · β1 |A| k=1 Pn ù p ¦ Ú ª k=1 bk Ck1 uÝ A1 1 1| . )•þ X Ù§‹I ,Ón¦Ñ. x1 = |A |A| 4.3.2 Ý •†1 ª ( Ï ® ² 3 ½ n 4.3.1 y ² ¥ ‰ Ñ). ¤ ±, ª ·‚®²• , éu˜‡• A, 1 ª |A| š"´ A ÷• ˜‡¿©7‡^‡. éu•˜„ œ¹, XJ A Ø´• , XÛlk'1 ª &EJ Ñ'u• &EQ ? • £‰ù‡¯K, ·‚k0 ˜ PÒ. (4.3.4) b I Ú J ©O´ [[1, m]] Ú [[1, n]] f8, |I| = |J| = r. b i1 , · · · , ir , J ¥ ƒUl Œ^S• j1 , · · · , jr . ·‚P (4.3.4.1) A n o I J =A n i i ··· i o 1 2 r j1 j2 ··· jr ai1 j1 ai2 j1 := . .. air j1 ··· ··· .. . ai1 j2 ai2 j2 .. . air j2 ··· I ¥ ƒUl Œ^S• ai1 jr ai2 jr . .. . air jr ù´‡ r 1 ª, ·‚¡ƒ• n o A ˜‡ r fª (minor of order r). I XJ I = J, ·‚¡ A ´ A ˜‡Ìfª (principal minor). XJ I = J = [[1, r]], ·‚¡ J n o I A •A r ^SÌfª½+qÌfª∗ (leading principal minor). J Ún 4.3.5: A ∈ Mr×n (K). K rank(A) = r ¿©7‡^‡´ A k˜‡ r fªš". y². e rank(A) = r, K•3 [[1, n]] ˜‡f8 J = {j1 , · · · , jr } ¦ n o A 1 j1 , · · · , jr ‰Ñ I •þ|‚5Ã'. ù˜ •þü¤˜‡• , Ù1 ª ŠT• A , Ù¥ I = [[1, r]]. cã •þ J n o n o I I | ‚5Ã'5`² A ´˜‡ ÷•• 1 ª, Ïd A 6= 0. J J n o I ‡L5, b A k˜‡ r fª A š". Ï• r ´ A ¤k1ê8, ¤±ùp •I8 I J n o I 7,´ [[1, r]]. •I8 J XJ ¤ J = {j1 , · · · , jr }, K A 6= 0 `² A 1 j1 , · · · , jr ‚5Ã J '. ¤± rank(A) ≥ r. rank(A) ≤ r o¤á (g•K 2.6), ·K 4.3.6: A ∈ Mm×n (K), r ≤ min{m, n}. K rank(A) = r ¿©7‡^‡´ A k˜‡ r dž rank(A) = r. fªš"… A ¤k r + 1 fªÑ u". y². ÏL A 1•þ|5n) A •, ·‚• rank(A) = r ¿©7‡^‡´•3 A ¥ r 1 ‚5Ã', … A ?¿ r + 1 1‚5ƒ'. éu•I8 [[1, m]] ?¿f8 I = {i1 , · · · , ik }, - BI L«d A 1 i1 , · · · , ik 1 ¤ fÝ . K•3 A ¥ r 1‚5Ã' …= •3 I ⊆ [[1, m]] ¦ |I| = r = rank(BI ). dÚn 4.3.6 Œ •, |I| = r ž, rank(BI ) = r ¿©7‡^‡´ BI k˜‡ r fªš". Tfª•´ A ˜‡ r ∗ “+qÌfª”´Šöé=©Lˆ leading principal minor =©‡“Minor (linear algebra)”c^ )º¥, ù˜Vg c„vk„ =©©z¥é“^SÌfª” Ù§`{. È{. ¥©©z¥DÚ ¡ •“^SÌfª”. 3‘Äz‰ ¡• leading principal minor ½ corner principal minor. Šö8 1oÙ 1 ª9ÙA^ eI I ¤A n o 212 fª, …A ?Û r fª A n o I J •˜½5gu,‡ BI (ò BI I J éA @‡ I =Œ). ¤±, •3 A ¥ r 1‚5Ã' ⇐⇒ •3 A ,‡ r fªš". aq/, A ùÒy² 4.3.3 ·‚I‡ A ?¿ r + 1 1‚5ƒ' ⇐⇒ A fª•". (Ø. õ‘ª!,†A y3·‚|^1 ?¿ r + 1 Š ª5½Â• A ·K 4.3.7: A ∈ Mn (K). K1 ª |λIn − A| ´˜‡'uC A A õ‘ª. õ‘ª, ù‡õ‘ªÚA n gÄ˜õ‘ª, § λ ŠkXš~ Xê • éX. Üáu K. ·‚¡ƒ• y². |^ Laplace Ðmªé n Š8B=Œ. •C S.þ<‚¬± λ L«Ý A ž,) Öò± PA (λ) L«• õ‘ª C , ØL7‡ž•Œ±†^Ù§i1. ( A A õ‘ª.† ·K 4.3.8: A ∈ Mn (K), λ0 ∈ K. K λ0 ´ A A Š …= |λ0 In − A| = 0. •Ò´`, Ý õ‘ª3 K ¥ Š. y². ŠâÚn 3.3.13, λ0 ´ A ‡´ |λ0 In − A| = 0. A Š …= ·K 4.3.9: A, B ∈ Mn (K). XJ A, B 3 C þƒq, =, •3Œ_Ý A, B A õ‘ªƒÓ. A3K ¥ A rank(λ0 In − A) < n. ·‚®²• P ∈ Mn (C) ¦ æ^ λ Š ŠT•§ A ö d^ B = P −1 AP 3 Mn (C) ¥¤á, K y². ÄkJ2Öö5¿˜‡{ü -‡ ¯¢: n • A Xê Ü5gu• K ž, ·‚Ó žŒ±@• A ´Xê Ü5gu C Ý . ÃØ´r A Š Mn (K) ¥ ƒ„´ Mn (C) ¥ ƒ, OŽA õ‘ª PA (λ) = |λIn − A| ž (J´ƒÓ . Ïd, • y²·K¤ã (Ø, ·‚Œ±† r A, B Ñ Š Mn (C) ¥ Ý . ŠâA õ ‘ª ½ÂÚ1 ª Ä 5Ÿ, PB (λ) = |λIn − B| = |λIn − P −1 AP | (Ï• λIn = P −1 (λIn )P ) = |P −1 (λIn − A)P | = |P −1 | · |λIn − A| · |P | = |λIn − A| · |P −1 | · |P | = |λIn − A| = PA (λ) . (ùp^ 5Ÿ |P −1 | · |P | = 1 ´3‰Öö {üöS.) ½ Â 4.3.10. V ´k•‘ K •þ˜m, A ∈ End(V ). Šâ·K 4.3.9, éu V ?¿ü|kSÄ 0 0 B, B , Ý MB (A ) Ú MB (A ) äkƒÓ A õ‘ª. ·‚òù‡õ‘ª¡•‚5C† A A õ‘ª, P• PA (λ). † éuA õ‘ªù‡-‡ Vg, ISþq ¿vkAO2• É Ú˜PÒ. 4.3 1 ª nØA^ 213 l·K 4.3.8 Ú“êÄ ½n (ë„½n A.4.3) Œ±• Mn (C) ¥ ?ÛÝ 3Eê•SÑk A Š. Ïd·‚Œ±”´ ·K 3.3.31 ¥(Ø: e V ´k•‘E•þ˜m, dim V ≥ 1, K V þ ?Û‚5C† A 3 C ¥kA Š (Ï A 3 V ¥kA •þ). (·‚ƒc‰Ñ y²Ø¦^1 ªÚA õ‘ª, ´éÙ§· À õ‘ªA^“êÄ ½n.) ~ 4.3.11. •ÄgêØ‡L n − 1 ± ‚5C† õ‘ª˜m V = K[X]≤n−1 . ÏL/ª¦ D : V −→ V ; (ë„~ 3.1.15) Œ f 7−→ f 0 . N´wÑ, f1 = 1 , f2 = x , f3 = x2 xn−1 , · · · , fn = 2! (n − 1)!  ù|õ‘ª ¤V ˜|Ä. 3ù|Äe‚5C† D Ý 0     L«• D =      1 0 0 1 .. ÏL{ü OŽŒ•, D A õ‘ª• PD (λ) = |λIn − D| = λn . ¤± D λ1 = 0 (•ØL§ -ê• n). (žÖög• λ1 éA A f˜m´Ÿo.)  ..     .. . . 0 .  ..  . 1 0 •Ã Š´ . 2 2 2 :^= θ I ! ^=C† A : R → R (ë„~ 3.1.9). 3 R cos θ − sin θ OÄe A Ý L«• A = . sin θ cos θ ¤± A A õ‘ª• PA (λ) = |λI2 − A| = λ2 − 2 cos θ · λ + 1. XJ | cos θ| = 6 1, K PA (λ) v k¢êŠ, ¤±dž A 3 R ¥vkA Š (•ÒvkA •þ). ù˜y– AÛ)ºQ3~ 3.3.25 ¥‰Ñ. ~ 4.3.12. •Ä²¡ R2 S7 ·‚UY?ØÝ A õ‘ª†ÙA Š éX. (4.3.13) A ∈ Mn (K) (Ù¥ n ≥ 1), PA (X) = |XIn − A| • A A õ‘ª. “êÄ ‡íØ (íØ A.4.9) `² PA (X) 3 C ¥k n ‡Š (-êO\) λ1 , · · · , λn , … ½n ˜ PA (X) = (X − λ1 ) · · · (X − λn ) . ù λi ÒÁ õ‘ª PA (X) 3 C ¥ ¤kŠ, λi Š•õ‘ª PA (X) Š • A A Š “ê-ê (algebraic multiplicity) ½{¡-ê. XJò PA (X) ¤ -ê•¤• λi Š PA (X) = X n + an−1 X n−1 + · · · + a1 X + a0 K Š â õ ‘ ª X ê Ú Š Ä ' X (ë „ · K A.4.2), PA (X) ~ ê ‘ a0 = PA (0) u¦È (−1)n λ1 · · · λn . ,˜•¡, lLˆª PA (X) = |XIn − A| Œ±wÑ, ± X = 0 “\õ‘ª PA (X) ž, ¤ Š u PA (0) = | − A| = (−1)n |A|. Ïd, (−1)n λ1 · · · λn = PA (0) = (−1)n |A|. ¤±·‚ ±eü‡'u1 ª |A| -‡(Ø: (4.3.13.1) (−1)n |A| uA A õ‘ª ~ê‘ , 1oÙ 214 1 ª9ÙA^ 9 |A| (4.3.13.2) ·‚2½Â A (4.3.13.3) u A 3 C ¥¤kA Š (-êO\) ¦È . , (trace) Tr(A) • Tr(A) := n X λi = A 3 C ¥¤kA Š (-êO\) Ú. i=1 l½Âª·‚•UwÑ Tr(A) ∈ C. õ‘ª PA (X) ¥ X n−1 Xê. A ∈ Mn (K) ž, Tr(A) ∈ K. Ï•ƒq ·K 4.3.14: • äkƒÓ ·K 4.3.15: A õ‘ª (·K 4.3.9), ¤±d (4.3.13.4) Œ± Xe·K: A, B ∈ Mn (K). XJ A, B 3 C þƒq, K Tr(A) = Tr(B). ÏL±þ·K·‚Œ± • A − Tr(A) := A (4.3.13.4) AO/, ´Šâ·K A.4.2 Œ• Tr(A) = −an−1 , = Ý ,˜‡{ü A = (aij ) ∈ Mn (K). K Tr(A) -‡ uA •x. é ‚ ƒƒÚ. y²·K 4.3.15 ·‚ky˜‡Ún. Ún 4.3.16: n ≥ 1. 1. é?¿ A, B ∈ Mn (K), AB 2. e P ∈ Mn (K) •Œ_ ƒƒÚƒ . é ‚ ƒƒÚ é ‚ ƒƒÚ. , Kéu?¿ A ∈ Mn (K), P −1 AP é ‚ y². é?¿ M ∈ Mn (K), 6ž± T (M ) L« M (bij ) ∈ Mn (K), ·‚k T (AB) = T (BA) = n X i=1 n X u BA é (AB)ii = (BA)jj = ‚ n n X X ƒƒÚ† A é ‚ ƒƒÚ. Kéu?¿ A = (aij ), B = aij bji , i=1 j=1 n n X X bji aij . j=1 i=1 j=1 dë\ÎÒ Σ ˜^Ä 5Ÿ (ë„ (A.3.2.1)) =• T (AB) = T (BA). ùÒy² Øä (1). é u Ø ä (2), • I ‡ 5 ¿ Ø ä (1) ( Ø Œ ± í Ñ T (P −1 · AP ) = T (AP · P −1 ) = −1 T (A(P P )) = T (A). ·K 4.3.15 y². ·‚Œ±ò A Š Mn (C) ¥ • 5OŽ Tr(A). ùƒ ub K = C. P T (A) • A é ‚ ƒƒÚ. Šâ·K 3.3.32, A Ú˜‡þn A0 ∈ Mn (C) 3 C þƒ q. Šâ·K 4.3.14 ÚÚn 4.3.16 (2), M ∈ Mn (C) 7−→ Tr(M ) ∈ C ùü‡¼êÑ3Ý þn , ¤±Ùé Tr(A) = T (A). Ú M ∈ Mn (C) 7−→ T (M ) ∈ C ƒ q C † e ± Ø C. ¤ ±, Tr(A) = Tr(A0 ), T (A) = T (A0 ). Ï • A0 ´ ‚ ƒ Ò ´ § ¤ k A Š, d½Â= Tr(A0 ) = T (A0 ). ù Ò y ² 4.3 1 ª nØA^ 215 g • K 4.10. Šâ·K 4.3.15 ÚÚn 4.3.16 (1), XJ A, B ´ƒÓŒ • , K Tr(AB) = Tr(BA). XJ m 6= n, A ∈ Mm×n (K), B ∈ Mn×m (K), @o Tr(AB) = Tr(BA) ´Ä˜½¤á ? e´, ž ‰Ñy². eÄ, žÞÑ‡~. ~ 4.3.17. A ∈ Mm×n (K), B ∈ Mn×m (K). ÏL©¬Ý A õ‘ªƒm ˜‡éX, = AB Ú BA λn · |λIm − AB| = λm · |λIn − BA| . (4.3.17.1) AO/, $ŽE|, ·‚Œ± m = n ž, ·‚w AB Ú BA äkƒÓ A ¯¢þ, ‡y² (4.3.17.1) •IÏL±eü‡©¬Ý ,, ·‚3ùp^ ! In −A 0 Im λIn λA B λIm ©¬þn · ! · λIn λA B λIm ! In −A 0 Im ! ½en 1 õ‘ª. m ª¦1 = λIn 0 B λIm − AB = λIn − BA 0 ª B λIm ª=Œ ! ! 5Ÿ (Ún 4.2.9). Šâ,Ú1 ª†A õ‘ª 'X (= (4.3.13.1) Ú (4.3.13.4)), ±9A C†e ØC5 (·K 4.3.9), ·‚Œ±‰±e½Â: õ‘ª3Ý ƒq ½ Â 4.3.18. n ≥ 1, V ´ n ‘ K •þ˜m, A ∈ End(V ). éu V ?¿ü|kSÄ B, B 0 , Ý MB (A ) Ú MB0 (A ) äkƒÓ 1 ªÚ,. ·‚ò |MB (A )| Ú Tr(MB (A )) ©O¡•‚5C† A 1 ªÚ,. cöP• det(A ) ½ |A |, öP• Tr(A ). Ï•• ,Œ±ÏLé ·K 4.3.19: ‚ ƒƒÚ5OŽ, ¤±±e(Ø e±Ý . n ≥ 1, V ´ n ‘ K •þ˜m. KN Tr : End(V ) −→ K ; ´ K-‚5N y²ÒC¤éN´ A 7−→ Tr(A ) . ŠóLã, =éu?¿ A, B ∈ Mn (K) Ú?¿ c ∈ K, Tr(A + B) = Tr(A) + Tr(B) , Tr(cA) = c Tr(A) . y². 3‰Öö. g • K 4.11. n ≥ 1, V ´ n ‘ K •þ˜m. y²: Ø•3‚5C† A , B ∈ End(V ) ¦ J«: •Ä‚5C† g • K 4.12. End(V ) Xe: A B − BA = I (ùp I L« V þ ð C†). ,. V = K[X] ´Xê3 K ¥ Nõ‘ª A : f (X) 7−→ f 0 (X) ; y A B − BA = I (ùp I L« V þ ð ¤ •þ˜m. ½Â‚5C† A , B ∈ B : f (X) 7−→ Xf (X) . C†). 1oÙ 216 4.3.4 1 ª9ÙA^ Cayley–Hamilton ½n ! Ì‡8Iý²e¡ -‡½n. ½n 4.3.20 (Cayley–Hamilton ½n): n ≥ 1, V ´ n ‘E•þ˜m, A ∈ End(V ). f (X) = PA (X) • A A õ‘ª. K f (A ) = 0. (ùp 0 ´"N .) 5 ¿ 3 ± þ ½ n ¥, f (A ) ´ ˜ ‡ ‚ 5 C †. ¤ ± T ½ n ´ `, ò A õ ‘ ª f (X) ¥ “ \ X=A ‚5C† f (A ) ´"C†. •, f (X) äkLˆª f (X) = det(XIn − A ), ´XJ † 3Lˆª det(XIn − A ) ¥± X = A “\, K¬ ˜‡1 ª Š ( ,T1 ª• 0), ù ´˜‡êŠ, Ø´½n 4.3.20 ¥ 9 ‚5C† f (A ). Ïd, 3n)½n 4.3.20 ž˜½‡5¿: k± X = A “\2OŽ1 ª, Úk± X •CþOŽ 1 ª‰Ñ õ‘ª, 2± X = A “\ ‚5C†, ùü«öŠ J´ ØÓ . ½n 4.3.20 y². ŠâíØ A.4.9, Œ±3 C[X] ¥òõ‘ª f (X) ‰Xe©) f (X) = (X − λ1 ) · · · (X − λn ) Ù¥ λ1 , · · · , λn ∈ C ´ A n ‡A Š (-ŠO\). u´ f (A ) = (A − λ1 I) · · · (A − λn I) . , ù , Šâ·K 3.3.32, •3 V λ1 , · · · , λn U^SfÐ´ A ˜|Ä v1 , · · · , vn ¦ A 3ù|Äe é ‚ ƒ, =   λ1 ∗ · · · ∗   λ2 · · · ∗    A= ..  .. . .  .  λn Ý A •þn ,  ( (ùp  λ1   0  A v1 = (v1 , · · · , vn ) ·   ..  = λ1 v1 , . 0   ∗ . . .   λj   j ≥ 2 ž) A vj = (v1 , · · · , vn ) ·   0  = ∗v1 + ∗v2 + · · · + ∗vj−1 + λj vj .   . . . 0 (Ò ∗ L«éA ~ê´ŸoŠÑÃ';‡.) eV0 = 0 , Vj = span(v1 , · · · , vj ) , j ∈ [[1, n]] Kéz‡ j ∈ [[1, n]] Œ±ÏL±þOŽ(J y •: •þ| (A − λj I)v1 , · · · , (A − λj I)vj−1 , (A − λj I)vj . Ø” 4.3 1 ª nØA^ 217 •¹u Vj−1 . •Ò´`, éz‡ j ∈ [[1, n]] þk (A − λj I)(Vj ) ⊆ Vj−1 . ‡E¦^ù‡(ØŒ f (A )(V ) =(A − λ1 I) · · · (A − λn I)(V ) = (A − λ1 I) · · · (A − λn I)(Vn ) ⊆(A − λ1 I) · · · (A − λn−1 I)(Vn−1 ) ⊆ · · · · · · ⊆ (A − λ1 I)(V1 ) ⊆ V0 = 0 . •Ò´` f (A ) ´ V þ "C†. ½ny.. þ¡·‚´^‚5C† ±e(Ø: Šó5•ã Cayley–Hamilton ½n. XJ^Ý ½n 4.3.21 (Cayley–Hamilton ½n): K f (A) = 0 (ùp 0 ´"Ý ). y². ·‚Œ±r A Š Mn (C) ¥ n ≥ 1, A ∈ Mn (K). Ý /ªLã, Œ± f (X) ∈ K[X] • A A õ‘ª. , Ïd§É•þ˜m Cn þ‚5C† A : v ∈ Cn 7−→ Av ∈ Cn 3IOÄe Ý . Šâ½n 4.3.20, f (A ) ´ Cn þ K 3.2.17 (2)), f (A) = 0. "C†. f (A ) Ý L«• f (A) (ë„· Cayley–Hamilton ½nd Hamilton§ u 1853 cÄkéãAÏ • (4 ¢Ý Ú 2 EÝ ) ‰Ñ y². êcƒ , Cayley¶ äóÓ (Øé¤k• Ñ¤á, ØL¦•éØ‡L 3 œ¹‰Ñ y². T½n˜„œ¹ 1˜‡y²5gu Frobeniusk . c¡·‚kéE•þ˜m ‚5C†y² Cayley–Hamilton ½n, , Ý /ª¥·‚w Ý XêŒ± g C f• K. ù¦ c¡½n 4.3.20 C ?¿f• K þ. íØ 4.3.22: n ≥ 1, V ´ n ‘ K-•þ˜m, A ∈ End(V ). ª. K f (A ) = 0. ÙÝ /ª. 3 (Ø•Œ±í2 f (X) = PA (X) • A A õ‘ y². ?¿ ½ V ˜|kSÄ, A ∈ Mn (K) • A 3ù|kSÄe Ý . K3ƒÓ Äe f (A ) Ý L«• f (A) (ë„·K 3.2.17 (2)). Šâ½n 4.3.21, f (A) ´"Ý . Ïd f (A ) 7,´ "C†. 4.3.5 SK S K 4.3.1. e ‚5•§|´Äk•˜)? e´, ž^ Cramer {K¦):   x1 + x2 + 2x3 + 3x4 = 1,      3x1 − x2 − x3 − 2x4 = −4, 1.  2x1 + 3x2 − x3 − x4 = −6,      x + 2x + 3x − x = −4; 1 2 3 4 § William Rowan Hamilton (1805–1865) O =êÆ[. ¶ Arthur Cayley (1821–1895), =IêÆ[. ¦u 1858 cuL nØ Ä:, Ù¥Äg‰Ñ Ý ¦{ ½Â. k Ferdinand Georg Frobenius (1849 – 1917), IêÆ[. ÍŠ A memoir on the theory of matrices C½ y“Ý 1oÙ 218 2.    2x1 − x2 + 3x3 + 2x4 = 4,     3x1 + 3x2 + 3x3 + 2x4 = 6,  3x1 − 2x2 − x3 + 2x4 = 6,      3x − x + 3x − x = 6. 1 2 3 4 S K 4.3.2. ^ Cramer {K¦)e ‚5•§|:   x1 + 2x2 + 3x3 − 2x4 = 6,      2x1 − x2 − 2x3 − 3x4 = 8, 1.  3x1 + 2x2 − x3 + 2x4 = 4,      2x − 3x + 3x + x = 8; 1 2 3 4   2x1 − x2 + 3x3 + 2x4 = 6,      3x1 + 3x2 + 3x3 + 2x4 = 5, 2.  3x1 − x2 − x3 + 2x4 = 3,      3x − x + 3x − x = 4. 1 2 3 4 S K 4.3.3. A, B ∈ Mn (R). y²: 1. Tr(AAT ) ≥ 0. 2. Tr(AAT ) = 0 ¿©7‡^‡´ A = 0. 3. e A, B Ñ´é¡Ý , K Tr((AB)2 ) ≤ Tr(A2 B 2 ). 4.4 !•ÀùSN, ± ·žÖ¿. * ü !˜††1 ª 1 ª9ÙA^ 220 1oÙ 1 ª9ÙA^ N¹ A ˜ A.1 •£Ö¿ êX êÆù€Æ‰• ©••Ä SN´ê Æ¯. lÖ„±5, ·‚®²Úê‹ éõc . ¤±·‚Œ Œ±`“ê”´êÆp•Nń) Vg . , , ý ‡^Ü6þ°( •ª )ºÄ¾Ÿo´ê, ùÙ¢´š~Ø²… ¯œ. Ö ÖöAT®²ég,ê! ê!knê!¢ ê±9Eêù VgkÄ ). l•Ð' g, š"g,ê, êÆ[‚u 18 -VâÅì' ²x Eê, <‚¤@• ê NX•ªªu {, Ù¥ L§K²{ êÆ¤þNõyž“ Œ¯‡. ~X, é¢ê (n)´{¤þ‡È©nØ î‚z '…˜Ú (ùáuÖöA ÆS êÆ©Û‘§SN). éŒJ, •uŸÌ ÖÃ{•[0 ù Å>F, {¤Ú°çª¥ êÆSN. Ž‡• \n)ù•¡êÆSN ÖöŒ±ë•êÆ©Ûa ²; á (~X [19]![20] ). •÷v ÖSN I‡, ·‚J2Öö5¿'ug,ê!knêÚEê ˜ -‡¯¢. 3e ©QãL§¥, ·‚¬¦^ þ5g8ÜØ Vg, •)8Ü (set)! ƒ (element)!f8 (subset)! ˜8 (empty set) . ·‚ ,‡b½ÖöÙGù •Ä Vg. (A.1.1) Äk, ·‚r¤kg,ê ¤ 8ÜP• N, Ù¥•¹ 0∗ . ly“ênÜ6 *:w, 8Ü N Ÿþ´d¤¢ Peano ún† 5•x . ·‚Ø‹Ž0 Peano ún, ·‚I‡rN§kXe -‡ ( wq²…) íØ: XJ S ´ N ˜‡f8, ¿… S Ø´˜8, @o S ˜½¹k˜‡• , =, •3 S ¥ ˜‡ ƒ m ¦ S ¥?Û ƒ a Ñ÷v a ≥ m. ±þ(ØÏ~¡•g,ê8 ûS5Ÿ (well ordering property). g • K A.1. XJ3þã(Ø¥r N †¤¢ê8, Ó Øä´ÄE¤á ? 3g,ê8Ü Ä:þŒ± ï ê8 Z (äNL§lÑ). e¡Qã (Ø´ ê8 Z ˜‡ aq5Ÿ, §Œ±dg,ê8 ûS5ŸíÑ. Ï•·‚¢Sþvk)ºXÛ E ê, ¤±ù˜ y²••UÑ . ·K A.1.2: S ´ Z ˜‡š˜f8, XJ S 3 Z ¥ke., =, •3 Z ¥ ÑŒu u c, @o S ˜½¹k• ƒ. e¡ ½n¢Sþ´·‚¥Æ~^ êÆ8B{ 1. c• ê. éuz‡ n = c ž, Øä P(n) ¤á. 61 ˜« ½. L •Q61LØr 0 † Giuseppe Peano (1858–1932), ¿Œ|êÆ[!Ü6Æ[!ŠóÆ[. 221 S nØ•â. ½n A.1.3 (êÆ8B n mathematical induction principle): P(n) •˜‡'u n Øä. b ±eü‡5Ÿ¤á: ∗ ò 0 8\g,ê, ùý“' ƒc¦ Šg,ê ½. ê n ≥ c, ƒ N¹ A ˜ 222 2. éu?Û •£Ö¿ ê k ≥ c, XJØä P(k) ¤á, KØä P(k + 1) •¤á. Kéu¤k n ≥ c Øä P(n) Ñ¤á. y². •Ä ê8 Z f8 S := {n ∈ Z | n ≥ c … P(n) Ø¤á}. ·‚F"y² ´ S ˜½´˜8. b Ø ,, K d · K A.1.2 Œ • S k ˜ ‡ • , ò Ù P • m. Š â 5 Ÿ (1), m 6= c. u ´ m − 1 ≥ c. du m ´ S • , ¤± m − 1 Øáu S. •Ò´`, éu k = m − 1, 5Ÿ P(k) ´¤ á . ´, Šâ5Ÿ (2), ùíÑ P(m) •¤á. ù† m áu S ƒgñ. „k˜«?1êÆ8B ½n A.1.4 (1 ‡ ê n ≥ c, •{´~^ . ÙnØ•âXe: êÆ8B n the second mathematical induction principle): P(n) •˜‡'u n Øä. b ±eü‡5Ÿ¤á: c• ê. éuz n = c ž, Øä P(n) ¤á. 1. 2. éu?Û •¤á. ê k > c, XJé÷v c ≤ m < k ¤k ê m Øä P(m) Ñ¤á, @oØä P(k) Kéu¤k n ≥ c Øä P(n) Ñ¤á. y². 3‰ÖööS. (A.1.5) 3 ê8 Z Ä:þŒ±3 Eknê8 Q. ÃØÖö±c‘ knê½ÂXÛ, 3 Ö ÖL§¥A«@±e¯¢: knê8 Q ¥ ƒTÐ´Œ± ¤©ê a/b /ª ê, Ù¥ a, b þ• ê… b 6= 0. ü‡©ê a/b Ú a0 /b0 ƒ ¿©7‡^‡´ ab0 Ú a0 b ùü‡ êƒ . lknê8 Q E¢ê8 R L§´êÆ©Û‘§ Ø%SNƒ˜. XJÖöéù SN n)k¯K, ½ŒQG‰©Ûa‘§ P“. ¢ê8 ˜‡A:´Ù¥?¿ü‡êŒ±' Œ . · ‚, #N¦^ +∞ Ú −∞ ùü‡ÎÒÚ¢êƒm' Œ . ÎÒ +∞ ÖŠ“ Ã¡”, ½§' ?Û¢êŒ, −∞ ÖŠ“KÃ¡”, §'?Û¢êÑ . kžÿ•r +∞ {P• ∞. e a ≤ b þ•¢ê½ ±∞, P ]a, b[ = {x ∈ R | a < x < b}. (êÆ©Û½p êÆ á¥ m« m~P• (a, b), • «Ou²¡¥: ‹IƒPÒ, ÖU^þ¡ù«{ŠêÆ©z¥~^ m «mPÒ.) aq/„k±e˜ PÒ: [a, b[ = {x ∈ R | a ≤ x < b} , ]a, b] = {x ∈ R | a < x ≤ b} , [a, b] = {x ∈ R | a ≤ x ≤ b} . , , a, b þ• êž, ·‚^±ePÒL«AÏ A‡ ê«m: [[a, b]] = {k ∈ Z | a ≤ k ≤ b} , ]] − ∞, b]] = {k ∈ Z | k ≤ b} , [[a, +∞[[ = {k ∈ Z | a ≤ k} . ~X, [[1, 5]] = {1, 2, 3, 4, 5}; ]] − ∞, 2]] = {k ∈ Z | k ≤ 2}; [[−6, +∞[[ = {k ∈ Z | − 6 ≤ k}. ±•{z ( ™7´Ü6þÃ ŒÂ ) •ª5`, l¢ê8 R EEê8 C •I‡Ú\˜ ‡PÒ i, ‡¦§÷v i2 = −1. <‚~¡TÎÒ i •Jêü (imaginary unit). 3“êÆ¥, <‚~ √ ~•#N^ −1 5L«Jêü i. ˜‡EêŒ±@•´˜‡/ªLˆª x + yi, Ù¥ x, y •¢ê, ©O¡•Eê x + yi ¢Ü (real part) ÚJÜ (imaginary part). ¢Ü• 0 Eê¡•XJê (purely imaginary number). Eê z = x + yi (Ù¥ x, y •¢ê) Ý (conjugate) z̄ ½Â• z̄ := x − yi, p § (modulus) ½ýéŠ (absolute value) |z| ½Â• |z| := x2 + y 2 . Ø z = x + yi ù«“ê /ª, Eê•Œ±^n /ª (trigonometric form) ½4‹I/ª (polar form) L«. äN5`, ˜‡Eê z o´Œ± ¤ z = ρ(cos θ + i sin θ) /ª, Ù¥ ρ ´šK ¢ê (¯¢þ ρ = |z|), θ ∈ R. z 6= 0 ž, θ 7L÷v x y cos θ = , sin θ = Ù¥ x , y ©O´ z ¢ÜÚJÜ. |z| |z| A.1 êX 223 éuš"Eê z, ?Û÷vþã^‡ θ ∈ R ¡• z ˜‡Ë (argument, amplitude). (0 Ë Ø ƒ½Â.) XJ·‚•› θ Š3 ] − π, pi] ù‡«m, @o θ ŠÒ z •˜(½ . Ï~r ] − π, π] ù‡«mS Ë Š¡• z Ë ÌŠ (principal value of the argument)‡ . |Êê 4‹I/ªŒ±é•B é?¿ θ, θ0 ∈ R , dd„Œ± Ãue¡ de Moivre úª§ (cos θ + i sin θ)(cos θ0 + i sin θ0 ) = cos(θ + θ0 ) + i sin(θ + θ0 ) . : é?¿ θ ∈ R , n ∈ N , (A.1.5.1) ó ?1¦{$Ž, ùÌ‡ ·‚¬I‡¦^˜ ŠâEC¼ênØ¥-‡ ©Û (cos θ + i sin θ)n = cos(nθ) + i sin(nθ) . •£, 'X±~ê e •. •ê¼ê, Ù¥ e = lim n→+∞ 1 + n1 n . Euler úª¶ , ·‚k é?¿ θ ∈ R , eiθ = cos θ + i sin θ . (A.1.5.2) ù‡úª †>Œ±üÕ‰Ñ½Â, I‡Öö )•õ ©Û•£. XJ6žØ+†> ¹Â, Ö ö•Œ±r (A.1.5.2) ª m> Š eiθ ù‡ÎÒ ½Â. |^ (A.1.5.2) ªŒ±éN´ (A.1.5.1) ª. ,, XJ x, y ∈ R, (A.1.5.2) ª„L² ex+yi = ex (cos y + i sin y) . Ïd, éuEê z = x + yi (x, y ∈ R), ez /ª. ýéŠ´ ex . •êLˆª ez ~~•¬ ¤ exp(z) (A.1.6) ·‚b Öö• EêƒmŒ±?1\~¦ØoK$Ž (‰Ø{žØêØ• 0). ·‚• rNEê8 C ‘koK$Žž, ~~¡ƒ•Eê• (field of complex numbers). Š•Eê• C f8, ¢ê8 R ´'u\~¦ØoK$Žþµ4 , •Ò´`, XJ a, b Ñ´¢ê, @o a + b, a − b Ú ab •Ñ´¢ê, b 6= 0, a/b •´¢ê. aq/, knê8 Q •´'u\~¦ØoK$Žþµ4 . Ïd, • rN R Ú Q ù«5Ÿ, ·‚Œ±©O¡ R Ú Q •¢ê• (field of real numbers) Ú knê• (field of rational numbers). •˜„/, b K ´ C ˜‡f8, §•¹ 0 Ú 1, ¿… K 'u\~¦ØoK$Žþµ4, @ o·‚` K ´ C ˜‡f• (subfield)k . XJ L •´ C f•, ¿… K •´ L f8, @o·‚ ` L ´ K (3 C ¥) ˜‡*• (extension field), •¡ K ´ L (3 C ¥) ˜‡f•. ~ A.1.7. P Q(i) := {a + bi | a, b ∈ Q}. ÖöØJ y, Q(i) ´ C √ √ {a + b 2 | a, b ∈ Q} •´ C ˜‡f•. ¢ê• R ´ Q( 2) *•, ·K A.1.8: Eê• C ?Ûf•Ñ´knê• Q √ ˜‡f•. aq/, Q( 2) := Ø´ Q(i) *•. *•. y². U½Â, ?Ûf•Ñ‡•¹ 0 Ú 1 ùü‡ ƒ. ¤kknêÑŒ±d 0 Ú 1 ²Lk•g\~ ¦Ø$Ž . Šâf•'uoK$Ž µ45, Q 7,•¹3Tf•ƒS. ‡ •k ÖŒU¬ [0, 2π[ ù‡«mŠ•Ë ÌŠ ‰Œ. § Abraham de Moivre (1667–1754), {IêÆ[. ¶ Leonhard Euler (1707–1783), a¬êÆ[. k ISNõp “ê á, ~X [22], [25], [35] , ÑrEê• C f•¡•ê•. ù«·¶ŒU´å u Hecke ÍŠ [10] (=©€È‡• [11]). ØL, 3y“ “êêØë•Ö¥, “ê•”˜c •ÊH/ Š““êê•” {¡, ““ê ê•”ù‡Vg ½ÂI‡N\þ•r •›^‡ (\þ@‡•›^‡ R Ú C ÑØ2´“êêØ¥¡ “ê•”). N¹ A ˜ 224 •£Ö¿ 8Ü†N A.2 ÖöAT3¥ÆêÆ®²ÆL (Èƒ) 8ÜØ† Ä •£. ØL, • •BÖöESý „´5é˜ Ä Vg‰˜ £ , ^Bò·‚æ^ PÒÚâŠ5‰˜e. , ·‚ (A.2.1) Xc©¤ã, ·‚b½Öön)•)“f8”Ú“˜8”3S ˜ 8ÜØÄ VgÚ~^P Ò. ~X, ·‚Ö 8Ü ½ÂÏ~kü«: ˜´† ÞÑ8Ü ¤k ƒ. ¦^ù«•ª˜„‡ ¦8Ü ƒ•kk•õ‡, ½ö•,´Ã•õ‡ Ek' N´ •ª Ü ÞÑÙ ƒ. ~X, X = {1, 2, 3} Ú N = {0, 1, 2, · · · } ùü« {Ò´^ù«•ª5£ã8Ü. ,˜«•ª´¦^˜‡ Ä–ÎÒ (~Xi1 a) “L8Ü ˜„ ƒ, , ÏL˜«5Ÿ (P• P(a)) 5£ãTÎÒU“L 8Ü¥˜‡ ƒ ¿©7‡^‡. ù •x 8Ü X Œ±^ X = {a | P(a)} ù«/ªÖ . ùp ÎÒ | å©… Š^, kžÿ•¬^ : ½öÙ¦Œ±å ©…Š^ ÎÒ“O. •k žÿ, Œ±r 5Ÿ P(a) ¥ Ü©Šé˜3©…ÎÒ | ƒc. ~X, ê8Ü N∗ Œ±^±e?Û˜«•ªL«: N∗ = {a | a ´ ê… a > 0} , N∗ = {a : a ´ ê… a > 0} , ∗ N = {a ∈ Z | a > 0} , N∗ = {a ´ ê : a > 0} , Xd . e X •˜‡8Ü, a ∈ X L« a ´ X ƒ. ü‡8Ü X Ú Y ƒ , ´• X Ú Y ¤¹ ƒ ƒÓ, = a ∈ X …= a ∈ Y . ˜8 PÒ´ ∅. éu?¿8Ü X, ˜8 ∅ Œ±À• X ˜‡ f8. XJ Y ´ X f8, Œ±^PÒ Y ⊆ X ½ X ⊇ Y 5L«. XJ Y ´ X f8, … Y 6= X, K¡ Y ´ X ýf8 (proper subset), P• Y ⊂ X, Y $ X ½ X ⊃ Y , X % Y . ‘kÄ½¹Â P Òa∈ / X ´• a Ø´8Ü X ƒ, Y * X ´• Y Ø´ X f8. Ù¦aqPÒ ¹Â•daí. XJ8Ü X •¹ ƒ=kk•õ‡ (Œ±´ 0 ‡), K¡ X ´k•8 (finite set), ÄK¡ X •Ã•8 (infinite). XJ X ´k•8, §¤¹ ƒ‡ê•¡• X Äê (cardinality), P• #X, Card(X) ½ö |X|. ~^ |X| = ∞ 5L« X ´Ã•8. é?¿ü‡8Ü X, X 0 , Œ±½Â§‚ (k È (Cartesian product) ½¦È (product) •Xe 8Ü X × X 0 := {(a, a0 ) | a ∈ X , a0 ∈ X 0 } . •Ò´`, X × X 0 ´±¤k÷v a ∈ X Ú a0 ∈ X 0 kSé (a, a0 ) • ƒ 8Ü. N´wÑ, X × X 0 ´k•8 …= X Ú X 0 Ñ´k•8, … §‚´k•8ž, |X × X 0 | = |X| · |X 0 |. 5¿, (k È X × X 0 ½Â•6u X, X 0 Ö c ^S. †óƒ, Øš X = X 0 , ÄK X × X 0 Ú X 0 × X ´ØÓ 8Ü. •˜„/, éuk•õ‡8Ü X1 , · · · , Xn , Œ±aq/½Â§‚ (k È• X1 × · · · × Xn := {(a1 , · · · , an ) | a1 ∈ X1 , · · · , an ∈ Xn } . e A, B Ñ´8Ü X f8, §‚ 8 (difference) A \ B ©O½ÂXe: 8 (intersection) A ∩ B!¿8 (union) A ∪ B Ú A ~ B A ∩ B : = {x ∈ X | x ∈ A … x ∈ B} , A ∪ B : = {x ∈ X | x ∈ A ½ x ∈ B} , A \ B : = {x ∈ X | x ∈ A † 'u• k.) únz8ÜØ, k, ÖöŒëw [26] ˜Ö x∈ / B} . 1˜Ù ( éuÐÆö5`, ·‚š~ØïÆ }ù‡Ú A.2 8Ü†N 225 •˜„/, éu X k•õ‡f8 A1 , · · · , An , Œ±aq/½Â 8 A1 ∩· · ·∩An Ú¿8 A1 ∪· · ·∪An . w,, ?¿N† A1 , · · · , An ü ^SÑØ¬UC A1 ∩ · · · ∩ An Ú A1 ∪ · · · ∪ An . éu˜‡‰½ 8Ü X, Œ±ò X z‡f8Š•˜‡ ƒ, òù ƒÂ8å5 ¤˜‡8 Ü, P• P(X), ¡• X ˜8 (power set). X = ∅ ž, P(X) = {∅}. 5¿, {∅} L« Ø´˜8, ´± ∅ ••˜ ƒ ˜‡ü:8 (singleton) ! (A.2.2) 8Ü ,ý“êÆ•Ø% Vg. ´êÆØU•r˜‡‡8Ü áå5?1ïÄ, ´ I ‡ • Ä 8 Ü † 8 Ü ƒ m ' é Ú p Ä. • d, · ‚ l Ø m“N ” V g. ˜ ‡ N (map, mapping) Ï~^ f : X → Y ù PÒ5L«. O(5`, ‡½Âù ˜‡N I‡•²n‡‡ƒ: Ù¥ü‡‡ƒ´ü‡ (ƒÓ½ØÓ ) 8Ü X Ú Y , • ˜‡‡ƒ´˜‡éA{K, P• f . ùp 8Ü X Ï~¡•N f ½Â• (domain), Y ¡• f • (codomain), TN •¡•l X Y ˜‡N . 'uùp éA{K f , ·‚k…=kù ‡¦ (·‚ …òù‡5Ÿ¡•°(é A5): éu X ¥ ?¿ ƒ a, TéA{KU (ƒ/û½•˜˜‡ Y ¥ ƒ b ¦ a 3T{ KeéAu b. ùé{wþ ¿ØJn), ´• •ÏÖö;m•õ n)Ø«, ·‚ŽrN±e A:: 1. Ø þ¡¤ã °(éA5‡¦, ·‚éuéA{K f vkÙ¦ •›. ÃØT{K £ ãkõo{üo !Ã Ã !‚¯Ùß!VrEŠ½ööîÃn, •‡§Ué X z‡ ƒ•˜(½/š Y ¥ ˜‡ ƒ†ƒéA, T{KÒŒ±^5Š•˜‡N . 'X, ·‚Œ ± X = {0, 1}, Y = {Šö <, Šö IŠ, Šö 1Š}. , ·‚Œ±r15½ù ˜‡ éA{K f , §5½ 0 éA Šö <, 1 éA Šö 1Š. ¦+ÖöŒU@•ù‡{K›½ /›©Vr, ½ö@•§#¶Ù©ÎÃnd, ù‡éA{KÒ´l X Y ˜‡N . ¤ ±, ·‚½Â˜‡N žØI‡ RÙéA{K´Ä)M Ã';‡ ¯K, •I‡•ÄT {K´Ää “°(éA5”. 2. °(éA5•‡¦éu½u• X ¥ z‡ ƒ, ÑU •˜(½ Y ¥˜‡ ƒ†ƒéA. § ¿Ø‡¦é Y ¥ z‡ ƒ b ˜½•3 X ¥,‡ ƒ a UéA b. ~X·‚c˜ãÞÑ ~fÒØ÷vù˜:. XJ˜‡N f : X → Y äkfâ¤` 5Ÿ, =, éu Y ¥ z‡ ƒ b, oUé X ¥ ƒ a Uìéu{K f UéA b, @o·‚`N f ´˜‡÷ (surjective map). • rN˜‡N ´÷ , kžÿ·‚¬3L«TN žr¤^ †Þ → U ¤ f. u ´, PÒ f : X Y L« f ´˜‡l X Y ÷ . 3. éu X ¥ü‡ØÓ ƒ a Ú a0 , °(éA5‡¦ f Uò a Ú a0 ©OéA Y ¥ü‡(½ ƒ b Ú b0 , ¿Ø‡¦ b Ú b0 •ØÓ. †óƒ, #N X ¥, ØÓ ƒUìéA{K f éAuƒÓ ƒ. XJ˜‡N f : X → Y r X ¥ØÓ ƒo´éA Y ¥ØÓ ƒ, @o·‚` f ´ ˜‡ü (injective map). 3L«ü ž, ·‚~~¬r¤^ †Þ → U ¤ ,→ f. • Ò´`, PÒ f : X ,→ Y L« f ´˜‡l X Y ü . XJ˜‡N Q´ü q´÷ , ·‚Ò`§´˜‡V (bijective map) ½ö˜˜éA (one∼ to-one correspondence). 3I‡rN˜‡N žV ž, ·‚¬r¤^ †Þ → U ¤ − → ∼ f. u´, PÒ f : X − → Y L« f ´˜‡l X Y V . ~ A.2.3. XJ f : Z → Z Uì x éAu x3 5½Â, @o f ´ü Ø´÷ ; XJ f : Z → Z U ì x éAu x2 5½Â, @o f Ø´ü •Ø´÷ ; XJ f : Z → Z ½Â´ x éA x + 2, K f ´V . XJ f : Z → N Uì x éAu |x| 5½Â, K f ´÷ Ø´ü . N¹ A ˜ 226 •£Ö¿ 5P A.2.4. k˜‡¯KŒUI‡•˜ AO %ÊÊ Öö‰‡`². ¦+3ýŒõêœ¹e, · ‚•Ä8Üm N žÑ••Äš˜ 8Ü, Oo´¬k<=Ø4‡¯: XJ X ½ Y ´˜8, @ol X Y N TXÛ½ÂQ ? bX·‚´3‰˜‡ ppt ü«© , @od?˜½¬‰˜‡6žÛõ‰Y òžÂ˜ J, 3 ‰Öö þžmgCg•˜e¯K ‰Y. ŒJ·‚ùp´3 ©iÖv, vk4úñ¡ ¹O Å¬. ¤±·‚•Ð† Ñ¯K ‰Y: Äk, XJ X ´˜8, @ol X ?¿ 8Ü Y (•) Y ´˜8 œ¹) k…=k˜‡N , TN éA{KÒ´“å•Ø‰” ˜{K; XJ X š˜, @ o X ˜8Ø•3?ÛN . Ù ¢ ù † Ù ` ´ ˜ ‡ £ ‰, Ø X ` ´ ˜ ‡ Ö ¿ 5 ½ ½ ½. • Ò ´ `, · ‚ Œ ± @ • ƒ c (A.2.2) ˜ã‰Ñ ½Â•é X Ú Y Ñš˜ žÿ¦^, ff‰Ñ ù‡‰Y^5Ö¿5½ X ½ Y •˜8 œ/. ,, XJ·‚M‡rc¡ ½Â@^3 X ½ Y •˜8 œ/, @o•Œ±^ „© i •ªžr‰þã‰Y‰‡)º. ‰Y ŒéATŃń) . •Ò´`, XJ8Ü X ¥¹k˜ ƒ, Y vk?Û ƒ, @o·‚3N ½Â¥5½ °(éA5Ÿw,Ã{¢y. ¤±dž·‚@•l X ˜8vk ?ÛN . y35)ºù‡‰Y cŒé. XJ·‚c[T ƒc'uN °(éA5Ÿ £ã, ¬uy·‚ `{´: éu X ¥ z‡ ƒ, ‡¦{KU•˜(½ Y ¥ ƒ†ƒéA. X • ˜8 žÿ, ·‚‡çyù‡`{k…=k •ªÒ´ŸoÑØ‰. ùÒÐ'˜‡[•é¯f«ì ù 5K: “zg\êÆ•Á 100 ©, ·Òøy\ 100 ¬a.” XJ¯fl5vk L 100 ©, @o·‚Œ±@•ù‡5K ¿gÒ´[•ŸoÑØ‰. ( ,, XJ[•E,‡‰¯føy, ·‚ Œ±`@´ÑuÒ, †þã5KÃ'.) XJÖöé·‚ )ºEú Ã{ É, @oØ”éd¯KkØ ƒ. ¢Sþ, 3 Ö ÆS L§¥ Ñù‡¯K´ 1 Ï . ŒU Ööò5ÆS˜ ‰ÆnØ (category theory) žÿ 2 n)ù‡¯K•Ø´. d , e©¥ g•K A.2 •Œ±Š•˜‡ y5<y·‚'u˜8m N ‰Y (½ö` ½) •Ÿo´k n . (A.2.5) f : X → Y •š˜8Üm N . éu a ∈ X, 3 Y ¥ÏLéA{K f †ƒéA ƒ Ï~P¹• f (a), § ¡• ƒ a 3N f Šê ” (image). L«N éu ƒ Š^~^ PÒ´ a 7→ f (a). éu X ˜‡f8 A, a HA¥ ƒž, ò f (a) ù 5g Y ƒ˜3 ˜å ¤ 8ÜP• f (A), §´ Y f8, ¡• A 3N f Šê ”. ^PÒLãžÏ~Œ±P f (A) = {f (a) | a ∈ A} . e A = ∅, K@• f (A) = ∅. 8Ü f (X) ¡•N f ”½öŠ• (range), § ,˜‡~^PÒ´ Im(f ). w,, f (X) = Y ¿©7‡^‡´ f •÷ . e b ∈ Y , ?¿÷v f (a) = b ƒ a ∈ X ¡• b 3N f Šê ˜‡ ” (preimage). 5 ¿, Šâ ƒ b ØÓœ¹, ù ƒ a ŒUØ•3, •ŒU•3 Ø•˜. éu Y ?¿f8 B, ·‚^ f −1 (B) L«±e8Ü f −1 (B) := {a ∈ X | f (a) ∈ B} . ¡• B 3 f Šê ”. B = ∅ ž, ·‚@• f −1 (B) = ∅. ÖöŒU±cÆL“_N ” Vg (ë„e© (A.2.9) ˜ã). f k_N ž, § _N ²~ −1 P• f . •Nk Öö±c‘ P“½ á¬kù `{: f _N Ø•3ž, ØU ¦^ f −1 L«˜‡N . ·‚¿ØÄ½ù‡`{, ØL·‚‡`² ´, 3¦^ f −1 (B) ù‡PÒž, ·‚r§ Š˜‡ N, ¿Ø‡¦ù‡PÒ ?Û˜Ü©Ñk¿Â. (ùÒÐ'˜‡Çi Ü ÄüÕÄ Ñ5 žÿ, kŒUE´‡Çi, •kŒUØ2´‡Çi.) ¤±, =¦ f _N Ø• 3 (l ØT^ f −1 L«˜‡N ), ·‚EŒ±^ f −1 (B) ù‡PÒL«þ¡½Â 8Ü. ù˜P Ò3êÆ.´š~~^ , ÏdF"Öö˜½‡n)ù˜:, Ø‡ÕYu_N ´Ä•3 ¯K. § A.2 8Ü†N 227 XJ B •¹˜‡ ƒ b, = B = {b}, @o ”8Ü f −1 (B) = f −1 ({b}) •¡•N f 3 ƒ b ƒþ n‘ (fiber). 3Ø–uÚåÜÂ žÿ, ·‚~ò f −1 ({b}) ù‡8Ü{P• f −1 (b). ˜‡Œ UÚå‡ ÜÂ |Ü´: f k_N , ¿…^ f −1 L«§ _N ž, f −1 (b) , ˜‡¹Â´ b ∈ Y 3N f −1 : Y → X Šê ”. ¦+Xd, ùpŒU· ü‡¹Â OÁ Œ± Ñ . Ï• f _N •3ž, b ∈ Y ”•U´ X ¥•˜˜‡ ƒ a. ¤±, f −1 ({b}) = {a}, b 3 f −1 Šê ”Ò´ ƒ a. •Ò´`, dž f −1 (b) ü«ŒU ¹Â˜‡´8Ü {a}, ,˜‡ ´ ƒ a. XJ`ù‡ O3ýŒõêœ¹eŒ± Ñ, ƒ&õêÖöØ¬kr ‡é. ØJuy, N f : X → Y ´ü …= éu?¿ b ∈ Y , Ùn‘ f −1 (b) –õ•¹˜‡ f ´÷ ¿©7‡^‡´éu?¿ b ∈ Y , Ùn‘ f −1 (b) – ¹k˜‡ ƒ. (A.2.6) XJ˜‡N f : X → Y S.þ` f ´˜‡¼ê (function)∗∗ . •Y ¥ ƒÑ´ê (= Y Éê8 C ƒ; f8), @o·‚ 'u“¼ê”Ú“½Â•”ùü‡c, ½NÖö3¥Æž n)Ú Ö .~k «O. ¥ÆžÖ p öŒU‘ Lù SK: ¦¼ê f (x) = x(x − 1) ½Â•. 3ù‡SK •ã¥, “¼ê”Ú“½ Â•” Vgî‚5`ÑÚ·‚3 Ö¥ ½Ñk«O. Äk K •ã@•¼êÒ´üX•˜ p ‡'uCþ x Lˆª (ùpÞ ~f´ x(x − 1)), , §%@TLˆª¥Ñy Cþ x •3¢ ê‰ŒS Š. dž, TSK¤` “¦½Â•”Ò´•3¢ê8 R ¥éÑ˜‡f8 X, ¦ ¤‰ Lˆª f (x) TÐéuf8 X ¥ ¢êk¿Â. u´, ¤‰ LˆªŒ±½ÂÑl X R ˜‡¼ ê. ù n)•ª3NõDÚ êÆ¯K¥E,' •B, ¤±¥Æ ãŒ[ŒU•I‡Uù« •ªn)“¼ê”Ú“½Â•” Vg. Ö0 ½ÂÚy“êÆÆâ ÊH É .~ƒÎ, =, ˜‡“N ” f : X → Y •¹ ‡ƒØ•´ŠûCþ éA{K f , „‡•)ü‡•½ 8Ü X Ú Y . •Ò´`, Uì½Â, X Ú Y ùü‡8Ü•´TN |¤‡ƒ, Ù¥ X Š•TN ˜‡| ¤Ü©, † •½ éA{K f –\u= ƒ, ·‚† r X ¡•TN ½Â•. ·‚J2Öö5¿, Uì Ö *: (ù•´“êÆ+•¥˜„æ^ *:), üXd“éA{ K”´ØUû½“½Â•” , Ï•kžÿÓ éA{KŒ±éØÓ 8Ü X Ú Y k . ~X, ÏL 3 éA{K x 7→ x ·‚QŒ±½Â˜‡l Z Z N f1 : Z → Z ; x 7→ x3 , •Œ±½Â˜‡l R 3 R N f2 : R → R , x 7→ x , „Œ±½Â˜‡l Z R N f3 : Z → R ; x 7→ x3 . ¦+¦ ^ éA{KƒÓ, î‚5`·‚„ÁT@• f1 , f2 , f3 ń‡pØƒÓ N , §‚3éõ• ¡¬LyÑØÓ 5Ÿ. ~X, f1 Ú f3 Ø´÷ , f2 ´÷ . 2'X, e- B • R ¥ m«m −1 ] − 1, 1[ := {a ∈ R | − 1 < a < 1}, K f2 (B) = B =] − 1, 1[ , , f3−1 (B) = {0}. 3 “ ê Æ ¥, ‡ y ü ‡ N f1 : X1 → Y1 Ú f2 : X2 → Y2 ƒ , 7 L … • L y ± e ^ ‡ Ñ¤á: X1 = X2 , Y1 = Y2 , f1 Ú f2 ¤^ éA{KƒÓ (=, éu?¿ a ∈ X1 = X2 þk f1 (a) = f2 (a) ∈ Y1 = Y2 ). ,, XJlþe©Ùé²w/• f1 Ú f2 ½Â•Ú •ÑƒÓ, @ o•‡ yff¤ã • ˜^, =, éz‡ ƒ a ∈ X1 = X2 , þk f1 (a) = f2 (a). ·‚3ùp^BrN˜e, Ó éA{KkžÿŒU¬±ØÓ LˆªLˆÑ5. ·‚u éA{K´ÄƒÓ, Ø´wÙLˆª´ÄƒÓ, ´‡w´Äéuz‡ ƒ Š^ JÑƒÓ. ~ X, Ó ´l R R N , XJ f1 æ^Lˆª f1 (x) = (sin x)2 + (cos x)2 Lˆ, f2 K^ f2 (x) ð u 1 5Lˆ, @o•, ö LˆªØÓ, duéuz‡ x ∈ R þk f1 (x) = f2 (x), ¤±·‚E, ` f1 Ú f2 (Š•l R R N ) ´ƒ . 2'X X = {0, 1}, Y = R, XJ½Â f1 : X → Y 3 • f1 (x) = x, f2 : X → Y ½Â• f2 (x) = x , @o¦+LˆªØÓ, f1 = f2 . éu?¿ü‡8Ü X, Y , ·‚^l X Y ¤kN Š• ƒ, dd EÑ˜‡8Ü, òÙP • Map(X, Y ). c¡ ?Ø‘3`²: X1 6= X2 ½ö Y1 6= Y2 ž, Map(X1 , Y1 ) Ú Map(X2 , Y2 ) ù ü‡8Üžvkú ƒ . ∗∗ •,Uì½Â5`, •k˜Ü©N ^“N ”Ú“¼ê”ù ¡•¼ê, Ööò53êÆ c. 3ž„I‡Öö”‘VÃ˜ ŒÝ, °NêÆ f·È Ò¬uy, péõ<ŒU¬?¿/· Phf‚ù«ÃúŒä ?51•. N¹ A ˜ 228 •£Ö¿ éu?¿š˜8Ü X, ÏLòz‡ ƒéA Ùg Œ± ½Âl X X ˜‡N , ù ‡N ¡• X ð N (identity map), P• IdX , ½ökž{P• Id. éA˜8 ∅, ·‚Q`L, l∅ ∅ k…=k˜‡N . ·‚rTN Š (½ö`·¶•) ˜8 ∅ ð N . y3 A ´8Ü X ˜‡f8. ÏLòz‡ ƒ a ∈ A éA a Œ± A ð N IdA : A → A. ØL, Ó éA{K•Œ±@•½Â l A X ˜‡N i : A → X ; a 7→ a, Ï • a ∈ A ž•Œ±r a Š X ƒ. ùÑ † IdA ´k«O , Ï•§ •† IdA ØÓ. ·‚Ï~rþãN i : A → X ¡•f8 A X ¹\N (inclusion map) ½g,¹\ (natural inclusion). ,, XJ A = X, @o¹\N i : A → X Ò´ IdX . ·‚• ˜8 ∅ Œ±@•´?¿8Ü X f8. Ïd, l ∅ X ATo´k˜‡¹\N . ·‚ƒcq`L ∅ X k…=k•˜˜‡N . ÏdTN ˜½´ùp¤` ¹\N . ù•`² ·‚AT@• ∅ ?Û8Ü X ÑkN , ÄK“lf8 8ok¹\N ”ù‡ØäÒ¬k~ . ¹\N o´ü , Ïd·‚~~^/X i : A ,→ X PÒ5L«˜‡¹\N . y 3 · ‚ Œ ± r 5 P A.2.4 ¥ ( Ø E ã X e: é u ? ¿ 8 Ü Y , 8 Ü Map(∅, Y ) T k ˜ ‡ ƒ, T ƒ ´ l ∅ Y ¹\N ; é u ? ¿ š ˜ 8 Ü X, 8 Ü Map(X, ∅) ´ ˜ 8, =, |Map(X, ∅)| = 0. g • K A.2. X, Y þ•š˜ k•8Ü, m = |X|, n = |Y |. y²: |Map(X, Y )| = nm . (Šâ (A.2.6) ˜ã"— o(, XJ·‚5½3OŽ8Ü ƒ‡êž 0 "g˜• u 1, @o dK (Ø3 X ½ Y •˜8ž•¤á.) (A.2.7) b kü‡N f : X → Y Ú g : Y → Z, Ù¥ f •Ú g ½Â•¬Ü. ·‚kb½ X š˜. (džN f Ú g •35 %¹X Y Ú Z •š˜.) éu X ¥ z‡ ƒ a, kÏL f Œ±•˜/éA Y ¥˜‡ ƒ b := f (a), , ¦^N g Œ±2r b = f (a) é A Z ¥•˜(½ ƒ g(b) = g(f (a)). nÜ f Ú g k Š^ J, ·‚ ˜‡éA{K, §ò X ¥z‡ ƒ a °(éAu Z ¥ ƒ g(f (a)). dd·‚ lX Z ˜‡N , P• g ◦ f , ¡• g Ú f EÜ (composition) ½EÜN (composite map). 3Ø—uÚåÜÂž, ·‚• ~^ gf ù‡{zPÒ5O“ g ◦ f . XJ X •˜8, @o g ◦ f : X → Z Œ±½Â•l˜8 Z @‡•˜ N . ù , ÃØ X ´Ä•˜8, f : X → Y Ú g : Y → Z EÜN g ◦ f o´Œ±½Â . Ï~^ g ◦ f : X −→ Z ; a 7−→ g(f (a)) ½ö f g g ◦ f : X −→ Y −→ Z PÒ5 /L« g Ú f EÜN . k Ö (~X [22]) •òEÜN g ◦ f ¡• g † f ¦È (product). • ~ ÐÆö (¾± 9;•˜ ŒU ÜÂ, Ö˜„¬;•ù«¡ . ØL, òN “ EÜ”ù«$Žn)•˜«“¦ {”, ù (´k n (¤± gf ù«PÒ ¦^•kÜn5). ¯¢þ, N EÜ$ŽÚê ¦{ $Ž– k±eü:aq 5Ÿ: (1) ü (identity element) •35: éu?¿N f : X → Y , ok (A.2.7.1) f ◦ IdX = f = IdY ◦f . ùL²8Ü ð N ƒéu“EÜ”ù«$Ž ó, Úê ¦{$Ž¥ 1 ù‡ê Š^›©ƒq. (A.2.7.1) y²Ù¢Ò´˜«Ä–¢{, •‡ÖöU ²x f ◦ IdX Ú IdY ◦f ùü‡EÜN ½ Â, Ò¬˜Ù§‚ ½Â•! •±9éA{KÑÚ f ˜—. ¤±ùn‡N ´ƒÓ . A.2 8Ü†N 229 (2) (ÜÆ (associativity, associative law): b kn‡N f : X → Y , g : Y → Z Ú h : Z → W . (dž g ◦ f , h ◦ (g ◦ f ), h ◦ g Ú (h ◦ g) ◦ f ù EÜN Ñk¿Â.) K±e ªo¤á: h ◦ (g ◦ f ) = (h ◦ g) ◦ f . (A.2.7.2) ‡y²ù‡ªf, Äk·‚* Ò†mÑ´l X a ∈ X, ò h ◦ (g ◦ f ) Ú (h ◦ g) ◦ f ùü‡N Šû a ½Â= : Z N . e5•‡ y: éu?¿ ƒ ”´8Ü Z ¥ƒÓ ƒ. ¯¢þ, d [h ◦ (g ◦ f )](a) = h(g ◦ f (a)) = h(g(f (a))) ; [(h ◦ g) ◦ f ](a) = [h ◦ g](f (a)) = h(g(f (a))) . ¤±, Šâ·‚3 (A.2.6) ˜ã¥¤` yN ƒ •{, (A.2.7.2) ªy3Ò y². k (A.2.7.2) ª¥ (ÜÆ, ·‚± Ö õ‡N EÜžÒØ7o´\þ)Ò5rNÙ¥ EÜ$Ž öŠgS . ,, N EÜÚê ¦{ƒ'„k˜:é-‡ «O, ùÒ´ g ◦ f (½ö gf ) ù‡PÒ¥ f Ú g Ö ^S´ØU‘¿•† . XJ X 6= Z, @o f ◦ g ù‡PÒ´vk¿Â , ½ö` f ◦ g ù ‡EÜN ´vk½Â . =¦ X = Y = Z (dž g ◦ f Ú f ◦ g Ñk¿Â¿…Ñ´l X Ùg N ), •kŒU g ◦ f 6= f ◦ g. ~X, X = Y = Z = R, f Ú g ©OUì f (x) = x + 1 Ú g(x) = x2 5½Âž, EÜN g ◦ f : R → R Ú f ◦ g : R → R Ñk½Â, g ◦ f 6= f ◦ g. (ÖöUÄ Ñ g ◦ f (x) Ú f ◦ g(x) ²wLˆª ?) XJü‡N f Ú g U¦ g ◦ f Ú f ◦ g Ñk½Â¿… öƒ , @o·‚` f Ú g Œ † (commute). Ï • ˜ „ ó ù « y – Ø ˜ ½ ¤ á, ¤ ± · ‚ ¬ `: N EÜØ÷v †Æ (commutativity, commutative law). lPÒ †*/–þw, N EÜk aqÄ—ƒë ˜«“ 9iZ”. ·‚•Äõ‡N EÜž, ù«iZkžÿŒ± s zÑ. Ù¥•Š ˜J ´¤¢“ †ãL” y–. b k˜ N ¤ Xe/ª ˜‡ãL: (A.2.7.3) > Y1 f0 f1 / Y2 f2 / ··· / Ym ··· fm X >Z f00 Y10 f10 / Y20 f20 / ··· / Yr0 ··· fr0 ·‚6…òù ãL¡•˜‡“V‚ ´ã”, ÙA:´ã¥“LN ¤k†ÞTÐŒ±©¤ü |, §‚3˜åŒ¤ ˜‡µ4 õ>/, …3z˜|S †ÞŒ±Ä—ƒë, •ª/¤üGN f0 f1 f2 fm f0 f0 f0 f0 X −→ Y1 −→ Y2 −→ · · · −→ Ym −→ Z , 0 1 2 r X −→ Y10 −→ Y20 −→ · · · −→ Yr0 −→ Z. (ù p, · ‚ # N m ½ r u 0. X J m = 0, · ‚ Ò r þ ¡ ˜ G N n ) • ü Õ ˜ ‡ N f0 : X → Z. aq/, XJ r = 0, @oe¡˜GN Òn)•˜‡N f00 : X → Z.) ÏLN EÜ •ªŒ±lùüGN Xeü‡½Â•Ú •ÑƒÓ EÜN . XJùü‡EÜN ƒ ,= fm ◦ · · · ◦ f1 ◦ f0 = fr0 ◦ · · · ◦ f10 ◦ f00 , @o·‚` (A.2.7.3) ´ † (commutative). N¹ A ˜ 230 •£Ö¿ XJ·‚•Ä'ãL (A.2.7.3) •˜„ œ¹, @o3k (dN ¤ ) ãL¥ŒUé õ ‡“V‚ ´ã”. dž, XJãL¥¤k V‚ ´ãÑ´ † , ·‚Ò`TãL´˜‡ †ãL (commutative diagram). ~X, Uìþã½Â /Z ? h X g f Y ´ †ãL …= h = g ◦ f; f0 X 0 −−−−→ Y 0   β  αy y X −−−−→ Y f ´ †ãL …= f ◦ α = β ◦ f 0; f0 g0 X 0 −−−−→ Y 0 −−−−→ Z 0     β γ αy y y (A.2.7.4) X −−−−→ Y −−−−→ Z g f ´ †ãL …= e ªþ¤á: β ◦ f0 = f ◦ α , ~X, éu?¿N γ ◦ g0 = g ◦ β , f : X → Y Ú g : Y → Z, ãL /Z ? g◦f X g f Y o´ † γ ◦ (g 0 ◦ f 0 ) = (g ◦ f ) ◦ α . Ú X IdX f Y /X g◦f f IdY /Y . /Z IdZ g /Z g • K A.3. y²: 3ãL (A.2.7.4) ¥, •‡ β ◦ f 0 = f ◦ α Ú γ ◦ g 0 = g ◦ β ùü‡ ª¤á, Òk γ ◦ (g 0 ◦ f 0 ) = (g ◦ f ) ◦ α •¤á. (Ïd, ‡ y (A.2.7.4) ´ †ãL, ¢SþŒ±• ycü‡ ª.) (A.2.8) c¡3 (A.2.6) ˜ã·‚rNL, =¦´ƒÓ éA{K, •‡½Â•½ •Ø ƒÓ, ƒ A N •´ØÓ . ØL, éu¤^éA{KƒÓ ü‡N , ·‚=BØUä½§‚´Ó˜‡N , •¬~~w §‚ƒmk' ;— éX. 'X, ·‚3c¡?ØLdéA{K x 7→ x3 ‰Ñ n‡N : f1 : Z −→ Z , f2 : R −→ R , f3 : Z −→ R . ùp, N f3 Œ±d f2 <• ½Â• , f1 Œ±3 f3 Ä:þ?˜Ú • . •O (/`, XJ i : Z ,→ R L« Z R ¹\N , @o f2 = f3 ◦ i, f3 = i ◦ f1 . ·‚Œ±` f3 ´d f2 ÏLò½Â••›uf8 Z ⊆ R , f2 Œ±@•´l f3 ÏLò½Â•òÿ–•Œ 8Ü . aq/, XJ‡£ã f1 Ú f2 'X, ·‚•Œ±` f1 ´d f2 ÏL•› , f2 Œ±d f1 ÏLòÿ . A.2 8Ü†N 231 ˜„œ¹e, b kü‡8Ü X, Y , ±9ˆg f8 A ⊆ X, B ⊆ Y . ^ i : A ,→ X Ú j : B ,→ Y L«ƒA ¹\N . XJü‡N f : A → B Ú F : X → Y U¦±eãL † / B f A _ i X /Y F _ j ½= F ◦ i = j ◦ f , K¡ f ´ F ˜‡•›N (restriction map) (½ö{¡•›), F ´f ˜ ‡òÿ (extension). AO/, éu?¿N F : X → Y ±9˜‡¹\N i : A ,→ X, EÜN f := F ◦ i ´ F ˜‡•› (ù´ B = Y œ¹). (A.2.9) f : X → Y •˜‡N . XJ•3N g ◦ f = IdX (A.2.9.1) g:Y →X ¦ … f ◦ g = IdY , @o·‚` f ´Œ_ (invertible), ¿…¡÷v (A.2.9.1) ª ?Û˜‡N g • f ˜‡_N (inverse). ¯¢þ, XJ f ´‡Œ_N , @o§k…=k•˜˜‡_N . •35´dŒ_N ½Â y , •˜5 Ÿþ´d (A.2.7) ˜ã Ñ ü 5ŸÚN EÜ (ÜÆ y . äN 0 5`, XJ g Ú g Ñ´ f _N , @oŠâ½Â g ◦ f = IdX , f ◦ g 0 = IdY . ¤±, Šâð N 5Ÿ±9(ÜÆ, ·‚k g = g ◦ IdY = g ◦ (f ◦ g 0 ) = (g ◦ f ) ◦ g 0 = IdX ◦g 0 = g 0 . f Œ_ž, ~^ f −1 5L«§ _N . ÖöI‡3¿, 3 (A.2.9.1) ª¥·‚‡¦ ü‡ ª¤á. XJ•‡¦Ù¥˜‡ ª, @o˜ „œ¹e¿ØU y,˜‡ ªÓž¤á. éu®‰ N f : X → Y , XJ•3N g1 : Y → X ¦ g1 ◦ f = IdX , @o·‚` f ´†Œ _ (left invertible), ½ö` f k†_ (left inverse), ¿…r÷vþã^‡ N g1 ¡• f ˜‡ †_. aq/, XJ•3N g2 : Y → X ¦ f ◦ g2 = IdY , @o·‚` f ´mŒ_ (right invertible) ½ö f km_ (right inverse), ¿…rù N g2 ¡• f ˜‡m_. y3·‚5`² f Œ_ (A.2.9.2) w,, XJ f ´Œ_N , @o§ †_ g1 •k˜‡m_ g2 , @o …= _N f Qk†_qkm_. g Q´ f †_, •´ f m_. ‡L5, XJ f k˜‡ g1 = g1 ◦ IdY = g1 ◦ (f ◦ g2 ) = (g1 ◦ f ) ◦ g2 = IdX ◦g2 = g2 . ¤± g1 Ú g2 ´˜‡ƒÓ N g, TN ~ A.2.10. ÚŒ_N ~X, •ÄXeN œ¹ØÓ, g ÷v (A.2.9.1) ª, ·‚• ˜‡N f : X := {0, 1} −→ Y := {0, 1, 2} ; ±9±eü‡N f Œ_. †Œ_ž, ØUä½§ f (0) = 0, f (1) = 1 : g : Y = {0, 1, 2} −→ X = {0, 1} ; g(0) = 0, g(1) = 1 , g(2) = 0 , g 0 : Y = {0, 1, 2} −→ X = {0, 1} ; g 0 (0) = 0, g 0 (1) = 1 , g 0 (2) = 1 . †_´•˜ . N¹ A ˜ 232 N´ y, g Ú g 0 Ñ´ f †_. aq/, XJ•• ˜‡N mŒ_, ·‚ØUä½TN ~f. ·K A.2.11: f : X → Y ´ü‡š˜8Üƒm 1. f k†_ …= f ´ü . 2. f km_ …= f ´÷ . 3. f Œ_ …= f ´V ˜‡N m_´•˜ •£Ö¿ . žÖögCÞÑù . . y². (1) k b f k ˜ ‡ † _ g : Y → X, 5 y ² f ´ ü . • Ò ´ `, · ‚ I ‡ y: é u ? ¿ ü ‡ ƒ a, a0 ∈ X, X J f (a) = f (a0 ), @ o 7 L k a = a0 . ¯ ¢ þ, d f (a) = f (a0 ) Œ g(f (a)) = g(f (a0 )). Š• f †_, N g U¦ g ◦ f = IdX . u´, a = IdX (a) = g(f (a)) = g(f (a0 )) = IdX (a0 ) = a0 . y3‡L5y² f ´ü ž, §˜½k†_. •d, ·‚k ½ X ¥˜‡ ƒ a0 , , · ‚Œ±UìXe•ª½Â˜‡N g : Y → X: éu?¿ b ∈ Y , XJ b ∈ / Im(f ) = f (X), @o−1 g(b) = a0 ; XJ b ∈ Im(f ) = f (X), @oŠâ f ´ü b , n‘ f (b) ¥TÐ¹k˜‡ ƒ (ë „ (A.2.5) ˜ã"—), †óƒ, b ∈ Im(f ) ž, 3 X ¥•3•˜˜‡ ƒ a ¦ b = f (a). dž· ‚Ò½Â g(b) = a. N´ y, ù ½ÂÑ5 N g Ò´ f ˜‡†_. (2) b g : Y → X ´ f ˜‡m_. K f ◦g = IdY . Ïd, éu?¿ b ∈ Y , f (g(b)) = IdY (b) = b, •Ò´`, ƒ a = g(b) U ÷v f (a) = b. ùÒ`² f ˜½´÷ . ‡L5, kb g ´÷ , ·‚5y² f km_. éu Y ¥ ?¿ ƒ b, Šâ f ´÷ b , o´Œ±é X ¥˜‡ ƒ a ¦ f (a) = b. ÏLéz‡ b ∈ Y À ˜‡ù a ∈ X, , <• /5½ g(b) = a, ·‚ ˜‡N g : Y → X. † yŒ•, f ◦ g = IdY , ½=, g ´ f ˜‡m _. (3) ò (1) Ú (2) ¥ (Ø† (A.2.9.2) (Ü= . g • K A.4. Ñ˜‡äN §km_ vk†_. N f1 : R → R, §k†_ vkm_. 2 Ñ˜‡N f2 : R → R, (A.2.12) b I Ú X ´ü‡8Ü. XJ‰½ ˜‡N f : I → X, @o·‚Œ±ÏLN f òz ‡ i ∈ I Œ±éA X ¥˜‡ ƒ xi := f (i). ‡L5, XJ·‚kéz‡ i ∈ I, •½ X ¥˜‡ ƒ†ƒéA¿òT ƒP• xi , @oÏL i 7→ f (i) := xi ù˜éA{KŒ±ïá˜‡N f : I → X. 8 ·‚~~r?¿˜‡N f : I → X ¡•˜‡± I ••I8 (index set)!d X ¥ (˜ ) ƒ ¤ ƒx (family of elements). ¦^ù˜¶cž, N ÎÒ f Ø´·‚•'% SN, · ‚ Ì‡'5:´ xi = f (i) ù ƒ. ¤±, ù ˜‡ ƒx~~¬ê PÒƒ˜5L«: xi , i ∈ I ; {xi }i∈I ; (xi )i∈I . 5¿, 3ù PÒ¥ØÓ •I i éA ƒ xi Œ±ØÓ•Œ±ƒÓ. (•Ò´`, N f : i 7→ xi Œ±´ü •Œ±Ø´.) 5Þ˜ ~^ ƒ x ~ f. X J X ¥ ƒ Ñ ´ , ‡ ˜ m ¥ “• þ”, @ o ƒ x xi , i ∈ I Œ± ¡•˜‡•þx, ½ö˜x•þ. 2'X X ¥ ƒÑ´˜‡‰½8Ü A f8 (=, X ´˜8 P(A) f8), @o ƒx xi , i ∈ I Œ± ¡•˜‡d A f8 ¤ 8x, ½ö˜ x A f8. •I8 I ´˜‡ke. (k•½Ã•) ê«mž, = I = [[a, b]] ½ I = [[a, +∞[[ (Ù¥ a, b ∈ Z), T ƒx•¡•˜‡S (sequence). ~X, I=N…X ¥ ƒÑ´êž, ± I A.3 ë\ÎÒ P Úë¦ÎÒ Q 233 ••I8 ƒxÒ´˜‡±g,ê8••I ê . ˜‡k•S Ï~Œ±^¡Þ •ªL«• xa , xa+1 , · · · , xb , Ã•S Œ±L«• xa , xa+1 , · · · •I8 I ´k•8ž, ±§••I8 ƒx•¡•˜‡ ƒ| (system of elements). X JŒ± ÞÑk•8 I ¥ ¤k ƒ, 'X I = {i1 , · · · , in }, @o ƒ| xi , i ∈ I ~~ L«• xi1 , · · · , xin . ë\ÎÒ A.3 Úë¦ÎÒ P Q \{Ú¦{´êÆ¥•Ä ü«$Ž. ·‚²~¬‘ ˜ëGêƒ\½öƒ¦. • ‡ ÒÚ! Ö¡˜m, êÆ[‚ Ú\ ë\ÎÒ Σ Úë¦ÎÒ Π. ·‚y30 §‚ Ä {. (A.3.1) b k˜ ê ±‘eI {zP ¦^• /ªIP• a1 , a2 , . . . , an . @o·‚Œ±P n X (A.3.1.1) ai := a1 + a2 + · · · + an . i=1 n P kžÿ·‚•¬rÎÒ Σ þ•Ú e•Ö SNN–mþ•Úme•, = ù« {Œ± i=1 Pn Òm>i1 i ´vk³¡ , †>PÒ¥ i1 i ==• •B i=1 “O. 5¿, 3 (A.3.1.1) ·‚£ã¦Úª¥¦Ú‘ À ‰Œ. ù‡?ÖŒ±dÙ¦?Û (3þe©¥Ø¬ )ÜÂ ) i Pn 1“O. •Ò´`, XJ·‚î1 j 3 i zgÑy /•“O§, PÒ j=1 aj Ó Œ±L « a1 + a2 + · · · + an ù‡ë\(J. = n X ai = a1 + a2 + · · · + an = i=1 n X aj . j=1 Pn âd·‚ò (A.3.1.1) Ñy i ¡•´å•I (dummy index). PÒ i=1 ¥Ñy 1 Ú n ^5•« P å•I CÄ‰Œ, U^PÒ 1≤i≤n Ä þ•U¦Ó ¹Â LˆÑ5, ¤±·‚•Œ±ù 5Ö . XJI‡UC 1 n ù‡CÄ‰Œ¥ þ• n ½öe• 1 ½ö öÓžUC, @o?Ué APÒ •{•´w, . ~X, n−1 X ai = a3 + a4 + · · · + an−1 . i=3 ÃØê ai Ú•I i äN•6'XXÛ, ·‚Œ±@•k˜‡N f ò i N • ai . ¤±• Pn Œ±æ^ i=1 f (i) ù«PÒ. XJ·‚• f äNLˆª, @oŒ±† ò f (i) äNLˆª P Pn Pn ˜u¦ÚÒ ¡. ~X, e ai = f (i) = 2i + 1, K i=1 ai Œ± ¤ i=1 (2i + 1). 5¿, ùp f Œ±´˜‡~Š¼ê, ~Xé¤k i Ñk f (i) = 1, @o n X f (i) = f (1) + f (2) + · · · + f (n) = 1 + 1 + · · · + 1 i=1 Š•PÒ·‚Œ± ¤ •‡²x ¦ÚÎÒ Pn i=1 1. ¹Â, ÖöéN´²xe¡ü‡úª´¤á n X (A.3.1.2) (ai + bi ) = i=1 ai + n X i=1 n X λai = λ · i=1 ‡ â•y, 1˜‡¦^ Σ •ë\ÎÒ n X ´î. (Euler). i=1 n X i=1 ai bi : N¹ A ˜ 234 ë¦ÎÒ Π ¦^•{ •£Ö¿ aq. =, n Y (A.3.1.3) ai := a1 · a2 · an . i=1 Ööé˜ aq Ö S.Ú{ü5ŸAT•´%•Ü² . ~X, Qn i=1 , n Q Ú 1≤i≤n ÑL«Ó Q i=1 ˜¹Â, , n Y ai bi = i=1 n Y (A.3.1.4) n Y n X · ai i=1 λai = λn · i=1 Xd ! ! bi i=1 n Y ai i=1 . (A.3.2) y3b k mn ‡±ü‡•I‰eI ê aij , Ù¥ i n. @oŒ±òù êü¤ m 1 n Ý/êL a11 a21 .. . amn1 òêL¥z˜1 êë\å5, Œ± A1 = n X a12 a22 .. . ··· ··· .. . am2 ··· Š‰Œ´ 1 a1n a2n .. . amn m ‡¦Ú(J a1j , A2 = j=1 n X a2j , . . . , Am = j=1 n X amj j=1 Pm 2òù m ‡êë\å5, i=1 Ai . ü‡Ú½Ü¿å5, •ª¤ ÚŒ±   m n X X  aij  . i=1 XJ·‚kò±þêL¥z˜ m X n ‡Ú ai1 , B2 = i=1 2òù n ‡êë\å5 j=1 ê\å5 B1 = m X ai2 , . . . , Bn = i=1 Pn j=1 Bj . Ü¿üÚ¤ j=1 mn ‡ê) \å5 m X i=1 m X (JŒ± ! aij . i=1 Ú, ¤±   ! n n m X X X   aij = aij . j=1 ain i=1 n m X X Ï••ªÑ´¤k aij ( m, j j=1 i=1 ¤ ¤ Š‰Œ´ 1 A.3 ë\ÎÒ P Úë¦ÎÒ Q 235 ùÒ´`, •Äk•õ‡êë\ž, ”þ¡ù Ñy S . þª¥·‚Œ±Ñ Ù¥ )Ò, † ¤ m X n X (A.3.2.1) aij = i=1 j=1 -¦Ú”´Œ± “ n X m X †üÚ¦Ú öŠ^ aij . j=1 i=1 ÆSL‡È©½êÆ©Ûƒa‘§ Öö• , ¦Ú‘k•´ yþ¡ ´Ã¡õ‘¦Ú, @o“ -¦Ú¥¦ÚÚ½ †^S”´˜«I‡ o´Ün . ª¤á -‡b . XJ % öŠ, ù«öŠ™7 (A.3.3) ±þ'u¦ÚÒ Σ (½ë¦Ò Π) ¦^5KŒ±2‰{ü í2. ¯¢þ, ·‚Ø˜½‡ é?ÒëY ˜|ê5¦Ú, •]ÀÙ¥˜Ü©5¦Ú. ùžÿkü«í2 œ¹Š 0 . b éz‡ i ∈ [[1, n]] ®²‰ ˜‡ê ai . Äk, ·‚Ø”b½ I ´ ê8 [[1, n]] ˜‡f8, @o·‚Œ±•À 5g I ¥ @ eI i 5òéA ê ai Ü\å5, ù ÚŒ±P• X ai . i∈I XJ I •˜8, ·‚ ½þª u 0. XJ J ´ [[1, n]] ,˜‡f8, w,k X X X X (A.3.3.1) ai = ai + ai − ai . i∈I∪J i∈I i∈J i∈I∩J AO/, XJ I ∩ J = ∅, K X (A.3.3.2) i∈I∪J , , bXk˜‡k•8 K, ai = X ai + i∈I X ai i∈J s : K → [[1, n]] ´®‰ X as(k) ˜‡N , @oPÒ k∈K Œ±^5L«Àg a1 , . . . , an |K| ‡ê¦Ú¤ (J. ùp¦Úª¥¦Ú‘ÀJÑy 5KÒ´: z k˜‡ k Ñy3 K ¥(U˜gO), a1 , . . . , an ƒ¥±êŠ s(k) •eI @‡êÒÑy3þ¡Ú ª¥˜g. XJkØÓ k ¦ s(k) ƒÓ Š, @oƒA êŒ±•âØÓ k êþÑyõg. ¢ Sþ, XJò•I i †ê ai éA'XwŠ•I8 [[1, n]] þ½Â ˜‡N a, ¤ a(i) = ai , @o Pn Pn ¤ i=1 a(i). ·‚qk,˜‡N s : K → [[1, n]] ž, Œ±ò a † s ‰EÜ # i=1 ai Œ± N a ◦ s. ù as(k) ÃšÒÉÜN a ◦ s 3 k ? Š. u´ X X as(k) = (a ◦ s)(k) k∈K k∈K ÒŃ a ◦ s 3 (k•) 8Ü K z‡ ƒ? ŠƒÚ. ~X, e K = {−1, 1}, s(k) = k 2 , @o X as(k) = as(−1) + as(1) = a1 + a1 . k∈K ˜‡•~„ ~f´: K = [[1, n]], s(k) = n + 1 − k. ù n X k=1 as(k) = n X k=1 an+1−k = an + an−1 + · · · + a1 . N¹ A ˜ 236 þª¦Ú(Jw, u •£Ö¿ Pn k=1 ak . =, n X (A.3.3.3) an+1−k = k=1 n X ak . k=1 â`êÆUâpd3 Æ)žÒuy l 1 100 ¤kg,ê\å5¦Ú ¯„Ž{, •ƒ uuy ê ¦Úúª. Ù¢ù‡6D¹2 ¯ÒÚúª (A.3.3.3) k'. ¯¢þ, XJ· Pn ‚b ai Ú i •6'X• ai = c + di, Ù¥ c, d •~ê, @o¦Úª S = i=1 ai Œ±UXe• {¦Ñ: Pn Pn Šâ (A.3.3.3), S = i=1 an+1−i = i=1 (c + d(n + 1 − i)). ù 2S = S + S = n X (c + di) + i=1 = n X (c + d(n + 1 − i)) i=1 n X n X c + di + c + d(n + 1 − i) = 2c + dn + d = n(2c + dn + d) i=1 i=1 ¤± n X n(2c + dn + d) . (c + di) = 2 i=1 ùÙ¢Ò´·‚Ù• ê ¦Úúª. ò (A.3.3.3) ª‰‡{ü í2, Œ±•Ä?¿˜‡V [[1, n]]. džEk n X (A.3.3.4) as(k) = k=1 dúª¤á ·‚2Þ n X (=, Q´ü q´÷ ) s : [[1, n]] → ak . k=1 ndžÖög1g•. ~f5`²ë\ÎÒÚë¦ÎÒ A^. Qn ê n, Ù ¦ (factorial) n! ½Â• n! = i=1 i. ·‚2 |Üê (number of combinations)   n! e k ∈ [[0, n]] n := k!·(n−k)! 0 k eØ, ~ A.3.4. é?¿ Œ±½Âš~k^ (A.3.4.1) ½ 0! = 1. ¦^ Šâ{ü ü |Ü•£, éu?Û ƒ‡ê• n 8Ü, Ù¥ ƒ‡êT• k n k . Í¶ Úî ‘ªúª (Newton’s binomial formula) = n n X n n−k k X n i n−i n (A.3.4.2) (a + b) = a b = ab k i i=0 ¦ f8‡êÒ´ k=0 Ï•ù‡úª, n k ù ê• ¡• ‘ªXê (binomial coefficients). ~ A.3.5. X J ê ak , k ∈ N∗ z˜‘5gu,˜‡ê bl , l ∈ N ¥ ë Y ü ‘ Pn bk − bk−1 é?¿ k ≥ 1 ¤á, @o¦Úª i=1 ai éN´ z{: (A.3.5.1) n X i=1 ai = n X i=1 , = ak = (bi − bi−1 ) = (b1 − b0 ) + (b2 − b1 ) + · · · + (bn − bn−1 ) = bn − b0 . A.4 õ‘ª9ÙÏª©) 237 ~X, e bk = k p , Ù¥ p • ½ ê, KŠ• (A.3.5.1) n X (A.3.5.2) ˜‡A~·‚ : (ip − (i − 1)p ) = np . i=1 3 (A.3.5.2) ¥ p = 2, 2Šâð 2 n X ª i2 − (i − 1)2 = 2i − 1 Œ i− n X i=1 ? n X 1= i=1 (i2 − (i − 1)2 ) = n2 i=1 • n X 1 i= 2 i=1 (A.3.5.3) 2 n + n X ! 1 i=1 = n(n + 1) 1 2 (n + n) = . 2 2 ùØÒ´pd•Ž 1 + 2 + · · · + 100 “üo ? XJ3 (A.3.5.2) ¥ p = 3, 2|^ð ª i3 − (i − 1)3 = 3i2 − 3i + 1 Úúª (A.3.5.3), ·‚Œ ±ŽÑ n X n(n + 1)(2n + 1) 1 1 1 (A.3.5.4) . i2 = n3 + n2 + n = 3 2 6 6 i=1 ùÕŽL§ [!·‚3‰ÖööS. Pn Š•g•K, ÖöØ”?˜Ú}ÁéÑ¦Úª i=1 i3 OŽúª. Q (A.3.6) XJÖö¿©n) ë\ÎÒ Σ Úë¦ÎÒ ^{‡Â, @oƒ&ØJn)Ó P Ò¦^5KŒ±@^ Ù¦aq œ/. 'X, XJ A1 , · · · , An Ñ´,‡‰½8Ü A f8, @o Tn Sn Œ±^ i=1 Ai Ú i=1 ©OL« 8 A1 ∩ · · · ∩ An Ú¿8 A1 ∪ · · · ∪ An . •˜„/, XJ Ai , i ∈ I T S ´ A ˜xf8, @o i∈I Ai Ú i∈I Ai Œ±©O^5L«ùxf8 8Ú¿8, =, \ Ai : = {a ∈ A | éuz‡ i ∈ I þk a ∈ Ai } , i∈I [ Ai : = {a ∈ A | •3,‡ i ∈ I ¦ a ∈ Ai } . i∈I 8 ·‚ŒU„¬‘ †Ù¦êÆé–ƒ' ˜ žÿ·‚ÒØ2Kã@ ÎÒ ¦^5K . A.4 $Ž, Ïd„¬kÙ¦˜ Ú Σ, Q aq PÒ, õ‘ª9ÙÏª©) !¥·‚ K ´ C ˜‡f•, K[X] L«Xê3 K ¥± X •™½ Ü, §´˜‡ K-•þ˜m (ë„ (2.3.22) ˜ã). (A.4.1) éu˜‡š"õ‘ª f (X) ∈ K[X], k•˜ •ªòÙ õ‘ª ¤ 8 ¤ f (X) = an X n + an−1 X n−1 + · · · + a1 X + a0 /ª, Ù¥ n ∈ N, an 6= 0. K f gê (P• deg(f )) Ò´ n, ¤¢ f Ä‘XêÒ´• an . (" õ‘ª Ä‘Xê@•´ 0, gê@•´ −∞.) ¤¢Ä˜õ‘ª´•Ä‘Xê• 1 õ‘ª. þ¡‰Ñ õ‘ª f ,•Œ±À• C[X] ¥ õ‘ª, Ïd3Ù¥Œ±ò X ±?¿Eê“\ Š. XJ˜‡Eê c ∈ C ÷v f (c) = 0, K¡ c • f 3 C ¥ ˜‡Š (root) ½ö": (zero). c ∈ K ¿…´ f Š, Ò¡ c ´ f 3 K ¥ ˜‡Š. ±e´‡{ü Ä ¯¢: N¹ A ˜ 238 ·K A.4.2: f ∈ K[X] ´ n gÄ˜õ‘ª, òÙ •£Ö¿ ¤Xe/ª: f (X) = X n + an−1 X n−1 + · · · + a1 X + a0 , Ù¥z‡ ai ∈ K . XJEê c1 , · · · , cn ∈ C ¦ ±e'uõ‘ª ª¤á f (X) = (X − c1 )(X − c2 ) · · · (X − cn ) , K a0 = (−1)n c1 c2 · · · cn , an−1 = −(c1 + · · · + cn ) . y². òõ‘ª¦È (X − c1 )(X − c2 ) · · · (X − cn ) U½ÂÐmOŽ, * Xê=Œ. e¡•ã -‡½n•, ¡•“êÄ ½n, Ùy²7L¦^©ÛÆ ¿¦^ù‡½n, Ñ § y² (Ï•ùØ´‚5“ê ÌZSN). •£. ·‚ò«@ ½n A.4.3 (“êÄ ½n Fundamental theorem of algebra): ª3 C ¥Ñk":. u1 ?ÛgêŒu½ EXêõ‘ (A.4.4) f, g ∈ K[X]. XJ•3õ‘ª h ∈ K[X] ¦ f = gh, K¡ g 3 K[X] ¥ (½ö3 K þ) Ø (divide) f , •¡ g ´ f ˜‡Ïª (factor). d^‡¤áž·‚P g | f . Ï•ü‡õ‘ª¦È gê u ögêƒÚ, ¤±XJ f ´š"õ‘ª, @o g | f ˜‡7‡^‡´ deg(g) ≤ deg(f ). ·‚3‚5“ê‘§¥¬^ õ‘ª Ïª©). e¡ (Ø´ù•¡•Ä (Øƒ˜. § Œ±Ø¦^‚5“ê •£5y², éu Ö ó·‚æ^‚5“ê •{Šy²A ´Ü· . Ún A.4.5 (õ‘ª ‘{Ø{ division algorithm for polynomials): f, g ∈ K[X], Ù¥ g •š" õ‘ª. K•3•˜˜éõ‘ª (Q , R) ∈ K[X] × K[X] ÷v±e^‡ f = gQ + R … deg(R) < deg(g) . (5¿, Q Ú R ÑŒ±´"õ‘ª.) ùp Q Ú R ¡• f g Ø (½ö` f Ø± g) ¤ û (quotient) Ú{ª (remainder). y². k5y²•˜5. eküéõ‘ª (Q, R) Ú (Q0 , R0 ) þ÷v¤I^‡, K g(Q0 − Q) + (R0 − R) = gQ0 + R0 − (gQ + R) = f − f = 0 , l g Ø R − R0 . ´ deg(R − R0 ) ≤ max{deg(R), deg(R0 )} < deg(g). ¤±T Ø'X•k3 R − R0 = 0 žâU¤á. R − R0 = 0 ž2dþªŒ• g(Q0 − Q) = 0. Ï• g ´š"õ‘ª, ¤± 0 0 7L Q − Q = 0. Ïd, (Q , R0 ) = (Q, R), ¤I•˜5¼y. e¡y²•35. Ï• g ´š"õ‘ª, Ùgê m = deg(g) ´‡g,ê (=, deg(g) 6= −∞). e n = deg(f ) u m = deg(g), K† Q = 0, R = f =Œ. ÏdØ” n = deg(f ) ≥ m = deg(g). •Ä K-•þ˜m V = K[X]≤n−m × K[X]≤m−1 , = V ´ü‡•þ˜m K[X]≤n−m Ú K[X]≤m−1 ‰†È •þ˜m (ë„ (2.3.38)). ÏL®² ½ õ‘ª g 5½ÂN T : V = K[X]≤n−m × K[X]≤m−1 −→ K[X]≤n ; N´ y T ´˜‡‚5N , …±þ'u•˜5 (q, r) 7−→ gq + r . ØyL² T ´˜‡ü . Šâ·K 2.3.39, dim V = dim K[X]≤n−m + dim K[X]≤m−1 = (n − m + 1) + m = n + 1 = dim K[X]≤n . ¤±, T 7,•´˜‡÷ (ë„·K 3.1.31). Ïd, ·‚I‡•Ä ”8. =, •3˜éõ‘ª (Q, R) ÷v¤I 5Ÿ. õ‘ª f ∈ K[X]≤n áu T A.4 õ‘ª9ÙÏª©) 239 5 P A.4.6. f, g ∈ K[X], g 6= 0 X Ú n A.4.5. Q, R ∈ K[X] © O ´ f Ø ± g X J L ´ K 3 C ¥ ˜ ‡ * •, @ o , • Œ ± ò f, g Ñ À • L[X] ¥ õ ‘ ª, Ï d • • 3 L[X] ¥ õ ‘ ª é (Q0 , R0 ) ¦ f = gQ0 + R0 3 L[X] ¥ ¤ á, … deg(R0 ) < deg(g). , ˜ • ¡, K[X] ¥ ª f = gQ + R • Œ ± Š L[X] ¥ ª. Š â ‘ { Ø { L ˆ ª 3 L[X] ¥ • ˜ 5, 7 , (Q, R) = (Q0 , R0 ). •Ò´`, 3 f, g Ñáu K[X] cJe, 3 K[X] ¥ŽÑ5 f Ø± g ûÚ {ª•´3 L[X] ¥?1Ó˜Ø{$Ž ûÚ{ª. AO/, f, g ∈ K[X] ž, Ø'X g | f 3 K[X] ¥¤áÚ3 L[X] ¥¤á´ d . íØ A.4.7: f ∈ K[X], λ ∈ K. K λ ´ f ": …= X −λ Ø f. y². f (X) = (X − λ)Q(X) + R(X) • f (X) Ø± X − λ ¤ Ø{Žª, Ù¥ Q ∈ K[X] ´û, R ∈ K[X] ´{ª. ^‡ deg(R) < deg(X − λ) = 1 L² R(X) ´‡~Šõ‘ª. ÏLò X = λ “\ ŠŒ R(λ) = f (λ) − (λ − λ)Q(λ) = f (λ) . Ïd f (λ) = 0 °(¹Â. ¿©7‡^‡´ (~Šõ‘ª) R •"õ‘ª, íØ A.4.8: e f ∈ K[X] ´ n gõ‘ª. K f 3 K ¥–õk n ‡ØÓ ˜Øä ´ X −λ Øf ":. y². e n = 0, K f ´š" ~Šõ‘ª, Ù":‡ê• 0, Ïd(Ø¤á. e n = 1, K f /X a0 + a1 X, Ù¥ a1 6= 0. džw, f •k˜‡":, (Ø•¤á. éu˜„ œ¹, ·‚¦^égê n = deg(f ) 8B{. y3 n > 1. e f 3 K ¥vk":, K":‡êg, ≤ n. e f 3 K ¥k˜‡": λ. KŠ âíØ A.4.5, f (X) = (X − λ)g(X), Ù¥ g ∈ K[X] ´ n − 1 gõ‘ª. Šâ8Bb , g 3 K ¥• õk n − 1 ‡ØÓ Š. ¤±, l f (X) = (X − λ)g(X) ù˜'XªŒ±wÑ f 3 K ¥•õk n ‡ Š. íØ A.4.9: f ∈ C[X] ´ n gõ‘ª, n ≥ 1, c • f Ä‘Xê. K•3 λ1 , · · · , λn ∈ C ¦õ‘ª f 3 C[X] ¥kXe©)ª (A.4.9.1) f (X) = c(X − λ1 )(X − λ2 ) · · · (X − λn ) . …, ±þù«©)ª3ØOÏªü ^S ¿Âe´•˜ , =, XJ,k˜«©) f (X) = c(X − λ01 )(X − λ02 ) · · · (X − λ0n ) K λ01 , · · · , λ0n ù|ê˜½Œ±ÏL-#üSC¤† λ1 , · · · , λn Å˜¬Ü. y². ·‚é f gê n ‰8B{. ÖöŒ±g1 y n = 1 ž (Ø¤á. y3 n > 1. kŠâ“êÄ ½nŒ± λn ∈ C • f ˜‡Š. u´íØ A.4.5 L² f Œ ©)• f (X) = (X − λn )g(X), Ù¥ g ∈ K[X] gê• n − 1, Ä‘Xê• c. Šâ8Bb , •3 λ1 , · · · , λn−1 ∈ C ¦ g(X) = c(X − λ1 ) · · · (X − λn−1 ) . “\ f (X = (X − λn )g(X) =Œ f ©)ª. • y²©) •˜5, b qk~ê λ01 , · · · , λ0n ∈ C ¦ c(X − λ1 ) · · · (X − λn ) = f (X) = c(X − λ01 ) · · · (X − λ0n ) . Äk·‚Œ±3þª†müàž š"~ê c. , * λ1 ´•†>õ‘ª ˜‡":, Ïd§ •´•m>õ‘ª ˜‡":. Ïd7,k˜‡ λ0i Ú λ1 ƒ . 'X´ λ01 = λ1 . Kòþª2ž ú Ïf X − λ1 , (Ügê• n − 1 ž 8Bb Œ• λ2 , · · · , λn Ú λ02 , · · · , λ0n ùü|ê-#üS 7,¬Ü. u´·‚ ¤ y². N¹ A ˜ 240 •£Ö¿ (A.4.10) l (A.4.9.1) ª·‚wÑ, ?Û gê EXêõ‘ª f ∈ C[X] Œ±•â§ EêŠ ©)•˜gÏª ¦È. (A.4.9.1) ª¤áž, ·‚~` f 3 C ¥ Ü n ‡Š • λ1 , · · · , λn æ ^ù«`{ž·‚#N λ1 , · · · , λn ¥k, êƒÓ. kž• rN, ·‚Œ±`3-ŠO\ •ª e (½ö3O\-ê ¿Âe) f 3 C ¥k n ‡Š λ1 , · · · , λn . XJr (A.4.9.1) ª¥ ŠƒÓ λi Ü¿å5, @oŒ± ¤ f (X) = c(X − λ1 )m1 · · · (X − λr )mr /ª, Ù¥ c ∈ K • f Ä‘Xê, λi ∈ C üüØÓ, mi ∈ N∗ . dž·‚~` λ1 , · · · , λr ´ f 3 C ¥¤kØÓ Š, f 3 C ¥ k r ‡ØÓ Š. éuz‡Š λi , ¡ mi •§ -ê (multiplicity). ÖöŒ± y: Eê λ ∈ C ´˜‡õ‘ª f ∈ C[X] m -Š …= (X − λ)m Øõ‘ª f (X − λ)m+1 Ø Ø f . ±þ?ØŒ±í2 •˜„• œ¹. =, éu˜„ • K, e f ∈ K[X] ´Ä‘Xê• c ∈ K n gõ‘ª, n ≥ 1, @o·‚` f 3 K ¥ (O\-êž!-ŠO\ž) k s ‡Š λ1 , · · · , λs , ´• •3©)ª f (X) = c(X − λ1 ) · · · (X − λs )g(X) (A.4.10.1) Ù¥ g ∈ K[X] 3 K ¥vk":. XJòÙ¥ ŠƒÓ λi Ü¿å5 ¤ f (X) = c(X − λ1 )m1 · · · (X − λr )mr g(X) (A.4.10.2) ·‚Œ±` f 3 K ¥ k r ‡ØÓ Š, z‡Š λi -ê (multiplicity) ½Â• mi . Ïd, λ ∈ K ´f m -Š …= (X − λ)m Ø f (X − λ)m+1 Ø Ø f . 5¿, λ ∈ K ž, 3 K ¥• Ä§Š• f Š -êÚ3 C ¥•Ä§ -ê´˜ . Pn g • K A.5. ·‚Qéõ‘ª f = i=0 ai X i ∈ K[X] ½ÂL/ª äNLˆª•  Pn i.a X i−1 = na X n−1 + · · · + 2a X + a i n 2 1 i=0 0 f (X) = 0 ê f 0 ∈ K[X] (ë„~ 3.1.15), e n = deg(f ) ≥ 1 e deg(f ) ≤ 0 . 1. y²: é?¿ f, g ∈ K[X] Ú?¿ m ∈ N, (f g)0 = f 0 g + f g 0 , (f n )0 = n.f n−1 .f 0 . 2. …= X −λ Ø f 0. K ŒUØ•¹u¢ê•, ÏdØUæ^¦4• ©ÛÆ •{y²±þ(Ø.) λ∈K ´f (5¿: ùp ˜‡Š. y²: (X − λ)2 Øf g • K A.6. éu?¿õ‘ª f ∈ K[X] Ú?¿ m ∈ N, ·‚U±e48•ª½Â f ê (m-th order formal derivative) f (m) : f (0) := f , f (1) := f 0 1. y²: éu?¿ a ∈ K Ú?¿ n ≥ 1, e (a) a ´ f /ª m ≥ 1 ž, f (m) := (f (m−1) )0 . •ã d: Š, a Ø´ f (n) n -Š; (b) éuz‡ m ∈ [[0, n − 1]], a ´ f (m) 2. éÑ¤kŒU m n ∈ [[1, 3]] 9Eê a, b ∈ C ¦ Š. õ‘ª X 4 + aX n + b 3 C ¥k˜‡n-Š. A.4 õ‘ª9ÙÏª©) 3. f ∈ K[X] 241 gê ≤ n. y²: éu?¿ c ∈ K, ±eúª (Taylor úª) 3 K[X] ¥¤á: f (X) = n X f (k) (c) k=0 k! (X − c)k . ¦5½• 0! = 1.) (5¿: 0 c¡ ?ØL²: gê EXêõ‘ªo´Œ±©)•˜gÏª ¦È, …ù«Ïª©) dõ‘ª Š±9ƒA -êû½. y325?Ø¢Xêõ‘ª Ïª©). Ún A.4.11: f ∈ R[X] ´š~Šõ‘ª (= deg(f ) ≥ 1), λ ∈ C. 1. e λ ´ f 3 C ¥ 2. e λ ´ f ˜‡Š, K§ E Š, … λ ∈ / R, KŠ• f Ý λ̄ •´ f 3 C ¥ ˜‡Š. EêŠ λ Ú λ̄ äkƒÓ -ê. y². (1) Ï• f ´¢Xêõ‘ª, ¤± f (λ̄) = f (λ), =, f 3 λ̄ ? Š f (λ̄) u f 3 λ ? Š E Ý. ddá=Œ ¤I(Ø. (2) Šâ (1) Œ• λ, λ̄ ´ f 3 C ¥ ü‡ØÓ Š, ¤± f (X) = (X − λ)(X − λ̄)h(X), Ù¥ h ∈ C[X]. (Ïd f gê n ≥ 2 Ø¬ u 1.) · ‚ æ ^ é g ê n = deg(f ) 8 B { 5 y ² λ Ú λ̄ - ê ƒ . e n = 2, K f (X) = c(X − λ)(X − λ̄). dž λ, λ̄ Ñ´ f 1 -Š. e¡b n > 2. OŽŒ• g(X) := (X − λ)(X − λ̄) = X 2 − 2Re(λ)X + |λ|2 ´‡¢Xêõ‘ª. þ¡·‚w f 3 C[X] ¥Ø± g ¤ û´ h, {ª´ 0. Ï• f, g Ñ´¢ Xêõ‘ª, Šâ5P A.4.6 ¥ ?Ø, 3 R[X] ¥OŽ f Ø± g ž, ¤ {ª7,•´ 0, û Q ∈ R[X] 7,† h ƒ . Ïd h = Q ´‡¢Xêõ‘ª. Šâ8Bb , λ Ú λ̄ ‡oÑØ´ h Š, ‡o ö´ h -êƒ Š. • , l f (X) = (X − λ)(X − λ̄)h(X) ù‡ ªŒ±• λ Ú λ̄ ´ f -êƒ Š. Ún–dy.. ·K A.4.12: (A.4.12.1) f ∈ R[X] •gê n ≥ 1 ¢Xêõ‘ª, Ä‘Xê• c ∈ R. K•3Ïª©) f (X) = c(X − λ1 ) · · · (X − λm )(X 2 + b1 X + c1 ) · · · (X 2 + bM X + cM ) Ù¥ λ1 , · · · , λm , b1 , · · · , bM , c1 , · · · , cM ∈ R, ¿…éz‡ j ∈ [[1, M ]] þk b2j − 4cj < 0. …, 3ØOÏªü ^S ¿Âe, (A.4.12.1) ª¥ ©)´•˜ . (ùp m ∈ N Œ±• 0, dž·‚@• (A.4.12.1) ª¥ ˜gÏªØ•3. aq/, M ∈ N Œ ±• 0, dž·‚@• (A.4.12.1) ª¥ gÏªØ•3.) y². dÚn A.4.11 Œ•, f 3 C ¥ Ïª©)Œ± ¤ Jê (=Ø´¢ê Eê) Š¤é/Ñy. ¤± f 3 C[X] ¥ f (X) = c(X − λ1 ) · · · (X − λm )(X − µ1 )(X − µ̄1 ) · · · (X − µM )(X − µ̄M ) Ù¥ λ1 , · · · , λm ´ f ¤k¢êŠ, µ1 , µ̄1 , · · · , µM , µ̄M ´ f ¤kJêŠ (þO\-ê). éz‡ j ∈ [[1, M ]] - bj = 2Re(µj ), cj = |µj |2 = (X − µj )(X − µ̄j ) = X 2 + bj X + cj . Ï• µj ∈ / R, ¤± 2 bj − 4cj < 0. ¤± (A.4.12.1) ¥ ©)ª´•3 . N¹ A ˜ 242 •£Ö¿ • ` ² (A.4.12.1) ª • ˜ 5, Ä k 5 ¿ c Š• f Ä‘Xê´•˜(½ . d , λ1 , · · · , λm ´ f 3 R ¥ ¤kŠ, Ïd§‚ù|¢ê•´•˜(½ . XJ,k˜|/X (A.4.12.1) ª ©), @oT©)7,Œ± ¤ f (X) = c(X − λ1 ) · · · (X − λm )(X 2 + b01 X + c01 ) · · · (X 2 + b0l X + c0l ) m + 2M = deg(f ) = /ª, Ù¥ b0j , c0j ∈ R ÷v (b0j )2 − 4c0j < 0. ÏL' õ‘ª gêŒ±k 2 m + 2l, M = l. e5, éz‡ j ∈ [[1, M ]], •Ä X + bj X + cj ü‡EêŠ µj , µ̄j . §‚7,´ ,‡Ïª X 2 + b0s X + c0s ü‡EêŠ, u´7, X 2 + bj X + cj = X 2 + b0s X + c0s . dd= ¤I •˜5(Ø. (A.4.13) 3Nõ¢SOŽ¯K¥, ·‚¬²~‘ Xê ´ ê õ‘ª. ù Xêõ‘ª ¦ Š¯KÏ~´êØ¥;€?Ø (Ø{ü ) ¯K. ·‚3ùp?Ø˜‡ƒé{ü 5Ÿ, §kžŒ ±•Ï·‚é ˜ Xêõ‘ª ":. 3gê' $ žÿ, ù«•{~~ék . ·‚I‡^ ˜ Ð êØ VgÚ•£. ·‚•ãù VgÚ5Ÿ, Ø\y². k, ÖöŒ±3?Û˜ Ð êØ á (~X [17]) ¥é ƒ' ØãÚy². é?¿ ê c, d, ·‚` d Ø c, ¿P d | c, ´••3 m ∈ Z ¦ c = dm. d 6= 0 ž, d | c Ò c ´` d ´ ê. a, b ´ü‡Ø •" ê, ·‚` a Ú b pƒ (coprime) ´•Ø•3î‚Œu 1 êd r Óž Ø a Ú b. XJ a, b pƒ, @o a Ú b ?¿g˜ b (Ù¥ r ∈ N) •pƒ. ˜‡š~-‡ k^ (Ø´: XJ a, b, c ń‡ ê, a | bc … a † b pƒ, K7, a | c. ù‡ (Økž ¡•pdÚn (Gauss’ lemma)‡ ·K A.4.14: n ≥ 1, f (X) = an X n + an−1 X n−1 + · · · + a1 X + a0 Ù¥ ai þ• ê… an 6= 0. α = pq , Ù¥ p, q ´š" XJ α ´ f ˜‡Š, @o7, q | an , p | a0 . y². ò α Š“\õ‘ª f , , Ï© ê… p, q pƒ. •Ä©fŒ an pn + an−1 pn−1 q + · · · + a1 pq n−1 + a0 q n = 0 . ùL² p | a0 q n , q | an pn . Ï• p † q pƒ, ¤±•† q n pƒ. d (A.4.13) ˜ãJ p | a0 q n Œ±í• p | a0 . aqŒy q | an . A.5 S K A.5.1. SK √ √ K = {a + b 3 2 + c 3 4 | a, b, c ∈ Q}. y² K ´ C S K A.5.2. éu C e f8, ä´Ä´ C pdÚn, l f•. f•¿`²nd. 1. Z; √ √ 2. {a + b 2 + c 3 | a, b, c ∈ Q}; 3. {0}; 4. {z ∈ C : |z| = 1}. ‡ ±pd Â. ¶i·¶ Únš~õ, ÏdïÆÖö¦^“pdÚn”ùac®ž, Ö7•Äþe©UÄ¦Öö (n)Ù¹ A.5 SK 243 S K A.5.3. A • C š˜f8, … A 'u~{´µ4 y²µ 0 ∈ A, … A 'u\{•´µ4 . S K A.5.4. A•C f8, … 0, 1 ∈ A. Þ~`²±ez«œ¹þkŒUu): 1. A 'u\~{Ú¦{Ñµ4, A Ø´ C 2. A 'u\{Ú¦{Ñµ4, 'u~{Øµ4. 3. A 'u\~{Ñµ4, S K A.5.5. 1. y²: K ∩ L •´ C S K A.5.6. b N ü‡f•. f•. f• K ⊆ L ½ K ⊇ L. x, y ∈ Q , þk Š• 0 ~ŠN f (x + y) = f (x) + f (y) , f (xy) = f (x)f (y) . , ‡o f ´ Q √ √ K = Q( 2) = {a + b 2 | a, b ∈ Q}. b S K A.5.7. é¤k x, y ∈ K , þk y²: f •U´±en‡N 1. f ´ …= f : Q → C ÷v±e^‡: é¤k y ²: ‡ o f ´ f (a) = a). f•. 'u¦{Øµ4. K, L ´ C 2. y²: K ∪ L ´ C Š• 0 . C N g,¹\N (=, é ? ¿ a ∈ Q, f : K → C ÷v±e^‡: f (x + y) = f (x) + f (y) , f (xy) = f (x)f (y) . ƒ˜: ~ŠN . √ √ g,¹\N , =, é?¿ a, b ∈ Q, f (a + b 2) = a + b 2. √ √ 3. é?¿ a, b ∈ Q, f (a + b 2) = a − b 2. 2. f ´ K S K A.5.8. C A, B •8Ü X ü‡f8, C, D •8Ü Y ü‡f8. y²: Š• X × Y f8, (A ∩ B) × (C ∩ D) = (A × C) ∩ (B ∩ D) . S K A.5.9. A, B, C, D þ•8Ü X y²; eØ (, žÞÑ‡~. f8. ée z‡Øä, äÙ (†Ä. e 1. X \ (A ∪ B) = (X \ A) ∩ (X \ B). 2. (A ∩ B) ∪ C = A ∩ (B ∪ C). 3. e A ∪ B = C ∪ D, … A ∩ D = ∅, B ∩ C = ∅, K A = C, B = D. S K A.5.10. A, B •8Ü X ü‡f8. y²: 1. XJéu X ?¿f8 C, þk C ∩ A = C ∪ B, K A = X, B = ∅. 2. XJ•3 X f8 D ¦ S K A.5.11. X = {1, 2, 3}. A ∩ D = B ∩ D … A ∪ D = B ∪ D, K A = B. Ñ˜8 P(X) ¤k ƒ. (, ž‰Ñ N¹ A ˜ 244 S K A.5.12. X = {1, 2, 3}, Y = {a, b}. S K A.5.13. X •˜‡8Ü, A = P(X), B = Map(X, {0, 1}). Á‰Ñl A S K A.5.14. X, Y þ•š˜k•8Ü, |X| = m, |Y | = n. ¦l X S K A.5.15. X ´˜‡š˜k•8, f ´l X 1. f ´ü . 2. f ´V . 3. f ´÷ . S K A.5.16. •Äü‡N Ñ8Ü Map(X, Y ) Ùg N ¤k . y²e ƒ. B ˜‡V ü ‡ê. •ã d: . f : X → Y Ú g : Y → Z. y²: 1. e f, g Ñ´ü , K g ◦ f •´ü . 2. e f, g Ñ´÷ , K g ◦ f •´÷ . 3. e f, g Ñ´V , K g ◦ f •´V , ¿… (g ◦ f )−1 = f −1 ◦ g −1 . S K A.5.17. •Äü‡N Y •£Ö¿ f : X → Y Ú g : Y → Z. 1. b g ◦ f ´ü , ´Ä f, g Ñ˜½´ü ? e´, ž‰Ñy². eÄ, ž‰Ñ‡~. 2. b g ◦ f ´÷ , ´Ä f, g Ñ˜½´÷ ? e´, ž‰Ñy². eÄ, ž‰Ñ‡~. 3. b g ◦ f ´V , ´Ä f, g Ñ˜½´V ? e´, ž‰Ñy². eÄ, ž‰Ñ‡~. S K A.5.18. • Ä 8 Ü X = {1, 2, 3, 4}, Y = {a, b, c, d} Ú § ‚ {b, c, d} ⊆ Y . UìXe{K½ÂN f : X → Y : f 8 A = {1, 2} ⊆ X, B = f (1) = a , f (2) = b , f (3) = b , f (4) = c . žÏL ƒ Þ •ªL«Ñe 8Ü: f (A) , f −1 (B) , f −1 (f (A)) , f (f −1 (B)) . S K A.5.19. C ⊆ Z, f : X → Y Ú g : Y → Z þ•š˜8Üƒm N . y²: éu?¿ A ⊆ X Ú?¿ (g ◦ f )(A) = g(f (A)) , (g ◦ f )−1 (C) = f −1 (g −1 (C)) . S K A.5.20. f : X → Y •ü‡š˜8Üƒm N , A, B • X f8, A0 , B 0 • Y f8. 1. y²: f −1 (f (A)) ⊇ A, f (f −1 (A0 )) ⊆ A0 . 2Þ~`²: kŒU f −1 (f (A)) 6= A, f (f −1 (A0 )) 6= A0 . 2. y²: f −1 (A0 ∩ B 0 ) = f −1 (A0 ) ∩ f −1 (B 0 ); f −1 (A0 ∪ B 0 ) = f −1 (A0 ) ∪ f −1 (B 0 ). 3. y²: f (A ∩ B) ⊆ f (A) ∩ f (B), f (A ∪ B) = f (A) ∪ f (B). 2Þ~`²: kŒU f (A ∩ B) ⊆ f (A) ∩ f (B). S K A.5.21. f : X → Y •ü‡š˜8Üƒm −1 8 B, þk f f (B) = B. N . y² f ´÷ …= éu Y ?¿f A.5 SK 245 S K A.5.22. 1. f ´ü f : X → Y •ü‡š˜8Üƒm N . y²e d: . 2. éu X ?¿f8 A, þk f −1 f (A) = A. 3. éu X ?¿f8 A, B, þk f (A ∩ B) = f (A) ∩ f (B). 4. éu X ?¿f8 A, B, f (A) ∩ f (B) š˜ S K A.5.23. y²±eãL •ã i : Z → R •l Z …= g,¹\N R A ∩ B š˜. , f, g ´Xeü‡N f : R −→ R ; ∀ x ∈ R , f (x) := sin(πx) , g : Z −→ R ; ∀ m ∈ Z , g(m) := 0 . : †: /R ? g Z i f R S K A.5.24. i : Q → R •l Q R i ,N Ú Q /R ? i Q g,¹\N R ÑŒ N . 2. y²: XJ f Ú g Ñ´ëY¼ê, @o˜½ f = g. S K A.5.25. X = {−1, 0, 1}, f : X → X ½Â• f (x) = x2 . 1. e g : X → X ½Â• g(x) = x3 . ±eãL´Ä †? /X > f X g f X 2. kõ ‡N g:X→X U S K A.5.26. •Ädš˜8ÜƒmN ¦þ¡ ¤ ãL †? XeãL f X −−−−→ Y    β αy y g X 0 −−−−→ Y 0 b TãL †. g R †. 1. Þ~`²: f Ú g ŒU´ØÓ /R ? i i f f : R → R, g : R → R ¦ N¹ A ˜ 246 1. b α Ú β Ñ´V . y²: f ´ü 2. y²: XJ α, g Ñ´ü 3. b f, β Ñ´÷ S K A.5.27. 1. f ´ü …= , K f •´ü g ´ü ; f ´÷ …= •£Ö¿ g ´÷ . . . ž¯ g ´Ä˜½´÷ ? α ´Ä˜½´÷ X, Y •š˜8Ü, f : X → Y Ń ? . y²e¡ü‡•ã d: . 2. éu?¿š˜8Ü Z Ú?¿N †: h : Z → Y , •õ•k˜‡l Z N X g¦ e¡ãL Z g X S K A.5.28. h ~ /Y f X, Y •š˜8Ü, f : X → Y Ń . 1. ½ÂN F : X → X × Y • F (x) = (x, f (x)). y² F ´ü 2. ½ÂN p : X × Y → Y • (x, y) 7→ y. y² p ´÷ 3. y²±þ½Â N F, p U¦e¡ ãL X; × Y S K A.5.30. ê. ¦e n X n X (i + 1)(i + 2) ; n X i=0 i=1 i=1 n• ê. ¦e n X 1 ; i(i + 1) i=1 Úª n X n (−1)i ; i i=1 /Y f n• ˆª p # X 8; . †: F S K A.5.29. . Š: (−1)i ; n X n X i . i + 1 i=1 (−1)i i ; i=1 Š: n n X X (i + j)2 ; n Y i+1 i=1 j=1 i=1 S K A.5.31. a, b, c ∈ R. éÑ'u a, b, c Ø X 4 + aX 2 + bX + c ¿©7‡^‡. i ; 2n Y (n + 1 − i) . i=1 ˜‡^‡¦Ù¤•õ‘ª X 2 + X + 1 3 R[X] ¥U S K A.5.32. f ∈ K[X] • gêõ‘ª. XJ•3õ‘ª g ∈ K[X] ÷v g | f … 1 ≤ deg(g) < deg(f ), ·‚¡ f ´ K[X] ¥ Œ õ‘ª. ÄK¡ f ´ K[X] ¥ ØŒ õ‘ª. 1. y²: C[X] ¥gê ≥ 2 2. y²: R[X] ¥ 3. ée È: ØŒ õ‘ªÑ´Œ . õ‘ªgê ≤ 2. z‡õ‘ª f , éÑ§3 C ¥ ¤kŠ, , òÙ3 R[X] ¥©)•ØŒ õ‘ª ¦ A.5 SK 247 (a) f (X) = X 6 + 1 (b) f (X) = X 2 + X + 1 (c) f (X) = X 8 + X 4 + 1 S K A.5.33. f, g ∈ Q[X]. é±ez«œ¹, ¦ f 3 Q[X] ¥Ø± g ¤ 1. f = 3X 5 + 4X 2 + 1, g = X 2 + 2X + 3; 2. f = 3X 5 + 2X 4 − X 2 + 1, g = X 3 + X + 2. ûÚ{ª: 248 N¹ A ˜ •£Ö¿ N¹ B ÿ †•ÁÁKÀ6 ÿ B.1 I ÿ FÏ: 2021 c 10 11 F žm: 50 ©¨; ÷©: 50 © ±eo K Ć f•, m, n • ê. 1. (12 ©) {‰K. ž† Ñe ¯K ‰Y. (ØI‡‰Ñ)KL§½)ºnd.)  2x + 3x − x = 0 1 2 3 (a) o ‚5•§| Ï)Œ±L«• x1 + 2x2 + x3 − x4 = 0 ! −1 2 0 (b) A= . ž Ñ A 1˜m¥ ƒ ˜„/ª. 0 1 −6 Ñü‡ƒÓŒ (c) 2. (12 ©) ä Ø. Œ_Ý ( A, B ¦ ž)ºnd, †Ø   0 0   b =  . Kéu?¿1 0 (a) F/ A + B ØŒ_. žÞÑ‡~. Ý A ∈ M4×3 (K), (n ) ‚5•§| AX = b ˜ 1 ½Ã). P, Q •ƒÓŒ (b) Ð Ý , K P Q E´Ð A ∈ Mm×n (K) km_. Kéu?¿Ý AX = B.   1 2 3 4 1 0 −2 −6   3. (6 ©) ¦Ý   _Ý . 2 3 1 2 (c) 1 (\ 1 1 Ý . B ∈ Mm×s (K), •3Ý X ∈ Mn×s (K) ¦ −1 )‰I‡•¹OŽL§¥ '…Ú½.) 4. (20 ©) A1 = (a) ±þo‡Ý 0 0 ¥= ! 1 , A2 = −1 1 0 ´1•{/Ý ! 1 , A3 = 0 0 0 1 0 ! , A4 = ? (eÙ¥vk1•{/Ý 249 0 −1 ! 1 . 0 , £‰“vk”=Œ.) N¹ B 250 (b) y²: M2 (K) ¥z‡Ý ÑŒ± ¤ A1 , A 2 , A 3 , A 4 ÿ †•ÁÁKÀ6 ‚5|Ü. (c) ´Ä•3•þ b ∈ K 2×1 ¦ •§| A1 X = b, A2 X = b Ú A3 X = b ˆgÑkÃ¡õ), A4 X = b =k•˜) ? e´, ‰Ñ˜‡ù •þ b. eÄ, ž)º•Ÿoù bØ •3. (d) ¦Ñ M2 (K) ¥† A1 , A2 , A3 , A4 þŒ B.2 † ¤kÝ ÿ II . ÿ FÏ: 2021 c 11 1F žm: 50 ©¨; ÷©: 50 © ±eo K Ć f•, m, n, s • 1. (12 ©) {‰K. ž† ‰Y. (ØI‡‰Ñ)KL§½)ºnd.)   −2 −1   A • n Œ_Ý , … A−1 ®•. •Ä n + 2 • B =  A  (ùp˜x? 1 Ý ƒ@•´ 0). K B _Ý Œ±L«• . (a) (b) Ñe ê. ¯K m × n Ý A •• n − s, •þ η1 , · · · , ηs , ηs+1 Ñ´‚5•§| AX = b . b 6= 0. K•þ| η1 , · · · , ηs , ηs+1 ••ŒŒUŠ´ ), Ù¥ v1 , v2 ´ K 10 ¥ ü‡‚5Ã'•þ, U ´ K 10 ¥ 4 ‘f˜m, … v1 ∈ / U, v2 ∈ / U. 10 W ´ K ¥•¹ {v1 , v2 } ∪ U • f˜m. K dim W ¤kŒU Š• . (c) 2. (12 ©) ä Ø. ( ž)ºnd, †Ø ƒÓ Ý žÞÑ‡~. (a) A, B •Œ Ó). . e rank(A) = rank(B), Kàg‚5•§| AX = 0 † BX = 0 (b) V ´ K n f˜m. b v1 , v2 , v3 ´ V ¥ •þ| v1 + v4 , v2 + v4 , v3 + v4 ‚5Ã'. ‚5Ã'•þ|, v4 ∈ K n v4 ∈ / V. K A ∈ M4×5 (K). ˜½Ø•3Ý B ∈ M5×4 (K) ¦ AB Œ_.     1 0 1 1 0 1     3. (10 ©) b Ý A = −1 −1 1 Ú B = 0 −1 2 ƒ-. 0 2 a 0 0 0 (c) (a) ¦Ñ a ¤kŒU (b) ´Ä•3Œ_Ý Ý Ø•3. Š. P ¦ P A = B ? e´, žéÑ˜‡ù Ý P , eÄž)º•Ûù 4. (8 ©) ± T Ú S ©OL« K n ¥ •þ| α1 , · · · , αr Ú β1 , · · · , βm . b T ´‚5Ã' , ¿ … T ¥ z‡•þþŒ •þ| S ‚5LÑ. y²: •3 [[1, m]] ˜‡f8 {i1 , · · · , ir } ¦ eI j 5g [[1, m]] \ {i1 , · · · , ir } m − r ‡•þ βj ëÓ T ˜å ¤ •þ| S 0 † S ‚5 d. (5: ±þ(Ø= Steinitz O†Ún (Steinitz exchange lemma). e†^,˜«•ª£ã, ùp S 0 Ò´3•þ| S ¥ò•þ βi1 , · · · , βir O†• α1 , · · · , αr ¤ •þ|.) B.3 ÿ III 251 5. (8 ©) Þ~`²: K 2 ¥•3÷v±e^‡ š˜f8 U, V : (a) U 'u\{Ú~{Ñµ4 (=, é?¿ u, v ∈ U þk u ± v ∈ U ), U Ø´ K 2 f˜m. (b) V 'uê¦µ4 (=, é?¿ v ∈ V Ú?¿ λ ∈ K þk λv ∈ V ), V Ø´ K 2 f˜m. ÿ B.3 III ÿ FÏ: 2021 c 11 29 F žm: 50 ©¨; ÷©: 50 © ±eo K Ć f•, n • 1. (12 ©) {‰K. ž† Ñe ê. ¯K ‰Y. (ØI‡‰Ñ)KL§½)ºnd.) (a) U • Mn (K) ¥ (b) f : R2 → R2 ´²1u†‚ U = span((1, −1)) Ý•†‚ W = span((4, −2)) , α = (2, 0). K f (α) = . (c) ògêØ‡L 2 ˜|Ä. î‚þn ¤ f˜m. K dim U = . EXêõ‘ª8Ü C[X]≤2 À• R-•þ˜m, ÝKN Ñ C[X]≤2 3 R þ 2. (6 ©) U, V, W þ•,‡•þ˜m¥ f˜m, … U + V, V + W Ú U + W Ñ´†Ú. ž¯ U + V + W ´Ä˜½´†Ú ? e´, ž‰Ñy². eÄ, žÞÑ‡~. 3. (12 ©) U = span(α1 , α2 ), W = span(β1 , β2 ), Ù¥ α1 = (1, 1, 0, 0), α2 = (1, 0, 1, 1), β1 = (0, 0, 1, 1), β2 = (0, 1, 1, 0) . ¦ U ∩W Ú U +W 4. (8 ©) ˜|Ä. (I‡Jøv U Ú W Ñ´•þ˜m V )KL§.) f˜m. y²±eØä d: (a) U ∪ W = U + W . (b) U ⊆ W ½ W ⊆ U . (c) U ∪ W ´‡f˜m. 5. (12 ©) •Ä¤k¢Xêõ‘ª ¤ ¢•þ˜m V = R[X] . (a) d"õ‘ªÚ V ¥¤kgê•óê o? õ‘ª ¤ f8 U ´Ä´ V (b) •ÄN ϕ : V → V : f (X) 7→ f (X + 1). ž¯ ϕ ´Ä´V N ? •Ÿo ? (c) ´Ä•3‚5N A : V → V , §´ü Ä, ž)º•Ÿoù A Ø•3. Ø´÷ f˜m ? •Ÿ ? •Ÿo ? ϕ ´Ä´‚5 ? e´, ž‰Ñ˜‡ù N . e N¹ B 252 ÿ B.4 ÿ †•ÁÁKÀ6 IV ÿ FÏ: 2021 c 12 20 F žm: 50 ©¨; ÷©: 50 © ±eo n• ê, K ´ C f•, ešAO`², •þ˜mÑ´• K-•þ˜m. 1. (6 ©) Ñ‚5N (a) "zÝÚ• /•ã'u‚5N (b) ½Â. •-"zÝ½n. 2. (6 ©) •Ä K 3 ü|kSÄ B = (v1 , v2 , , v3 ) Ú C = (η1 , η2 , η3 ), Ù¥             0 1 1 0 1 1             v1 = 1 , v2 = −1 , v3 =  2  , η1 = 1 , η2 = 2 , η3 =  1  . −2 5 3 −1 0 0   2 0 3   3 b A ∈ End(K ) 3kSÄ B e Ý ´ 0 −2 −1, •þ α 3kSÄ C e ‹I´ 1 −1 −4 T (1, 0, −1) . ¦ A α 3 C e ‹I. (ž Ñ•[ ¦)L§!) 3. (12 ©) ä Ø. ( ž)ºnd, †Ø žÞÑ‡~. (a) A : V → W Ú B : W → U Ñ´•þ˜mƒm K B ˜½´÷ . (b) V ´ n ‘ • þ ˜ m, K é u V MB (IdV ) ˜½ u n ü In . 4. (10 ©) (b) . e BA ´÷ ? ¿ ˜ | k S Ä B, ð N A ´•þ˜m V þ , A ´ü IdV 3 B e V ´k•‘•þ˜m, b ‚5C† A ∈ End(V ) 3,˜|kSÄe . @o A 3 V ?¿˜|kSÄe Ý þ•Œ_Ý . (c) (a) ‚5N Ý , Ý ´Œ_Ý ‚5C†. r ∈ N∗ , •þ α ∈ V ÷v A r−1 α 6= 0, A r α = 0. y²: •þ| α, A α, A 2 α, · · · , A r−1 α ‚5Ã'. dim V = n. b A n−1 6= 0, A n = 0. ¦ y: V k ˜ | k S Ä B ¦  0 1   0 1     .. ..   . .  .     .. . 1   5. (16 ©) ½ÂN 0 ! 0 −1 P = . •Ä V = M2 (K) ¥ kSÄ E = (e1 , e2 , e3 , e4 ), Ù¥ 2 0 ! ! ! ! 1 0 0 1 0 0 0 0 e1 = , e2 = , e3 = , e4 = . 0 0 0 0 1 0 0 1 T : V → V ; X 7→ P −1 XP . MB (A ) = 2019 c¢GÆÏÏ¥•ÁÁK B.5 (a) y² T ´ V þ 253 ‚5C†, ¿¦Ñ T 3kSÄ E e Ý . (b) η1 = ! 1 , η2 = 0 0 −1 y² (η1 , η2 , η3 , η4 ) (c) ¦ E LÞÝ B ¤V 1 −1 0 −1 ! , η3 = 0 1 1 0 ! , η4 = 0 1 1 1 ! . ˜|kSÄ B. . (d) ´Ä•3š"Ý X ∈ M2 (K) Ú λ ∈ K ¦ T (X) = λX ? e´, ž‰Ñù X Ú~ê λ. eÄ, ž)º•Ÿoù X Ú λ Ø•3. B.5 ±eo ˜‡Ý 2019 c¢GÆÏÏ¥•ÁÁK K •ê•, m, n L« ê. êÆPÒ†‘§ùÂƒÓ, k¦¯Œ±Î¯i•P“. 1˜Ü© { ‰ K ( 55 ©) For each of the following questions, you can just write down your answer without any further explanation. éuù˜Ü© z˜‡¯K, •)•I† Ñz K ‰Y, Ø7‰?Û)º. S K B.5.1 ( K 40 ©). 1. Ñ˜‡ê• K ¦ÙŠ• Q-•þ˜m÷v dimQ K = 2. 2. Ñ˜‡N 3. ½ÂN f : N∗ → N∗ , §k†_ f : R → R • f (x) = cos(x), I •4«m [1, 2]. ¦ f −1 (I) Ú f (f −1 (I)). 4. Ñü‡Œ 5. Ñ˜‡Ý A, §km_ vk†_.   1 2 0 1 0 −1 −2 1   A= . ¦ A−1 . 0 0 −1 0 0 0 −1 1 6. 7. vkm_.  0 0  A= 0 0 ƒÓ 1 0 0 0 0 0 0 0 • A Ú B, ¦  0 1  . ¦÷v An = 0 1 (A + B)2 6= A2 + 2AB + B 2 . • ê n. 0 A, B, C þ•Œ_• (Œ Ø˜½ƒÓ). M •Xe©¬Ý   A 0 0   M =  0 B D.¦ M −1 . 0 0 C   1 2 1   9. A=2 4 2 . éÑü‡Œ_Ý P, Q ∈ M3 (R) ¦ P AQ ¤•/X −1 −2 −1 ¬Ý , Ù¥ r = rank(A). 8. Ir 0 ! 0 0 © N¹ B 254 V = Mn (K), - G • V ¥Œ_Ý 10. ¤ A, B ∈ Mm×n (K). K A † B ƒ- 2. …= †•ÁÁKÀ6 f8, U = span(G). ¦ dimK U . S K B.5.2 ( K 15 ©). äé†. ée¡z‡Øä, `²Ù ŒIP• F. (ØI‡)ºnd.) 1. ÿ (†Ä, ( ŒIP• T, †Ø rank(A) = rank(B). e1 = (1, 0, · · · , 0), · · · , en = (0, · · · , 0, 1) • K n •ã d: IOÄ, α1 , . . . , αn • K n ¥•þ. Ke (a) •þ| α1 , . . . , αn ‚5Ã'. (b) éuz‡ i ∈ [[1, n]], •þ ei Ñ´ α1 , . . . , αn α1 , . . . , αm Ú β1 , . . . , βn • K s×1 ¥ 3. ‚5|Ü. •þ. b •3Ý A = (aij ) ∈ Mm×n (K) ¦ (β1 , · · · , βn ) = (α1 , · · · , αm ) (aij ) . XJ•þ| α1 , . . . , αm ‚5ƒ', K•þ| β1 , . . . , βn •‚5ƒ'. P ∈ Mm×n (K), A ∈ Mn×p (K). e P k†_, K rank(P A) = rank(A). 4. 5. 8Ü U = {A ∈ M3 (K) | A2 = 0} ´ K-•þ˜m M3 (K) éuù˜Ü© S K B.5.3 ( 1 Ü© Ä : K z˜‡¯K, •)I‡¦ŒU•¦/ K f˜m. ( 45 ©) Ñ‰K[!, ±¦z 6 ©). ¦e¡•§| ˜‡Ä:)X   x2 − x3 + x4      −7x2 + 3x3 + x4 =0 − 3x4 =0 2x2 + 3x3 − 4x4 =0  x1 +     x − 1 3x2 A = (aij ) ∈ Mn (K) •þn S K B.5.5 ( W ´ K-•þ˜m, V, U ⊆ W •š˜f8. 8 ©). , ¿…é ‚þ 1. b éu¤k u, v ∈ V Ú¤k λ ∈ K, þk u + λv ∈ V . y² V ´ W 2. b éu¤k α, β ∈ U , þk α + β ∈ U 9 −α ∈ U . ´Ä U ˜½´ W )‰˜ß . =0 S K B.5.4 ( K 6 ©). š". y² A Œ_. K K f˜m ? e˜½´, ž‰Ñy². ÄKžÞÑ‡~. ƒ a11 , . . . , ann þ f˜m. 2019 c¢GÆÏÏ¥•ÁÁK B.5 S K B.5.6 ( K 10 ©). 255 L •˜m R3 ¥÷v±e¤k^‡ †‚: • L ²L: P (2, −3 , −1). • L †±e†‚²1 : ` : 1. ¦ L •§ ( y+1 z x−1 = = . −2 −1 1 Ñ?Û˜«/ª=Œ, ~X˜„•§, ëê•§ 2. (½†‚ L Ú²¡ Π : x − y + z − 10 = 0 ). ˜'X (•ƒ !²1½•¹). S K B.5.7 ( K 15 ©). ˜‡Ý L ∈ Mn (R) ¡•´ Ñen , XJ L én ¿… § é ‚þ ƒÑ´ 1. (e n = 1, Ž˜ Ñen ´ Š 1 × 1 Ý w– ~ê 1.) · ‚`˜‡Ý A ∈ Mn (R) k LU-©), XJ•3˜‡Ñen L ∈ Mn (R) Ú˜‡þn U ∈ Mn (R) ¦ A = LU .   1 4 7   1. Á¦Ý 2 5 8  ˜‡ LU-©). 3 6 10   1 2 3   2. y²Ý 2 4 7 vk LU-©). 3 5 3 3. y²: XJ A ∈ Mn (K) Œ_¿…k LU-©), @o§ 4. Ñ˜‡š"• 5. A = (aij ) ∈ Mn (R). b LU-©)´Ž˜ , §k¿…kØ•˜‡ LU-©). a11 6= 0 …e Ý þŒ_:  A2 = a11 a21 .  ! a11 a12  , A3 = a21 a22 a31 a12 a22 a32 a11  a13 a21   a23  , · · · , An−1 =   ..  . a33 an−1,1 a12 a22 .. . ··· ··· .. .  a1,n−1  a2,n−1   ..   . an−1,2 ··· an−1, n−1  y² A k LU-©). éuù˜Ü© 1nÜ© k ] Ô 5 ¯ K ( 10 ©) z˜‡¯K, •)I‡¦ŒU•¦/ Ñ‰K[!, ±¦z S K B.5.8 (10 points y²: é?¿ K K 10 ©). •Ÿ6¥u<¬ ÚI¤á 70 ±c, •Ä•   1 1949 70 2019 0 1 1949 70    M =  . 0 0 1 1949 0 0 0 1 ê n, ˜½•3Ý A ∈ M4 (R) ¦ An = M . )‰˜ß . N¹ B 256 S K B.6.1 ( †•ÁÁKÀ6 2019 c¢GÆÏÏ"•ÁÁK B.6 ±eo ÿ ÁòêÆPÒ†‘§ùÂƒÓ, k¦¯Œ±Î¯i•P“. K •ê•, m, n L« ê. ešAO`², •þ˜mo´• K þ K 40 ©). ž† Ñ±e¯K •þ˜m. ‰Y. ØI‡‰?˜Ú)º.      0 x 1 2 1      1. ‰Ñ˜‡¢ê t ¦ ‚5•§| 2 3 t  y  = 1 Ã). 2 z t 1 0   1 0 2   2. A = −1 1 0. ¦ A N‘Ý A? . 1 3 4 3. 4. Ñ˜‡šé A ∈ M3 (R) ÷v A2 = I3 . Π •˜m R3 ¥²L A(0, 1, 2), B(1, −1, 1) ü:…²1u†‚ x2 = y+1 3 = z ²¡ Π •§ (?¿˜«/ªþŒ). 5. Ñ˜‡Ý 6. U = {(x, y, z, t) ∈ R4 | x − 3y = 2t , y = z − t}. ¦ R4 7. , §k†_ 8. f (x) = 9.  0 1  A= 0 0 0 0 1 0 10. Ñü‡þn 0 0 0 1 Ñ vkm_. ˜‡f˜m W ¦ R4 = U ⊕ W . f (X) = X 6 − X 4 + 2X 3 − 4, g(X) = X 2 + 2X − 1. ¦‘{Ø{ f = gQ + R ¥ ª R, Ù¥ deg(R) < deg(g). 1 1 1 2 1 4 1 8 ²¡. 1 −2 4 −8 1 x . éÑ f 3 C ¥¤k x2 x3  2018 2019  . ¦ A 2020 A û Q Ú{ Š. õ‘ª. 1 A, B ∈ M3 (R), Ù¥ A ØŒé z, B Œé S K B.6.2 ( K 16 ©). äé†. ée¡z‡Øä, `²Ù ŒIP• F. (ØI‡)ºnd.) ! A 0 1. A, B, C •Œ Ü· Ý , M = . C B z (†Ä, Ø´é ( . ŒIP• T, †Ø K rank(M ) = rank(A) + rank(B). 2. b η1 , η2 , η3 , η4 ´àg‚5•§| AX = 0 ´ AX = 0 ,˜‡Ä:)X. ˜‡Ä:)X. K η1 +η2 , η2 +η3 , η3 +η4 , η4 +η1 B.6 2019 c¢GÆÏÏ"•ÁÁK 3. ±g, •ªò C2 À• R þ •þ˜m. K (1, 1), (i, i) ´ C2 ¥ 4. U Ú V • K þ T :U →V. 5. U1 , U2 , U3 ••þ˜m V •´†Ú. 6. A •k•‘•þ˜m V þ A A •þ. 7. U ´•þ˜m V 257 k • ‘ • þ ˜ m. K dim U ≤ dim V ‚5C†. K A Œé f˜m. • 3 K-‚ 5 z …= V ¥¤kš"•þÑ´ vm+1 , · · · , vn ∈ / U. d: (b) v1 , · · · , vm ´ U ˜|Ä. ˜|Ä, … [vm+1 ]U = vm+1 + U, · · · , [vn ]U = vn + U ´ V /U V ´ C þ k•‘•þ˜m, A , B ∈ End(V ). b A Ú B 3 C ¥Ø ¦A Š. @o§‚ EÜN A B 3 C ¥•vk 0 ƒ Ù¦A Š. S K B.6.3 ( K ü v1 , · · · , vm , vm+1 , · · · , vn • V ¥•þ, …÷v (a) v1 , · · · , vm , vm+1 , · · · vn ´ V 8. …= f˜m. XJ U1 +U2 , U1 +U3 , U2 +U3 Ñ´†Ú, K U1 +U2 +U3 v1 , · · · , v m ∈ U K±eØã ‚5Ã'•þ|. 8 ©). •Ä R3 ¥äkXe•§ ü^†‚ L1 Ú L2 :   3x − 2y + z = 0 x − 3y + 2 = 0 L1 : 9 L2 : x − 3y + 5 = 0 x + y + z + 1 = 0 y² L1 Ú L2 •É¡†‚, ¿¦Ñ§‚ƒm S K B.6.4 ( K 10 ©). 1. y²e •ã d: 0ƒ ˜|Ä. ÑvkÙ . ål. m, n, s ∈ N∗ , A ∈ Ms×m (R), B ∈ Mm×n (R). (a) àg‚5•§| ABX = 0 Ú BX = 0 (Ù¥ X = (x1 , · · · , xn )T ) Ó). (b) rank(AB) = rank(B). 2. y² rank(A) = rank(AAT ) = rank(AT A). 3. M ∈ Ms×m (C). 3˜„œ¹e y². eÄ, žÞÑ‡~. S K B.6.5 ( K 10 ©). (xn )n≥0 , (yn )n≥0 Ú (zn )n≥0 •÷v±e^‡ n‡ê    xn+1 = xn + yn + 3zn x0 = 1, y0 = 0 , z0 = 1 ¦ xn , yn , zn ùn‡ê ª rank(M ) = rank(M T M ) ´ÄE,¤á ? e´, ž‰Ñ Ï‘ ( …é¤k n ∈ N , ¤¹ n yn+1 = xn + 3yn + zn    zn+1 = 3xn + yn + zn Lˆª). . : N¹ B 258 S K B.6.6 ( K ÿ †•ÁÁKÀ6 n ≥ 1, V = R[X]≤n . ½Â‚5C† A ∈ End(V ) • 10 ©). A : f (X) 7−→ A (f (X)) := f (X + 1) . ~X, A (X 2 ) = (X + 1)2 . 1. b n = 2, B • V = R[X]≤2 ¥ ¦ A 3kSÄ B e 2. é˜„ MB (A ), ¿OŽ A Ý n, ¦Ñ A kSÄ (1 , X − 1 , (X − 1)2 ). ¤kØÓ ¢A ŠÚƒA S K B.6.7 ( K 8 ©). A ••þ˜m V þ ØCf˜m, ·‚¡ A ´ØŒ . 1. ÞÑ˜‡‚5C† A ∈ End(V ) 1 ªÚ,. A •þ. ‚5C†. XJ A Ø ¢~, Ù¥ dim V > 1 … A ØŒ 0ÚV ƒ . 2. y A ∈ End(V ) Ú B ∈ End(W ) ´ ü ‡ š " • þ ˜ m V Ú W T : V → W •‚5N . b XeãLŒ vkO ØŒ ‚ 5 C †. †: T V −−−−→ W     Ay yB T V −−−−→ W y²: ‡o T = 0 ‡o T ´Ó S K B.6.8 ( K 8 ©). 1. y²: XJ A Œé 2. . A ∈ Mn (C), f (X) ∈ C[X] … deg(f ) ≥ 1, B = f (A). z, K B •Œé z. λ1 , · · · , λr ∈ C • A 3 C ¥¤kØÓ ,‡ i ∈ [[1, r]] ¦ µ = f (λi ). B.7 ±eo P“. K•C S K B.7.1 ( K A Š. y²: Eê µ ´ B f•, m, n, s L« …= •3 ê. êÆPÒ†‘§ùÂƒÓ, k¦¯Œ±Î¯i• Ñ±e¯K ‰Y. ØI‡‰?˜Ú)º.  X ¦ Š 2021 c¢GÆÏÏ¥•ÁÁK 32 ©). ž† 1. éÑ˜‡ 3 × 2 Ý A 0  X = AX + B, Ù¥ A = −1 −1   1 1 0   1 1 , B = 2 5 0 −1  −1  0 . −3 B.7 2021 c¢GÆÏÏ¥•ÁÁK  3 −2  t 4 9 −6 •´ 1. K t = A, §k†_ vkm_.  2. b Ñ˜‡Ý 3. 4. 1  −2 3 Ý 259 . α1 , α2 , α3 ´Œ_Ý P •þ|, Q = (α1 +α2 , α2 , α3 ). b  ∈ M3 (K)  1 0 0   ÷v P −1 AP = 0 1 0. K Q−1 AQ = . 0 0 2     a1 , · · · , an •š"~ê, A =    a1 5. an−1 an−2 . .. Ý A ∈ M3 (K)  1 1  ..  −1  .  ∈ Mn (K). K A =  1 . an  2  4  A= −1  3 −2  2  2  −1 . žl A 1•þ|¥éÑ1˜m R(A) ˜|Ä.  1 0  x + y − z = 0 ` ´˜m¥˜„•§• †‚, L ´²L: (1, 0, −1) …²1u ` x − y + 2 = 0 6. 7. KL 8. 4 7 −2 5 −3 ˜„•§´ A= −1 0 †‚. .  ! A 1  , M = 0 −1 0 A + I2 A + 2I2 0  0  0 . éÑ˜‡ 4 gõ‘ª f ¦ A S K B.7.2 ( K 16 ©). äé†. ée¡z‡Øä, `²Ù ŒIP• F. (ØI‡)ºnd.) (†Ä, f (M ) = 0. ( ŒIP• T, †Ø A ∈ Mm×n (K), b ∈ K m×1 . e•§| AX = b kÃ¡õ‡), K AX = 0 kš"). 1. 2. b •þ| v1 , v2 , v3 ‚5ƒ'. K v1 , v2 Ú v1 , v3 ùü‡•þ|‚5 d. 3. η1 , · · · , ηs • K n×1 ¥ ‚5Ã'•þ|. K˜½•3Ý àg‚5•§| AX = 0 ˜‡Ä:)X. 4. e1 , · · · , en • K n IOÄ, α1 , · · · , αn • K n ¥ •þ|. XJ e1 , · · · , en ¥ Œ α1 , · · · , αn ‚5LÑ. K α1 , · · · , αn ´ K n ˜|Ä. , B ∈ Mn (K) •‡é¡ A ∈ Mn (K) ¦ 5. A ∈ Mn (K) •é¡ . K AB − BA ´‡é¡ 6. A •é¡¢Ý 7. A, B ∈ Mn (K) ÷v AB = 0. - r = rank(A). K rank(B) = n − r. . XJ A2 = 0, K A = 0. η1 , · · · , ηs . ¤ z‡•þþ N¹ B 260 A ∈ M4×3 (K). ˜½Ø•3Ý 8. éue¡ z˜‡¯K, •)I‡¦ŒU•¦/ S K B.7.3 ( K †•ÁÁKÀ6 AB Œ_. Ñ‰K[!, ±¦z K )‰˜ß . 10 ©).  1  A= 0 −3 ž B ∈ M3×4 (K) ¦ ÿ ä A Ú B ´Ä1   1 0 0 2   0 0 −1 , B = 0 0 0 0 0 d ? ´Ä d ? (ž•\ 0 0 0 0 1 0  0  0 . 0 Øä‰Ñ¿© nd.) S K B.7.4 ( K 8 ©). α1 , α2 , α3 , α4 • Ý A ∈ M4 (K) (• g ü ) • þ |, β = α1 + α2 + α3 + α4 . b α2 , α3 , α4 ‚5Ã', α1 = 2α2 − α3 . ¦‚5•§| AX = β Ï). S K B.7.5 ( K 10 ©). L •˜m R3 ¥÷v±e¤k^‡ †‚: • L ²L: P (1, 0 , −2). • L †±e²¡²1 : Π : 3x − y + 2z + 2 = 0 . • L †±e†‚ƒ : ` : ¦L •§ ( x−1 y−3 z = = . 4 −2 1 Ñ?Û˜«/ª=Œ, ~X˜„•§!ëê•§ ). ! A C S K B.7.6 ( K 6 ©). A ∈ Mm×s (K), B ∈ Mn×s (K), C ∈ Mm×r (K). - M = . B 0 y ²: rank(M ) ≥ rank(B) + rank(C), … rank(B) = s ½ rank(C) = m ž, rank(M ) = rank(B) + rank(C). S K B.7.7 ( K 12 ©). ¤¢ P ´ n ˜†Ý ´• P ´ n • ¿…§ ˆ‡1•þŒ± dü Ý In ˆ1-#ü , •Ò´`, XJØ•Äˆ‡1 ü ^S, @o P 1•þ| † In 1•þ|´˜ . ~X, 1ãÐ Ý , =ÏL † In ,ü1 Ý , Ñ´˜† Ý . ü Ý In g •´˜†Ý . 1. ¦ n ˜†Ý ‡ê. 2. y²: P ´˜†Ý 3. P •n ˜†Ý (a) y² P P T = In . …= . P Œ± ¤k•õ‡1ãÐ Ý ¦È. B.8 2021 c¢GÆÏÏ"•ÁÁK (b) y²: Ý 261 P − In ØŒ_. S K B.7.8 ( K 16 ©). K¥·‚UYæ^c¡1 B.7.7 ŒK‰Ñ ˜†Ý Vg. 3)‰e ¡ Kž, XkI‡, #Ngd/¦^c˜ŒK¥Ñy (Ø (ÃØ\´ÄU y²@ (Ø). ^ GLn (K) L« Mn (K) ¥¤kŒ_Ý ¤ 8Ü. ½Â N • GLn (K) ¥¤k÷veã^ ‡ Ý G ¤ f8: é?¿é D ∈ GLn (K), GDG−1 Ú G−1 DG •Ñ´é . 1. éu?¿ M ∈ GLn (K), y²: M ∈ N M = P A. …= 2. y²: M ∈ N ž, =k•˜ M = P A. (P, A), Ù¥ P •˜†Ý P1 , · · · , Pr •¤k 3. ˜éÝ ˜†Ý n •3˜†Ý P ÚŒ_é . ^ B L« GLn (K) ¥¤k Ý A ¦ , A •Œ_é ,¦ Œ_þn Ý ¤ f 8. y²: (a) GLn (K) = Sr (b) éuØÓ •I i, j ∈ [[1, r]], 8Ü BPi B Ú BPj B 5: i=1 BPi B, =, BP1 B, · · · , BPr B ù K1 3 ¯´“‚5“ê+”nØ¥-‡ B.8 ±eo S K B.8.1 ( 1. 2. 3. 4. 5. 6. 7. f8 ¿ u GLn (K). 8•˜. Bruhat ©)½néu GLn AÏœ¹. 2021 c¢GÆÏÏ"•ÁÁK ÁòêÆPÒ†‘§ùÂƒÓ, k¦¯Œ±Î¯i•P“. K •ê•, m, n L« ê. ešAO`², •þ˜mo´• K þ K 40 ©). ž† Ñ±e¯K •þ˜m. ‰Y. ØI‡‰?˜Ú)º. A ∈ Mm×3 (K). b α = (1, 2, −1)T Ú β = (2, 1, 4)T :)X. e (1, 0, a)T ´ AX = 0 ˜‡), K a = . ¤‚5•§| AX = 0 U = {(x, y, z, t) ∈ R4 | x − y − z = t , y = z − t}. ¦ R4 ˜‡f˜m W ¦   1 0 2   A =  0 −1 0 . K§ N‘Ý • A? = . −2 3 −4 A ∈ M3 (K) ¿b det(A) = 2. K det(5A−1 − A? ) = ¤ f˜m. K dim U = R4 = U ⊕ W . . f : R2 → R2 ´²1u†‚ U := span((1, 1)) Ý•†‚ W := span((3, −4)) α = (10, 3). K f (α) = . U • Mn (K) ¥¤ké¡Ý ˜‡Ä ÝKN . . Π •˜m R3 ¥²L A(1, 0, 2), B(0, −1, 1) Ú C(−1, 1, 0) n: •§ (= ax + by + cz + d = 0 /ª •§). ²¡. Ñ²¡ Π ˜„ N¹ B ÿ †•ÁÁKÀ6 Ñ˜‡Œ_Ý P ∈ M4 (R) ¦ 262 8. OŽ1 ª:  a a a a 2022   A =   9. b c d a+b a+b+c a+b+c+d 2a + b 3a + 2b + c 4a + 3b + 2c + d 3a + b 6a + 3b + c 10a + 6b + 3c + d    1    5 1    .  , A0 =     19 5 2022 19 = . P −1 AP = A0 . Ñ˜‡Ã•‘•þ˜m V Ú˜‡‚5C† A ∈ End(V ), ¦ 10. S K B.8.2 ( K 16 ©). äé†. ée¡z‡Øä, `²Ù ŒIP• F. (ØI‡)ºnd.) 1. A ∈ Mm×n (K), v1 , · · · , vr • Ã' …= rank(A) = r. 2. ò C À• R þ 3. 5. b 6. •þ˜m. KÏL¦Eê U, V, W þ•,•þ˜m ´†Ú. 4. •þ˜m K n ¥ A ´ü (†Ä, ( Ø´÷ . ŒIP• T, †Ø ‚5Ã'|. K•þ| Av1 , · · · , Avr ‚5 Ý½Â N C → C ; z 7→ z̄ ´‡‚5N . f˜m. e U + V + W ´†Ú, K U + V, U + W Ú V + W Ñ V Ú W Ñ´k•‘•þ˜m, … dim V ≥ dim W . K7•3÷ f0 , f1 , f2 , f3 ´¢•þ˜m R[X]≤3 ‚5N T : V → W. ˜|Ä. K˜½•3 i ∈ [[0, 3]] ¦ deg(fi ) = 1. V ´k•‘•þ˜m, A ∈ End(V ). e 0 ´ A ˜‡A Š, K A Ø´÷ . ü‡•þ˜m V, W ÷v dim V = 2 9 dim(V × W ) = 6. K dim W = 3. ! 1 c 8. Ý 3 C þŒé z …= c = 0. 0 2 7. S K B.8.3 ( K 10 ©). •Ä R3 ¥äkXe•§ L1 : x−1 y−1 z = = 2 −1 2 Ú 1. y² L1 Ú L2 •É¡†‚, ¿¦Ñ§‚ƒm 2. ²¡ Π † L1 , L2 þ²1¿… ùü^†‚ ü^†‚ L1 Ú L2 :  x + y − 1 = 0 L2 : z = 0 . ål. ålƒ . ¦²¡ Π ˜„•§.  S K B.8.4 ( K 6 ©). OŽ1 ª |An |, Ù¥ An ∈ Mn (K) 2 −1    ½Â• An =     −1 2 .. .  −1 .. . .. . .. . 2 −1     .   −1 2 B.8 2021 c¢GÆÏÏ"•ÁÁK ±þÝ Lˆª¥˜x? S K B.8.5 ( K (1, X 2 − 1, X + 2). 1. ¦ E 2. ƒþ• 0, A1 = 2, A2 = ! −1 . 2 2 −1 8 ©). • Ä ¢ • þ ˜ m V = R[X]≤2 Ú § LÞÝ B 263 k S Ä E = (1, X, X 2 ) Ú B = P. g = X 2 − 4X − 2. ½ÂN A : V −→ V ; (a) y² A ´‚5N . (b) ¦ A 3kSÄ E Ú B e S K B.8.6 ( K f (X) 7−→ f (1)g(X) + f 0 (X) . Ý ME (A ) Ú MB (A ). 12 ©). •Ä¢•þ˜m V = M2 (R) ÚN f : V →V ; 1. y² f ´‚5C†, ¿éÑ Ker(f ) f (A) = A − AT . ˜|Ä. 2. y² V = Ker(f ) ⊕ Im(f ). 3. ¦ f ¤kA Š¿éz‡A 4. (½ f ´ÄŒé S K B.8.7 ( K Š λ, ‰ÑA f˜m E(λ, f ) ˜|Ä. z. 12 ©). A , B ´ n ‘•þ˜m V þ ‚5C†, … A 2 = B 2 = 0. 1. y²: rank(A ) ≤ n2 . A B + BA = I (ùp I L« V 2. ?˜Úb g ð N ). (a) y²: V = Ker(A ) ⊕ Ker(B). (b) V = Im(A ) ⊕ Im(B) ´Ä˜½¤á ? e´, ž‰Ñy², eÄ, žÞÑ‡~. (c) y² n = dim V ˜½´óê. (d) b n = 2. y² V k˜|kSÄ E ¦ S K B.8.8 ( K 4 ©). ¥ š"•þ. ½ÂN ME (A ) = 0 0 ! 1 , ME (B) = 0 f 7−→ f (ε) . ¦ rank(ϕ). K ! 0 . 0 U, W ©O• n ‘Ú m ‘•þ˜m. - V = Hom(U, W ). ϕ : V −→ W ; S K B.8.9 ( 0 1 2 ©). ø©K: ž% ÆÏ‘§ùÂ¥Ñy üÄ˜. εÚ 264 N¹ B ÿ †•ÁÁKÀ6 ë•©z [1] Tom M. Apostol, Linear Algebra: A First Course with Applications to Differential Equations, John Wiley & Sons, Inc., 1997. ¥©€È‡Ö¶5‚5“ê9ÙA^ Ø6, !§!!Z˜ È, <¬ e>Ñ‡ , 2010. [2] Michael Artin, Algebra, (Second edition), Pearson, 2010. ISkÅ ó’Ñ‡ K<‡. [3] Sheldon Axler, Linear Algebra Done Right, (Third edition), Springer, 2015. ¥©€È‡Ö¶5‚ 5“êATù Æ6, Úy&!4Œý!ê¬ È, <¬e>Ñ‡ , 2016. [4] Sheldon Axler, Down with determinants!, American Mathematical Monthly, 102, 1995, no. 2, pp. 139–154. [5] Marcel Berger, Geometry I, Translated from the 1977 French original by M. Cole and S. Levy, Springer-Verlag, Berlin, 2009. 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McDonald, Linear Algebra and Its Applications (Fifth edition), Pearson Eduction, Inc., 2016. ISkÅ ó’Ñ‡ K<‡. [15] Steven J. Leon, Linear Algebra with Applications (Ninth edition), Pearson Eduction, Inc., 2015. I SkÅ ó’Ñ‡ K<‡. [16] Steven Roman, Advanced Linear Algebra (Third edition), Graduate Texts in Mathematics 135, Springer, 2008. 265 ë•©z 266 [17] Kenneth H. Rosen, Elementary Number Theory and Its Applications (Sixth edition), Pearson, 2011. ISkÅ ó’Ñ‡ K<‡. [18] Gilbert Strang, Linear Algebra and Its Applications (Fourth edition), Thomson Learning, Inc., 2006. [19] Terence Tao, Analysis I, (Third edition), Springer, 2016. ¥©€È‡Ö¶5>óZ¢©Û (1 3 ‡)6, où È, <¬e>Ñ‡ , 2018. [20] Vladimir A. Zorich, Mathematical Analysis I, Springer, 2004. [21] •ÏW, A^‚5“ê; ‰ÆÑ‡ [22] 7±¥, p “ê{² [23] iµŸ, x {, ±Iz, p [24] oÿ“, ‚5“ê, p , 2011. §, þ!eþ (1 ‡); ‚5“êÆ, p ˜Ñ‡ ®ŒÆÑ‡ ˜Ñ‡ , 2014. , 2011. [25] oÿ“, ‚5“ê (êÆ;’^), p ˜Ñ‡ , 2006. [26] o©%; “êÆ•{ (ò˜: Ä:e ); p ˜Ñ‡ ; 2019. [27] m(ê!¶œš; “êÆÚØ (1 ‡); p ˜Ñ‡ , 2000. [28] £‘(, {²‚5“ê; ®ŒÆÑ‡ [29] £‘(, )ÛAÛ (1n‡); [30] ?2Z, [31] h, E !ÜåS! , 2002. ®ŒÆÑ‡ }•, ‚5“ê , 2015. AÛ¿Â; ÜS>f‰EŒÆÑ‡ )Ù, ŒÆ“ê†AÛ, ˜uŒÆÑ‡ “ê)K•{ (1 2 ‡), ˜uŒÆÑ‡ [33] | d [34] Üñ (ÜÇ) , 2005. , ‚5“ê†AÛ, þ!eþ (1 2 ‡), ˜uŒÆÑ‡ 6>, Q°9 È), ÊÙŽâ; ô€<¬Ñ‡ [35] Üp‰!N7u, p , 2015. , 2012. [32] N7u!Üp‰, p 1!°g+! , 2007. “êÆ (1 2 ‡), ˜uŒÆÑ‡ , 2004. , 2011. , 2014.

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