{"id":8657,"date":"2009-08-22T15:51:00","date_gmt":"2009-08-22T15:51:00","guid":{"rendered":"http:\/\/melotopia.net\/b\/?p=8657"},"modified":"2009-08-22T15:51:00","modified_gmt":"2009-08-22T15:51:00","slug":"%ec%bd%94%ec%8b%9c-%ec%8a%88%eb%b0%94%eb%a5%b4%ec%b8%a0-%eb%b6%80%eb%93%b1%ec%8b%9d","status":"publish","type":"post","link":"http:\/\/melotopia.net\/b\/?p=8657","title":{"rendered":"\ucf54\uc2dc-\uc288\ubc14\ub974\uce20 \ubd80\ub4f1\uc2dd"},"content":{"rendered":"
\n Cauchy-Schwarz \ubd80\ub4f1\uc2dd(\uc774\ud558, \ucf54\uc2dc \ubd80\ub4f1\uc2dd)\uc740, \uc138\uacc4\uc5d0\uc11c \uac00\uc7a5 \uc720\uba85\ud55c \ubd80\ub4f1\uc2dd \uc911\uc758 \ud558\ub098\ub77c\uace0 \ud574\ub3c4 \ub420 \uac83\uc774\ub2e4.
\n
\n (\ubb3c\ub860 \uac1c\uc778\uc801\uc73c\ub85c\ub294 \ubd88\ud655\uc815\uc131 \uc6d0\ub9ac\uc5d0 \ub098\uc624\ub294 \ubd80\ub4f1\uc2dd\uc744 \ub354 \uc88b\uc544\ud558\uc9c0\ub9cc, \uadf8 \ubd80\ub4f1\uc2dd\ub3c4 \ucf54\uc2dc \ubd80\ub4f1\uc2dd\uc5d0\uc11c \uc720\ub3c4\ub41c\ub2e4\uace0 \ud560 \uc218 \uc788\uc73c\ub2c8 \uc5b4\uca0c\ub4e0 \uc720\uba85\ud558\ub2e4\ub294 \uac74 \uc0ac\uc2e4\uc774\ub77c\uace0 \ud574\uc57c\uaca0\ub2e4. \uc5b4\uca0c\uac70\ub098 \uc720\uba85\ud558\uac74 \ub9d0\uac74 \ubcc4 \ubb38\uc81c\ub294 \uc5c6\ub2e4.)<\/p>\n

\uc774\ubc88 \uc2dc\uac04\uc5d4 \uc774\uac78 \uc99d\uba85\ud574 \ubcf4\uc790. \uc74c…\uad73\uc774 \uc99d\uba85\ud558\ub294 \uc774\uc720\ub294, \ub0b4\uac00 \uc2a4\ud2b8\ub808\uc2a4\uac00 \uc313\uc5ec\uc11c \uc880 \uc26c\uc6b4 \uc218\ud559\uc744 \ud574\ubcf4\uace0 \uc2f6\uae30 \ub54c\ubb38\uc774\ub2e4. \uc694\uc998 “\uc591\uc790\uc5ed\ud559\uc758 \uc218\ud559\uc801 \uac1c\ub150(Mathematical Concepts of Quantum Mechanics)”\ub77c\ub294 \ucc45\uc744 \uc77d\uace0 \uc788\ub294\ub370 \uc218\ub9ce\uc740 \uc99d\uba85\uc774 \ucf54\uc2dc \ubd80\ub4f1\uc2dd\uc5d0\uc11c \uc720\ub3c4\ub418\uace0 \uc788\ub2e4. \uace8\uce58\uc544\ud508 \uc218\ud559\uc740 \uc880 \uc9d1\uc5b4\ub358\uc9c0\uace0, \uc26c\uc6b4 2\ucc28 \ubc29\uc815\uc2dd\ubd80\ud130 \uc2dc\uc791\ud574 \ubcf4\uc790.<\/p>\n

\uace0\ub4f1\ud559\uad50\ub54c\uc778\uac00 \uc911\ud559\uad50\ub54c\uc778\uac00, 2\ucc28\ubc29\uc815\uc2dd\uc744 \ucc98\uc74c \ubc30\uc6b0\uace0 \ub098\uba74 \ub2e4\uc74c\uacfc \uac19\uc740 \uc5f0\uc2b5\ubb38\uc81c\ub97c \uc790\uc8fc \ubcf4\uac8c \ub41c\ub2e4.<\/p>\n

\n “\ub2e4\uc74c\uacfc \uac19\uc740 \uc774\ucc28\ubc29\uc815\uc2dd\uc774 \uc788\ub2e4.
\n
\n $ax^2 + bx + c = 0$
\n
\n \uc774 \uc774\ucc28\ubc29\uc815\uc2dd\uc774 \uc2e4\uc218\uc778 \ud574\ub97c \uac16\uc9c0 \uc54a\uae30 \uc704\ud55c $a, b, c$\uc758 \uc870\uac74\uc740?”\n <\/div>\n

\n \ubb50, \uc774\uac74 \uadfc\uc758 \uacf5\uc2dd\uc5d0\uc11c \uc81c\uacf1\uadfc \uae30\ud638 \uc548\uc5d0 \ub4e4\uc5b4\uac00 \uc788\ub294 \ud310\ubcc4\uc2dd\uc774 \uc74c\uc218\uac00 \ub418\uba74 \ub41c\ub2e4. \uace0\ub4f1\ud559\uad50\ub54c \uacf5\ubd80\ub97c \uc5f4\uc2ec\ud788 \ud574\ubcf8 \uc0ac\ub78c\uc774\ub77c\uba74 \uc54c \uac83\uc774\uace0, \ubaa8\ub974\uaca0\uc73c\uba74 \uc9c0\ub098\uac00\ub294 \uace0\ub4f1\ud559\uc0dd\uc744 \ubd99\ub4e4\uace0 (\uc774\uacfc\uc0dd\uc774 \uc88b\uaca0\uc9c0\ub9cc \ubb38\uacfc\uc0dd\ub3c4 \uc54c \uac83 \uac19\ub2e4.) \ubb3c\uc5b4\ubcf4\uc790.
\n <\/p>\n

\n $b^2 – 4ac < 0$\n <\/div>\n

\n \uc704\uc758 \uc870\uac74\uc744 \ub9cc\uc871\ud558\ub294 $a, b, c$\ub77c\uba74 \uc5b4\uca0c\ub4e0 \uadf8\ub7f0 \uc774\ucc28\ubc29\uc815\uc2dd\uc740 \uc2e4\uc218\uc778 \ud574\ub97c \uac16\uc9c0 \uc54a\ub294\ub2e4. \uc99d\uba85\uc740 \uac01\uc790 \ud574\ubcf4\uc790. \ubc18\ub840\ub97c \ucc3e\uace0 \uc2f6\uc73c\uba74 \ucc3e\uc544\ubcf4\ub294 \uac83\ub3c4 \uc88b\uaca0\uc9c0\ub9cc, \uc2dc\uac04 \ub0ad\ube44\uc5d0 \ubd88\uacfc\ud558\ub2e4\ub294 \uac83\uc744 \uc790\uc2e0\uc788\uac8c \ub9d0\ud574\uc8fc\uace0 \uc2f6\ub2e4.<\/p>\n

\ucf54\uc2dc \ubd80\ub4f1\uc2dd\uc758 \uc99d\uba85\uc740 \uc704\uc758 \ubcf4\uc870\uc815\ub9ac\uac00 \uc911\uc694\ud558\uac8c \uc0ac\uc6a9\ub41c\ub2e4.<\/p>\n

\ucf54\uc2dc \ubd80\ub4f1\uc2dd\uc744 \uc99d\uba85\ud558\ub824\uba74, \uc77c\ub2e8 \uc6b0\ub9ac\uac00 \uac16\uace0 \ub180\uac8c \ub420 \ub300\uc0c1\uc744 \uc815\ud574\uc57c \ud55c\ub2e4. \uadf8\ub0e5, \uc27d\uac8c \ub9d0\ud574\uc11c \ubca1\ud130 \uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. \ub9d0\uc774 \uc26c\uc6cc\uc11c \ubca1\ud130 \uacf5\uac04\uc774\uc9c0, \ubca1\ud130 \uacf5\uac04\uc758 \uc608\ub294 \uc5c4\uccad\ub098\uac8c \ub9ce\ub2e4. \ub530\ub77c\uc11c \ucf54\uc2dc \ubd80\ub4f1\uc2dd\uc758 \uc801\uc6a9 \ubc94\uc704\ub3c4 \uc5c4\uccad\ub098\uac8c \ub113\ub2e4.<\/p>\n

\n \ubca1\ud130 \uacf5\uac04\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uc131\uc9c8\uc744 \uac16\uace0 \uc788\ub2e4. (\uc815\ud655\ud55c \uc815\uc758\ub294 \uc120\ud615\ub300\uc218\ud559 \ucc45\uc744 \ucc3e\uc544\ubcf4\ub3c4\ub85d \ud558\uc790.)
\n
\n 1. \ubca1\ud130 \uacf5\uac04 \uc548\uc5d0 \uc788\ub294 \uc6d0\uc18c\ub294 \ubca1\ud130\ub2e4.
\n
\n 2. \ubca1\ud130\ub97c \ub354\ud55c \uac83\ub3c4 \ubca1\ud130\ub2e4. \uc774\ub54c, \ubca1\ud130\ub4e4\ub07c\ub9ac\uc758 \ub367\uc148\uc740 \uad50\ud658\ubc95\uce59\uacfc \uacb0\ud569\ubc95\uce59\uc774 \uc801\uc6a9\ub41c\ub2e4.
\n
\n 3. \ubca1\ud130\uc758 \uae38\uc774\ub97c \ubc14\uafbc \uac83\ub3c4 \ubca1\ud130\ub2e4. \ubca1\ud130\uc758 \uae38\uc774\ub97c \ubc14\uafb8\uace0 \uc5ec\uae30\uc5d0 \ub2e4\ub978 \ubca1\ud130\ub97c \ub354\ud558\ub294 \uc5f0\uc0b0 \uc0ac\uc774\uc5d0\ub294 \ubd84\ubc30\ubc95\uce59\uc774 \uc801\uc6a9\ub41c\ub2e4.
\n \n<\/div>\n

\n \ubca1\ud130 \uacf5\uac04 \uc911\uc5d0\uc11c, \uc880 \ud2b9\ubcc4\ud55c \uacf5\uac04\uc774 \uc788\ub294\ub370, \ubca1\ud130\ub4e4 \uc0ac\uc774\uc758 \uac01\ub3c4\uc640 \uae38\uc774\ub97c \uc7b4 \uc218 \uc788\ub294 \uacf5\uac04\uc774 \uc788\ub2e4. \uc774\ub7f0 \uacf5\uac04\uc744 \ud2b9\ubcc4\ud788 \ub0b4\uc801 \uacf5\uac04(Inner product space)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub2f9\uc2e0\uc758 \ub0b4\uba74\uc744 \ub9d0\ud558\ub294 \uac83\uc774 \uc544\ub2d8\uc5d0 \uc720\uc758\ud558\uc790.<\/p>\n

\ub0b4\uc801\uc774\ub77c\ub294 \uac83\uc740 \ub450\uac1c\uc758 \ubca1\ud130\ub97c \uc815\ud574\uc92c\uc744 \ub54c \uc801\ub2f9\ud55c \uc218\ub97c \uc54c\ub824\uc8fc\ub294 \ud568\uc218\ub2e4.
\n
\n \ubca1\ud130\ub4e4 \uc0ac\uc774\uc758 \ub0b4\uc801\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \ud45c\uc2dc\ud558\uc790. a\uc640 b\ub97c \ub450 \ubca1\ud130\ub77c\uace0 \ud55c\ub2e4\uba74 \uadf8 \uc0ac\uc774\uc758 \ub0b4\uc801\uc740<\/p>\n

\n \ub0b4\uc801(a, b)
\n \n<\/div>\n

\n \ucc98\ub7fc \ud45c\uc2dc\ud574 \ubcfc \uc218 \uc788\uaca0\ub2e4. \uadf8\ub7fc, \uc774 \ub0b4\uc801\uc774\ub77c\ub294 \ud568\uc218\ub294 \ubca1\ud130 \ub450\uac1c\ub97c \ub123\uc5b4\uc8fc\uba74 \uc801\ub2f9\ud55c \uc218 \ud558\ub098\ub97c \uc54c\ub824\uc900\ub2e4. \uadf8 \uc218\ub294 \uc77c\ubc18\uc801\uc73c\ub85c \ubcf5\uc18c\uc218\ub2e4. \ubcf5\uc7a1\ud55c \uc598\uae30\ub294 \ub410\uace0, \ub0b4\uc801 \ud568\uc218\uc5d0\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \ud2b9\uc9d5\uc774 \uc788\ub2e4.
\n <\/p>\n

\n \ub0b4\uc801(a,b) = \ub0b4\uc801(b,a)
\n
\n \ub0b4\uc801(a+b,c) = \ub0b4\uc801(a,c) + \ub0b4\uc801(b,c)
\n
\n \ub0b4\uc801(ka,b) = k\ub0b4\uc801(a,b)
\n
\n \ub9cc\uc57d a\uac00 0\ubca1\ud130\uac00 \uc544\ub2c8\uba74, \uadf8\ub9ac\uace0 \uadf8\ub7f0 \uacbd\uc6b0\uc5d0\ub294 \uc5b8\uc81c\ub098 \ub0b4\uc801(a,a) > 0\uc774 \ub41c\ub2e4.
\n \n<\/div>\n

\n \ub0b4\uc801 \ud568\uc218\ub97c \uc774\uc6a9\ud558\uba74 \ubca1\ud130\uc758 \uae38\uc774\uc640 \ub450 \ubca1\ud130 \uc0ac\uc774\uc758 \uac01\ub3c4\ub97c \uc54c \uc218 \uc788\ub2e4.<\/p>\n

\n \ubca1\ud130a\uc758 \uae38\uc774\uc758 \uc81c\uacf1 = \ub0b4\uc801(a, a)\n <\/div>\n

\ubb50, \uc5ec\uae30\uc120 \uadf8\ub2e4\uc9c0 \uc758\ubbf8\ub294 \uc5c6\uc9c0\ub9cc \uac01\ub3c4\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \uc54c \uc218 \uc788\ub2e4.<\/p>\n

\n \uac01\ub3c4(a,b) = \ub0b4\uc801(a,b)\/(\uae38\uc774(a)*\uae38\uc774(b))\n <\/div>\n

\uc5b4\uca0c\ub4e0. \uc5b4\ub5a4 \ub450 \ubca1\ud130 a, b\uc5d0 \ub300\ud574\uc11c, \ub2e4\uc74c\uacfc \uac19\uc740 \ubca1\ud130\ub97c \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4.
\n <\/p>\n

\n c = a – tb\n <\/div>\n

\uc5ec\uae30\uc11c t\ub294 \uadf8\ub0e5 \uc5b4\ub5a4 \uc218\uc774\uace0, b\uc758 \uae38\uc774\ub97c t\ub9cc\ud07c \ubc14\uafe8\ub2e4\ub294 \ub73b\uc774\ub2e4. (t\uac00 \ubcf5\uc18c\uc218\uac00 \ub420 \uc218\ub3c4 \uc788\uc9c0\ub9cc, \uadf8\ub0e5 \uadf8\ub7ec\ub824\ub2c8 \ud558\uc790.)<\/p>\n

c\uc758 \uae38\uc774\ub97c \uc7ac\ubcf4\uba74
\n <\/p>\n

\n \uae38\uc774\uc758 \uc81c\uacf1(c) = \ub0b4\uc801(c, c) = \ub0b4\uc801(a-tb, a-tb) = \ub0b4\uc801(b,b)*t*t – 2t(\ub0b4\uc801(a,b)+\ub0b4\uc801(b,a))+\ub0b4\uc801(a,a)\n <\/div>\n

\ubcf5\uc7a1\ud574 \ubcf4\uc774\uc9c0\ub9cc \uc0ac\uc2e4\uc740 \uadf8\ub0e5 t\uc5d0 \uad00\ud55c 2\ucc28\ubc29\uc815\uc2dd\uc774\ub2e4. \uc5ec\uae30\uc11c, \uae38\uc774\uc758 \uc81c\uacf1\uc740 c\uac00 0\uc774 \uc544\ub2cc \ud55c \uc5b8\uc81c\ub098 \uc591\uc758 \uc2e4\uc218\uac00 \ub418\uae30 \ub54c\ubb38\uc5d0, t\ub97c \uc5b4\ub5bb\uac8c \uace0\ub974\ub354\ub77c\ub3c4 \uc591\uc218\uac00 \ub098\uc640\uc57c\ub9cc \ud558\uace0, \ub530\ub77c\uc11c \uc774 2\ucc28\ubc29\uc815\uc2dd\uc758 \uadfc\uc774 \uc874\uc7ac\ud558\uba74 \uc548\ub41c\ub2e4.
\n
\n \uc774\uc81c \ucf54\uc2dc \ubd80\ub4f1\uc2dd\uc774 \uc99d\uba85\ub41c\ub2e4.<\/p>\n

\n\"\"\n<\/div>\n

\uba4b\uc788\uac8c \ubcf4\uc774\ub824\uace0 \uc704\ud0a4 \ubc31\uacfc\uc758 \uc218\uc2dd\uc744 \uae01\uc5b4\uc654\ub2e4.<\/p>\n

\uc5b4\uca0c\ub4e0 \uc774 \ubd80\ub4f1\uc2dd\uc740 \uc99d\uba85\uc740 \ubcf4\ub2e4\uc2dc\ud53c \uc758\uc678\ub85c \uc26c\uc6b4\ub370, \uadf8 \ud65c\uc6a9\uc740 \ub108\ubb34\ub098 \uad11\ubc94\uc704\ud558\ub2e4. \uc774\ub8e8 \ub9d0\ud560 \uc218 \uc5c6\uc774 \uc911\uc694\ud55c \ubd80\ub4f1\uc2dd\uc774\ub2e4. \uc624\ub298 \ub0b4\uac00 \uc591\uc790\uc5ed\ud559\uc758 \uc218\ud559\uc801 \uac1c\ub150 \ucc45\uc5d0\uc11c \ubcf8 \uac83\ub9cc \ud574\ub3c4 \uc138\uad70\ub370\uc778\uac00 \uc788\ub294 \uac83 \uac19\ub2e4.<\/p>\n

\uc704\uc758 \ub0b4\uc6a9\uc740 \uc77c\ubc18\uc801\uc73c\ub85c \uc0ac\uc6a9\ud558\ub294 \uc99d\uba85\uc774\ub2e4. \uac80\uc0c9\ud574 \ubcf4\ub2c8\uae4c \uc880 \ub354 \uc6b0\uc544\ud55c \uc99d\uba85\ub3c4 \uc788\ub2e4.
\n
\n<\/p>\n

CauchySchwarzInequalityProof.pdf\uc5d0 \uc561\uc138\uc2a4\ud558\ub824\uba74 \ud074\ub9ad\ud558\uc138\uc694.<\/a><\/p>\n

<\/a><\/p>\n

\uc608\ub97c \ub4e4\uc5b4\ubcf4\uc790.<\/p>\n

a=(1,4)\uc640 b=(4,2)\uac00 \uc788\ub2e4.<\/p>\n

\ub0b4\uc801(a,b) = a1*b1+a2*b2 \ub77c\uace0 \ud574 \ubcf4\uc790.
\n
\n \ub0b4\uc801(a,b) = 1*4+4*2 = 12
\n
\n \ub0b4\uc801(a,a) = 1*1+4*4 = 17
\n
\n \ub0b4\uc801(b,b) = 4*4+2*2 = 20
\n
\n \ub0b4\uc801(a,b)\uc758 \uc81c\uacf1 = 144
\n
\n \ub0b4\uc801(a,a)*\ub0b4\uc801(b,b)=340<\/p>\n

\ub530\ub77c\uc11c, 144<340\n
\n \ubd80\ub4f1\uc2dd\uc740 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

\ub0b4\uc801 \ud568\uc218\uac00 \ub108\ubb34 \ub9ce\uc774 \uc4f0\ub294 \uac70\ub77c\uc11c \uc6d0\ub798 \uadf8\ub7f0\uac70\ub77c\uace0 \uc0dd\uac01\ud55c\ub2e4\uba74, \ub2e4\uc74c\uacfc \uac19\uc774 \ub0b4\uc801\uc744 \uc7a1\uc544\ubcf4\uc790.
\n
\n \ub0b4\uc801(a,b) = 5*a1*b1 + 2*a2*b2 + 3*a1*b2 + 3*a2*b1<\/p>\n

\uc5ec\uae30\uc5d0 \uc704\uc758 \ubca1\ud130\ub97c \ub123\uc790. (\uc704\uc758 \ub0b4\uc801 \ud568\uc218\uac00 \uc55e\uc5d0\uc11c \ub0b4\uac00 \ub9d0\ud55c \uc870\uac74\uc744 \ub9cc\uc871\ud55c\ub2e4\ub294 \uac74 \uc27d\uac8c \ud655\uc778\ud560 \uc218 \uc788\ub2e4.)<\/p>\n

\ub0b4\uc801(a,b) = 5*1*4 + 2*4*2 + 3*1*2 + 3*4*4 = 20+16+6+48 = 90
\n
\n \ub0b4\uc801(a,a) = 5*1*1 + 2*4*4 + 3*1*4 + 3*4*1 = 5+32+12+12 =61
\n
\n \ub0b4\uc801(b,b) = 5*4*4 + 2*2*2 + 3*4*2 + 3*2*4 = 80+8+24+24 = 136<\/p>\n

\ub0b4\uc801(a,b)\uc758 \uc81c\uacf1 = 8100
\n
\n \ub0b4\uc801(a,a)*\ub0b4\uc801(b,b) = 8296<\/p>\n

\uc5b4\uca0c\ub4e0 8100<8296 \uc774\ub2e4.\n \n\n\n \ubb50, \uc544\ubb34 \ubca1\ud130\ub77c\ub3c4 \uc88b\uace0, \ub0b4\uc801\uc744 \uc704\uc640 \uac19\uc774 \uc7a1\uc9c0 \uc54a\uace0 \ub2e4\ub974\uac8c \uc7a1\uc544\ub3c4 \uc88b\ub2e4. (\uc55e\uc5d0\uc11c \ub0b4\uac00 \ub9d0\ud55c \ub0b4\uc801\uc774\ub77c\ub294 \ud568\uc218\uc758 \uc870\uac74\uc744 \ub9cc\uc871\ud558\uae30\ub9cc \ud558\uba74 \ub41c\ub2e4.) \uc5b4\uca0c\ub4e0 \uc131\ub9bd\ud55c\ub2e4. \u314b\u314b\u314b\n \n\n\n\n\n

\n
\n
\n\"\uc2e0\uace0\"
\n<\/a>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

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