{"id":6516,"date":"2021-06-02T15:22:29","date_gmt":"2021-06-02T06:22:29","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=6516"},"modified":"2021-06-02T22:47:59","modified_gmt":"2021-06-02T13:47:59","slug":"linear-algebra-determinats-volume","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/linear-algebra-determinats-volume\/","title":{"rendered":"\ud589\ub82c\uc2dd\uacfc \uccb4\uc801"},"content":{"rendered":"
\n

\uc774 \uae00\uc740 \ud604\uc7ac\ub294 \uc808\ud310\ub41c Serge Lang\uc758 \uc120\ud615\ub300\uc218 \uad50\uc7ac \uc81c2\ud310([1], VII, \u00a76)\uc758 \ub0b4\uc6a9\uc744 \ud1a0\ub300\ub85c \uc4f4 \uac83\uc774\ub2e4. \ud37c\uac00\uc9c0 \ub9c8\uc2dc\ub77c.\n<\/p>\n<\/div>\n

\n\uc774 \uae00\uc5d0\uc11c\ub294 \ud589\ub82c\uc2dd\uc744 \ud55c \ub3c4\ud615\uc758 \uccb4\uc801\uc73c\ub85c \uc774\ud574\ud558\ub294 \uc774\uc57c\uae30\ub97c \uc18c\uac1c\ud55c\ub2e4. \uba3c\uc800 2-\ucc28\uc6d0\uc758 \uacbd\uc6b0\ub97c \ub17c\ud560\ud150\ub370, ‘\uccb4\uc801(volume)’\uc774\ub77c\ub294 \uc6a9\uc5b4\ub97c 2-\ucc28\uc6d0 \ub3c4\ud615\uc758 \ub113\uc774\ub97c \uc77c\uceec\uc744 \ub54c\uc5d0\ub3c4 \uadf8\ub0e5 \uc0ac\uc6a9\ud558\uace0\uc790 \ud55c\ub2e4. \ub610\ud55c ‘\\(\\operatorname{Vol}\\)’\uc640 \uac19\uc740 \uae30\ud638\ub97c \uc774\uc6a9\ud558\uc5ec \ub113\uc774\ub97c \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud560 \uac83\uc774\ub2e4. \ubb3c\ub860 \uc774 \uae30\ud638\ub97c \uc77c\ubc18\uc801\uc778 \uace0\ucc28\uc6d0 \ub3c4\ud615\uc758 \uccb4\uc801\uc744 \ub098\ud0c0\ub0b4\ub294 \uae30\ud638\ub85c\ub3c4 \uc4f8 \uac83\uc774\ub2e4.\n<\/p>\n

\n\uba3c\uc800 \ub450 \ubca1\ud130 \\(v, w\\)\ub85c \uc0dd\uc131\ud55c \ud3c9\ud589\uc0ac\ubcc0\ud615\uc744 \uc0dd\uac01\ud574\ubcf4\uc790. \ubb3c\ub860 \uc774 \ud3c9\ud589\uc0ac\ubcc0\ud615\uc740
\n\\[
\nt_{1}v+t_{2}w\\quad (0\\leq t_{i}\\leq 1)
\n\\]
\n\uaf34\uc758 \uc77c\ucc28\uacb0\ud569\uc73c\ub85c \ud45c\ud604\ub418\ub294 \ubaa8\ub4e0 \ubca1\ud130\ub97c \ubaa8\uc544\ub193\uc740 \uc9d1\ud569\uc744 \uc77c\uceeb\ub294\ub2e4.\n<\/p>\n

\"\" <\/p>\n

\n\ub450 \ubca1\ud130 \\(v, w\\)\ub97c \uc5f4\ubca1\ud130\ub85c \uc0dd\uac01\ud558\uc5ec \ud589\ub82c\uc2dd \\(\\det(v, w)\\)\ub97c \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4. \uc774 \ud589\ub82c\uc2dd\uc740 \ub450 \ubca1\ud130\uac00 \ub098\ub780\ud558\uc9c0\ub9cc \uc54a\ub2e4\uba74, \uc591\uc218 \ud639\uc740 \uc74c\uc218\uc758 \uac12\uc744 \uac00\uc9c8 \uac83\uc774\uba70 \ub2e4\uc74c \ub4f1\uc2dd\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.
\n\\[
\n\\det(v, w)=-\\det(w, v)
\n\\]
\n\ub530\ub77c\uc11c \ud589\ub82c\uc2dd \uadf8 \uc790\uccb4\ub97c \uc9c0\uae08 \uc0dd\uac01\ud558\ub294 \ud3c9\ud589\uc0ac\ubcc0\ud615\uc758 \ub113\uc774\ub77c\uace0 \ubc14\ub85c \ub9d0\ud560 \uc218\ub294 \uc5c6\ub2e4. \uc65c\ub0d0\ud558\uba74 \uc6b0\ub9ac\ub294 \ub113\uc774 \ud639\uc740 \ubd80\ud53c\ub85c \uc0dd\uac01\ud558\ub294 \uac12\uc744 \ud56d\uc0c1 \uc74c\uc774 \uc544\ub2cc \uc2e4\uc218\uac12\uc73c\ub85c \uc0dd\uac01\ud558\uae30 \ub54c\ubb38\uc774\ub2e4. \uadf8\ub7ec\ub098 \ub2e4\uc74c\uc744 \ubcf4\uc77c \uc218 \uc788\ub2e4.\n<\/p>\n

\n

\n\t\t Theorem.1. <\/span>
\n\ub450 \ubca1\ud130 \\(v, w\\)\ub85c \uc0dd\uc131\ud55c \ud3c9\ud589\uc0ac\ubcc0\ud615\uc758 \ub113\uc774\ub294 \uadf8 \ub450 \ubca1\ud130\ub85c \ub9cc\ub4e0 \ud589\ub82c\uc2dd\uc758 \uc808\ub313\uac12\uc778 \\(\\vert\\det(v, w)\\,\\vert\\)\uc640 \uac19\ub2e4.\n<\/p>\n<\/div>\n

\n\uc815\ub9ac 1\uc744 \ubcf4\uc774\uae30 \uc704\ud574 \ubc29\ud5a5\uc774 \uc788\ub294 \ub113\uc774(oriented area)\ub77c\ub294 \uac1c\ub150\uc744 \uc18c\uac1c\ud55c\ub2e4. \ub450 \ubca1\ud130 \\(v\\)\uc640 \\(w\\)\ub85c \uc0dd\uc131\ud55c \ud3c9\ud589\uc0ac\ubcc0\ud615\uc744 \\(P(v, w)\\)\ub85c \ub098\ud0c0\ub0b4\uc790. \uadf8\ub9ac\uace0 \uc2e4\uc22b\uac12 \\(\\operatorname{Vol}_{\\mathrm{O}}(v, w)\\)\ub97c \uc815\uc758\ud558\ub418, \\(\\det(v,w)\\geq 0\\)\uc77c\ub54c\ub294 \uadf8 \uac12\uc744 \\(P(v, w)\\)\uc758 \ub113\uc774\ub85c, \\(\\det(v, w)<0\\)\uc77c\ub54c\ub294 \uadf8 \uac12\uc744 \\(P(v, w)\\)\uc758 \ub113\uc774\uc5d0 \\(-1\\)\uc744 \uacf1\ud55c \uac12\uc73c\ub85c \uc815\uc758\ud558\uc790. \uadf8\ub7ec\uba74 \\(\\operatorname{Vol}_{\\mathrm{O}}(v, w)\\)\uac00 \ud589\ub82c\uc2dd \\(\\det(v, w)\\)\uc640 \ub3d9\uc77c\ud55c \ubd80\ud638\ub97c \uac00\uc9d0\uc744 \ubc14\ub85c \uc54c \uc218 \uc788\ub2e4. \uc2e4\uc218 \\(\\operatorname{Vol}_{\\mathrm{O}}(v, w)\\)\ub97c \ubc29\ud5a5\uc774 \uc788\ub294 \ub113\uc774<\/span>\ub77c\uace0 \ubd80\ub974\uc790. \ub610\ud55c \\(v, w\\)\ub85c \uc0dd\uc131\ub41c \ud3c9\ud589\uc0ac\ubcc0\ud615\uc758 \ub113\uc774\ub294 \\(\\operatorname{Vol}(v, w)\\)\ub85c \ub098\ud0c0\ub0b4\uc790. \ub530\ub77c\uc11c \\(\\operatorname{Vol}_{\\mathrm{O}}(v, w)=\\operatorname{Vol}(v, w)\\) \ud639\uc740 \\(\\operatorname{Vol}_{\\mathrm{O}}(v, w)=-\\operatorname{Vol}(v, w)\\)\uc774\ub2e4.\n<\/p>\n

\uc815\ub9ac 1\uc744 \uc99d\uba85\ud558\uae30 \uc704\ud574\uc11c\ub294 \ub2e4\uc74c\uc744 \ubcf4\uc774\uba74 \ucda9\ubd84\ud558\ub2e4.<\/p>\n

\n

\n\t\t Proposition. <\/span>
\n\ubc29\ud5a5\uc774 \uc788\ub294 \ub113\uc774\ub294 \ud589\ub82c\uc2dd\uc758 \uac12\uacfc \uc77c\uce58\ud55c\ub2e4. \uc989
\n\\[
\n\\operatorname{Vol}_{\\mathrm{O}}(v, w)=\\det(v, w)
\n\\]
\n\uac00 \uc131\ub9bd\ud55c\ub2e4.\n<\/p>\n<\/div>\n

\n\uc774\uc81c \uc704\uc758 \uba85\uc81c\ub97c \uc99d\uba85\ud558\uae30 \uc704\ud574 \\(\\operatorname{Vol}_{\\mathrm{O}}\\)\uc774 \ud589\ub82c\uc2dd\uc744 \uaddc\uc815\ud558\ub294 \uc138 \uac00\uc9c0 \uc131\uc9c8\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4\ub294 \uac83\uc744 \ubcf4\uc774\uc790. \uc989\n<\/p>\n

    \n
  1. \ubcc0\uc218 \\(v\\)\uc640 \\(w\\) \uac01\uac01\uc5d0 \ub300\ud558\uc5ec \\(\\operatorname{Vol}_{\\mathrm{O}}\\)\uc740 \uc120\ud615\uc774\ub2e4.<\/li>\n
  2. \uc784\uc758\uc758 \\(v\\)\uc5d0 \ub300\ud558\uc5ec \\(\\operatorname{Vol}_{\\mathrm{O}}(v, v)=0\\)\uc774\ub2e4.<\/li>\n
  3. \\(e_{1}, e_{2}\\)\uac00 \ud45c\uc900\uae30\uc800\uc77c \ub54c, \\(\\operatorname{Vol}_{\\mathrm{O}}(e_{1}, e_{2})=1\\)\uc774\ub2e4.<\/li>\n<\/ol>\n

    \n\uc784\uc744 \ubcf4\uc774\uc790. \uc120\ud615\ub300\uc218\ub97c \uacf5\ubd80\ud588\ub2e4\uba74, \uc704\uc758 \uc138 \uac00\uc9c0 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \\(\\mathbb{R}^{2}\\times\\mathbb{R}^{2}\\)\uc5d0\uc11c \\(\\mathbb{R}\\)\ub85c\uc758 \ud568\uc218\uac00 \uc720\uc77c\ud558\uac8c \uc874\uc7ac\ud558\uba70, \uadf8 \ud568\uc218\ub97c \\(\\det\\)\ub85c \uc815\uc758\ud568\uc744 \uc54c\uace0 \uc788\uc744 \uac83\uc774\ub2e4. \uc774 \uae00\uc744 \uc77d\ub294 \ub3c5\uc790\ub97c \uc704\ud574 \uac04\ub7b5\ud788 \uc774 \ub0b4\uc6a9\uc744 \uac04\ub7b5\ud788 \uc18c\uac1c\ud574\ubcf8\ub2e4. \uc704\uc758 \uc138 \uac00\uc9c0 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ud568\uc218 \\(G\\)\uac00 \uc788\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc784\uc758\uc758 \ub450 \ubca1\ud130
    \n\\[
    \nv=ae_{1}+ce_{2},\\quad w=be_{1}+de_{2}
    \n\\]
    \n\uc5d0 \ub300\ud558\uc5ec,
    \n\\[
    \nG(ae_{1}+ce_{2}, be_{1}+de_{2}) = abG(e_{1}, e_{1})+ adG(e_{1}, e_{2})+cbG(e_{2}, e_{1}) + cdG(e_{2}, e_{2})
    \n\\]
    \n\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc5ec\uae30\uc11c \uc6b0\ubcc0\uc758 \uccab \ud56d\uacfc \ub9c8\uc9c0\ub9c9 \ud56d\uc740 \\(0\\)\uc784\uc744 \ubc14\ub85c \uc54c \uc218 \uc788\ub2e4. \ub610\ud55c \uac04\ub2e8\ud55c \uad00\ucc30\uc744 \ud1b5\ud574
    \n\\[
    \nG(e_{2}, e_{1})=-G(e_{1}, e_{2})
    \n\\]
    \n\uc784\uc744 \ud655\uc778\ud560 \uc218 \uc788\uace0 \ub530\ub77c\uc11c
    \n\\[
    \nG(v, w)=(ad-bc)G(e_{1}, e_{2})=ad-bc
    \n\\]
    \n\ub97c \uc5bb\ub294\ub2e4. \uc989 \\(\\mathbb{R}^{2}\\times\\mathbb{R}^{2}\\)\uc5d0\uc11c \\(\\mathbb{R}\\)\ub85c\uc758 \ud568\uc218 \\(G\\)\ub294 \\(\\det\\)\uc640 \uac19\ub2e4.\n<\/p>\n

    \n\uc774\uc81c \\(\\operatorname{Vol}_{\\mathrm{O}}\\)\uc774 \uc704\uc758 \uc138 \uac00\uc9c0 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4\ub294 \uac83\uc744 \ubcf4\uc774\uae30 \uc704\ud574 \ub113\uc774(\ud639\uc740 \ubd80\ud53c)\uac00 \uac16\ub294 \ub2e4\uc74c\uc758 \uac04\ub2e8\ud55c \uc131\uc9c8 \uba87 \uac00\uc9c0\ub97c \ud65c\uc6a9\ud558\ub824 \ud55c\ub2e4.\n<\/p>\n

      \n
    • \uc120\ubd84\uc758 \ub113\uc774\ub294 \\(0\\)\uacfc \uac19\ub2e4. <\/li>\n
    • \ub9cc\uc77c \\(A\\)\uac00 \uc5b4\ub5a4 \uc601\uc5ed\uc77c \ub54c, \\(A\\)\ub97c \ud3c9\ud589\uc774\ub3d9\ud558\uc5ec \uc5bb\uc740 \uc601\uc5ed \\(A_{w}=\\{ v+w\\mid v\\in A\\}\\)\uc758 \ub113\uc774\ub294 \\(A\\)\uc758 \ub113\uc774\uc640 \uac19\ub2e4. <\/li>\n
    • \ub450 \uc601\uc5ed \\(A, B\\)\uac00 \uc11c\ub85c \ub9cc\ub098\uc9c0 \uc54a\uac70\ub098 \ud639\uc740 \ub9cc\ub098\ub354\ub77c\ub3c4 \\(A\\cap B\\)\uc758 \ub113\uc774\uac00 \\(0\\)\uc774\uba74<\/li>\n

      \\[
      \n\\operatorname{Vol}(A\\cup B)=\\operatorname{Vol}(A)+\\operatorname{Vol}(B)
      \n\\]
      \n\uc774\ub2e4.\n<\/ul>\n

      \n\uc774\uc81c \\(\\operatorname{Vol}_{\\mathrm{O}}\\)\uc744 \uc0dd\uac01\ud574\ubcf4\uc790. \ud589\ub82c\uc2dd\uc744 \uaddc\uc815\ud558\ub294 \uc138 \uac00\uc9c0 \uc131\uc9c8 \uc911 \ub450 \ubc88\uc9f8, \uc138 \ubc88\uc9f8 \uc131\uc9c8\uc744 \\(\\operatorname{Vol}_{\\mathrm{O}}\\)\uc774 \ub9cc\uc871\uc2dc\ud0a8\ub2e4\ub294 \uac83\uc740 \uc790\uba85\ud558\ub2e4. \uc989 \\(v, v\\)\ub85c \uc0dd\uc131\ub41c \ud3c9\ud589\uc0ac\ubcc0\ud615\uc740 \ud558\ub098\uc758 \uc120\ubd84\uc774\ubbc0\ub85c \uc774\uc758 \\(2\\)-\ucc28\uc6d0 \ub113\uc774\ub294 \\(0\\)\uacfc \uac19\uace0 \ub530\ub77c\uc11c \\(\\operatorname{Vol}_{\\mathrm{O}}(v, v)=0\\)\uc774\ub2e4. \uc138 \ubc88\uc9f8 \uc131\uc9c8\uc744 \ud655\uc778\ud574\ubcf4\uc790. \ud45c\uc900\uae30\uc800 \\(e_{1}, e_{2}\\)\ub85c \uc0dd\uc131\ud55c \ud3c9\ud589\uc0ac\ubcc0\ud615\uc740 \ub2e8\uc704\uc815\uc0ac\uac01\ud615\uc73c\ub85c \ub113\uc774 \\(1\\)\uc744 \uac16\ub294\ub2e4. \ub530\ub77c\uc11c
      \n\\[
      \n\\operatorname{Vol}_{\\mathrm{O}}(e_{1}, e_{2})=1
      \n\\]
      \n\uc784\uc744 \uc5bb\ub294\ub2e4. \\(\\operatorname{Vol}_{\\mathrm{O}}\\)\uc774 \ud589\ub82c\uc2dd\uc744 \uaddc\uc815\ud558\ub294 \uccab \ubc88\uc9f8 \uc131\uc9c8\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4\ub294 \uac83\uc740 \ub2e4\ub978 \ub450 \uc131\uc9c8\uc744 \ud655\uc778\ud558\ub294 \uac83\uc5d0 \ube44\ud558\uc5ec \ud488\uc774 \ub354 \ub9ce\uc774 \ub4e0\ub2e4. \uc720\ud074\ub9ac\ub4dc \uacf5\uac04\uc758 \ubca1\ud130\ub97c \ub2e4\ub8e8\ub294\ub370 \uc870\uae08\uc740 \uc775\uc219\ud55c \uac83\uc774 \ub2e4\uc74c \ub0b4\uc6a9\uc744 \uc77d\ub294\ub370 \ub3c4\uc6c0\uc774 \ub420 \uac83\uc774\ub2e4.\n<\/p>\n

      \n\uba3c\uc800 \ub2e4\uc74c\uc758 \ubcf4\uc870\uc815\ub9ac\ub97c \ubcf4\uc774\uc790.\n<\/p>\n

      \n

      \n\t\t Lemma.2. <\/span>
      \n\ub9cc\uc77c \\(v, w\\)\uac00 \uc77c\ucc28\uc885\uc18d\uc774\uba74 \\(\\operatorname{Vol}_{\\mathrm{O}}(v, w)=0\\)\uc774\ub2e4.\n<\/p>\n<\/div>\n

      \n\tProof.<\/span>
      \n\uac00\uc815\uc5d0 \uc758\ud574 \\(a\\neq 0\\) \ud639\uc740 \\(b\\neq 0\\)\uc778 \ub450 \uc2a4\uce7c\ub77c \\(a, b\\)\ub97c \uc774\uc6a9\ud558\uc5ec
      \n\\[
      \nav+bw=0
      \n\\]
      \n\ub85c \uc4f8 \uc218 \uc788\ub2e4. \uc77c\ubc18\uc131\uc744 \uc783\uc9c0 \uc54a\uace0 \\(a\\neq 0\\)\uc774\ub77c \ud558\uba74
      \n\\[
      \nv=-\\frac{b}{a}w=c w
      \n\\]
      \n\uaf34\ub85c \uc4f8 \uc218 \uc788\uc73c\uba70 \uc774\ub294 \\(v\\)\uc640 \\(w\\)\uac00 \ub3d9\uc77c\ud55c \uc9c1\uc120 \uc0c1\uc5d0 \uc788\ub2e4\ub294 \ub73b\uc774 \ub41c\ub2e4. \ub530\ub77c\uc11c \\(v, w\\)\ub85c \uc0dd\uc131\ub41c \ud3c9\ud589\uc0ac\ubcc0\ud615\uc740 \ud558\ub098\uc758 \uc120\ubd84\uc774 \ub41c\ub2e4(\uadf8\ub9bc 2). \ub530\ub77c\uc11c \\(\\operatorname{Vol}_{\\mathrm{O}}(v, w)=0\\)\uc774\ub2e4.
      \n<\/span>\n<\/p>\n

      \"\" <\/p>\n

      \n\ud55c\ud3b8 \\(v, w\\)\uac00 \uc77c\ucc28\uc885\uc18d\uc77c \ub54c\ub294 \\(\\det(v, w)=0\\)\uc784\uc744 \uc54c\uace0 \uc788\uc73c\ubbc0\ub85c \uc77c\ucc28\uc885\uc18d\uc778 \ub450 \ubca1\ud130\uc5d0 \uad00\ud574\uc11c\ub294 \uc6b0\ub9ac\uac00 \uc99d\uba85\ud558\uace0\uc790\ud558\ub294 \uba85\uc81c\ub294 \uc99d\uba85\uc774 \ub41c \uac83\uacfc \uac19\ub2e4. \uc774\uc81c \ub2e4\uc74c\uc73c\ub85c \uc0b4\ud3b4\ubcfc \ubcf4\uc870\uc815\ub9ac\ub4e4\uc5d0\uc11c\ub294 \\(v, w\\)\uac00 \uc77c\ucc28\ub3c5\ub9bd\uc784\uc744 \uac00\uc815\ud55c\ub2e4.\n<\/p>\n

      \n

      \n\t\t Lemma.3. <\/span>
      \n\uc77c\ucc28\ub3c5\ub9bd\uc778 \\(v, w\\)\uc640 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
      \n\\[
      \n\\operatorname{Vol}(nv, w)=n\\cdot\\operatorname{Vol}(v, w)
      \n\\]
      \n\uc774 \uc131\ub9bd\ud55c\ub2e4.\n<\/p>\n<\/div>\n

      \n\tProof.<\/span>
      \n\ub450 \ubca1\ud130 \\(nv, w\\)\ub85c \uc0dd\uc131\ub41c \ud3c9\ud589\uc0ac\ubcc0\ud615\uc740 \uadf8\ub9bc 3\uacfc \uac19\uc774 \\(n\\)\uac1c\uc758 \ud3c9\ud589\uc0ac\ubcc0\ud615\uc73c\ub85c \uc774\ub8e8\uc5b4\uc838\uc788\ub2e4.\n<\/p>\n

      \"\" <\/p>\n

      \n\uc774 \\(n\\)\uac1c\uc758 \ud3c9\ud589\uc0ac\ubcc0\ud615\uc740 \uac01\uac01 \\(P(v, w)\\)\ub97c \\(v, 2v, \\ldots, (n-1)v\\)\ub97c \uc774\uc6a9\ud558\uc5ec \ud3c9\ud589\uc774\ub3d9\ud568\uc73c\ub85c\uc368 \uc5bb\uc740 \uac83\uc774\ubbc0\ub85c \uc774\ub4e4 \uac01\uac01\uc758 \ub113\uc774\ub294 \ubaa8\ub450 \\(P(v, w)\\)\uc758 \ub113\uc774\uc640 \uac19\ub2e4. \ub610\ud55c \uc774 \ud3c9\ud589\uc0ac\ubcc0\ud615\ub4e4\uc774 \ub2e4\ub978 \ud3c9\ud589\uc0ac\ubcc0\ud615\ub4e4\uacfc\ub294 \uae30\uaecf\ud574\uc57c \uc120\ubd84\ub4e4\ub9cc \uacf5\uc720\ud558\ubbc0\ub85c \uc6d0\ud558\ub294 \uacb0\ub860\uc778
      \n\\[
      \n\\operatorname{Vol}(nv, w)=n\\cdot\\operatorname{Vol}(v, w)
      \n\\]
      \n\ub97c \uc5bb\ub294\ub2e4.
      \n<\/span>\n<\/p>\n

      \n

      \n\t\t Corollary.4. <\/span>
      \n\uc77c\ucc28\ub3c5\ub9bd\uc778 \\(v, w\\)\uc640 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
      \n\\[
      \n\\operatorname{Vol}\\left(\\frac{1}{n}v, w\\right)=\\frac{1}{n}\\operatorname{Vol}(v, w)
      \n\\]
      \n\uac00 \uc131\ub9bd\ud55c\ub2e4. \ub610\ud55c \\(m, n\\)\uc774 \ubaa8\ub450 \uc790\uc5f0\uc218\uc774\uba74
      \n\\[
      \n\\operatorname{Vol}\\left(\\frac{m}{n}v, w\\right)=\\frac{m}{n}\\operatorname{Vol}(v, w)
      \n\\]
      \n\uc774 \uc131\ub9bd\ud55c\ub2e4.\n<\/p>\n<\/div>\n

      \n\tProof.<\/span>
      \n\\(v_{1}=(1\/n)v\\)\ub85c \ub450\uba74 \ubc14\ub85c \uc704\uc758 \ubcf4\uc870\uc815\ub9ac\ub85c\ubd80\ud130
      \n\\[
      \n\\operatorname{Vol}(nv_{1}, w)=n\\operatorname{Vol}(v_{1}, w)
      \n\\]
      \n\ub97c \uc5bb\ub294\ub2e4. \uc774\ub294, \ub4f1\uc2dd \\(nv_{1}=v\\)\ub97c \uc0dd\uac01\ud560 \ub54c, \uc9c0\uae08 \uc99d\uba85\ud558\uace0 \uc788\ub294 \uccab \ubc88\uc9f8 \ub4f1\uc2dd\uc744 \ub2e4\uc2dc \uc4f4 \uac83\uacfc \ub3d9\uc77c\ud55c \uc2dd\uc784\uc744 \uc54c \uc218 \uc788\ub2e4. \ub450 \ubc88\uc9f8 \ub4f1\uc2dd\uc744 \ubcf4\uc774\uae30 \uc704\ud574 \\(m\/n=m\\cdot 1\/n\\)\uc73c\ub85c \ub450\uace0 \ubc29\uae08 \ubcf4\uc778 \uccab \ubc88\uc9f8 \ub4f1\uc2dd\uc744 \ud65c\uc6a9\ud558\uba74
      \n\\begin{align*}
      \n\\operatorname{Vol}\\left(m\\cdot\\frac{1}{n}v, w\\right)&=m\\operatorname{Vol}\\left(\\frac{1}{n}v, w\\right)\\\\
      \n&=m\\cdot\\frac{1}{n}\\operatorname{Vol}\\left(v, w\\right) \\\\
      \n&=\\frac{m}{n}\\operatorname{Vol}\\left(v, w\\right)
      \n\\end{align*}
      \n\ub97c \uc5bb\ub294\ub2e4.
      \n<\/span>\n<\/p>\n

      \n

      \n\t\t Lemma.5. <\/span>
      \n \\(\\operatorname{Vol}(-v, w)=\\operatorname{Vol}(v, w).\\)\n<\/p>\n<\/div>\n

      \n\tProof.<\/span>
      \n\ub450 \ubca1\ud130 \\(-v, w\\)\ub85c \uc0dd\uc131\ub41c \ud3c9\ud589\uc0ac\ubcc0\ud615\uc740 \ud3c9\ud589\uc0ac\ubcc0\ud615 \\(P(v, w)\\)\ub97c \\(-v\\)\ub85c \ud3c9\ud589\uc774\ub3d9\ud55c \uac83\uacfc \uac19\ub2e4. \ub530\ub77c\uc11c \\(P(v, w)\\)\uc640 \\(P(-v, w)\\)\ub294 \ub3d9\uc77c\ud55c \ub113\uc774\ub97c \uac16\ub294\ub2e4(\uadf8\ub9bc 4).
      \n<\/span>\n<\/p>\n

      \"\" <\/p>\n

      \n

      \n\t\t Lemma.6. <\/span>
      \n\uc784\uc758\uc758 \uc591\uc758 \uc2e4\uc218 \\(c\\)\uc5d0 \ub300\ud558\uc5ec
      \n\\[
      \n\\operatorname{Vol}(cv, w)=c\\cdot\\operatorname{Vol}(v, w)
      \n\\]
      \n\uac00 \uc131\ub9bd\ud55c\ub2e4.\n<\/p>\n<\/div>\n

      \n\tProof.<\/span>
      \n\ub450 \uc720\ub9ac\uc218 \\(r, r’\\)\uc774 \\(0< r < c < r'\\)\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4\uace0 \ud558\uc790(\uadf8\ub9bc 5). \uadf8\ub7ec\uba74\n\\[\nP(rv, w)\\subset P(cv, w)\\subset P(r'v, w)\n\\]\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \ub530\ub77c\uc11c \ub530\ub984\uc815\ub9ac 4\ub85c\ubd80\ud130\n\\begin{align*}\nr\\cdot\\operatorname{Vol}(v, w)&=\\operatorname{Vol}(rv, w) \\\\\n&\\leq \\operatorname{Vol}(cv, w) \\\\\n&\\leq \\operatorname{Vol}(r'v, w) \\\\\n&=r'\\cdot\\operatorname{Vol}(v, w)\n\\end{align*}\n\ub97c \uc5bb\ub294\ub2e4. \\(r\\)\uacfc \\(r'\\)\ub294 \uac01\uac01\uc740 \\(c\\)\uc640 \uc5bc\ub9c8\ub4e0\uc9c0 \uac00\uae4c\uc6cc\uc9c8 \uc218 \uc788\uc73c\ubbc0\ub85c \\(r\\)\uacfc \\(r'\\)\uc774 \\(c\\)\ub85c \uac00\ub3c4\ub85d \uadf9\ud55c\uc744 \ucde8\ud558\uc5ec\n\\[\n\\operatorname{Vol}(cv, w)=c\\cdot\\operatorname{Vol}(v, w)\n\\]\n\ub97c \uc5bb\ub294\ub2e4.\n<\/span>\n<\/p>\n

      \n\ubcf4\uc870\uc815\ub9ac 5\uc640 \ubcf4\uc870\uc815\ub9ac 6\uc73c\ub85c\ubd80\ud130 \uc784\uc758\uc758 \uc2e4\uc218 \\(c\\)\uc640 \uc784\uc758\uc758 \ub450 \ubca1\ud130 \\(v, w\\)\uc5d0 \ub300\ud558\uc5ec
      \n\\[
      \n\\operatorname{Vol}_{\\mathrm{O}}(cv, w)=c\\cdot\\operatorname{Vol}_{\\mathrm{O}}(v, w)
      \n\\]
      \n\uac00 \uc131\ub9bd\ud568\uc744 \ubcf4\uc77c \uc218 \uc788\ub2e4. \ub9cc\uc77c \\(v, w\\)\uac00 \uc77c\ucc28\uc885\uc18d\uc774\uba74 \uc704 \uc2dd\uc758 \uc591\ubcc0\uc740 \\(0\\)\uc774\ub2e4. \ub9cc\uc77c \\(v, w\\)\uac00 \uc77c\ucc28\ub3c5\ub9bd\uc774\ub77c\uba74 \\(\\operatorname{Vol}_{\\mathrm{O}}\\)\uc758 \uc815\uc758\uc640 \ubcf4\uc870\uc815\ub9ac 5\uc640 \ubcf4\uc870\uc815\ub9ac 6\uc744 \uc774\uc6a9\ud55c\ub2e4. \\(\\det(v, w)=c\\)\uc774\uace0 \\(c\\)\uac00 \uc74c\uc218\uc77c \ub54c, \\(c=-d\\)\ub85c \ub450\uc790. \uadf8\ub7ec\uba74 \\(\\det(cv, w)\\leq 0\\)\uc774\uba70 \ub530\ub77c\uc11c
      \n\\begin{align*}
      \n\\operatorname{Vol}_{\\mathrm{O}}(cv, w)=-\\operatorname{Vol}(cv, w)&=-\\operatorname{Vol}(-dv, w) \\\\
      \n&= -\\operatorname{Vol}(dv, w) \\\\
      \n&=-d\\operatorname{Vol}(v, w) \\\\
      \n&=\\operatorname{Vol}(v, w)=c\\operatorname{Vol}_{\\mathrm{O}}(v, w)
      \n\\end{align*}
      \n\ub97c \uc5bb\ub294\ub2e4. \uc720\uc0ac\ud55c \ubc29\ubc95\uc744 \ud1b5\ud574 \\(\\det(v, w)\\leq 0\\)\uc778 \uacbd\uc6b0\ub97c \uc99d\uba85\ud560 \uc218 \uc788\ub2e4. \ub530\ub77c\uc11c \\(\\operatorname{Vol}_{\\mathrm{O}}\\)\uc758 \uc120\ud615\uc131\uc758 \uc870\uac74 \uc911 \ud558\ub098\ub97c \ubcf4\uc600\ub2e4. \ubb3c\ub860 \\(\\operatorname{Vol}_{\\mathrm{O}}(v, w)\\)\uc758 \ubcc0\uc218 \\(w\\)\uc5d0 \uad00\ud574\uc11c\ub3c4 \ub9c8\ucc2c\uac00\uc9c0\uc774\ub2e4. \uc989
      \n\\[
      \n\\operatorname{Vol}_{\\mathrm{O}}(v, cw)=c\\cdot\\operatorname{Vol}_{\\mathrm{O}}(v, w)
      \n\\]
      \n\uac00 \uc131\ub9bd\ud55c\ub2e4.\n<\/p>\n

      \n\uc774\uc81c \uc120\ud615\uc131\uc5d0 \uad00\ud55c \ub098\uba38\uc9c0 \uc870\uac74\uc744 \ubcf4\uc774\uae30 \uc704\ud574 \ub2e4\uc74c\uc758 \ubcf4\uc870\uc815\ub9ac\ub97c \ubcf4\uc774\uc790.\n<\/p>\n

      \n

      \n\t\t Lemma.7. <\/span>
      \n\ub450 \ubca1\ud130 \\(v, w\\)\uac00 \uc77c\ucc28\ub3c5\ub9bd\uc774\ub77c \ud558\uc790. \uadf8\ub7ec\uba74
      \n\\[
      \n\\operatorname{Vol}(v+w, w)=\\operatorname{Vol}(v, w)
      \n\\]
      \n\uc774 \uc131\ub9bd\ud55c\ub2e4.\n<\/p>\n<\/div>\n

      \n\tProof.<\/span>
      \n\ub450 \ubca1\ud130 \\(v\\)\uc640 \\(w\\)\ub85c \uc0dd\uc131\ub41c \ud3c9\ud589\uc0ac\ubcc0\ud615\uc758 \ub113\uc774\uac00 \\(v+w\\)\uc640 \\(w\\)\ub85c \uc0dd\uc131\ub41c \ud3c9\ud589\uc11c\ubcc0\ud615\uc758 \ub113\uc774\uc640 \uc77c\uce58\ud568\uc744 \ubcf4\uc5ec\uc57c\ud55c\ub2e4.\n<\/p>\n

      \"\" <\/p>\n

      \n\uadf8\ub9bc 6\uc5d0\uc11c \uc54c \uc218 \uc788\ub4ef\uc774 \\(v, w\\)\ub85c \uc0dd\uc131\ub41c \ud3c9\ud589\uc0ac\ubcc0\ud615\uc740 \ub450 \uc0bc\uac01\ud615 \\(A, B\\)\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\ub2e4. \uadf8\ub9ac\uace0 \\(v+w, w\\)\ub85c \uc0dd\uc131\ub41c \ud3c9\ud589\uc0ac\ubcc0\ud615\uc740 \uc0bc\uac01\ud615 \\(A\\)\ub97c \\(w\\)\ub85c \ud3c9\ud589\uc774\ub3d9\ud558\uc5ec \uc5bb\uc740 \uc0bc\uac01\ud615 \\(A+w\\)\uc640 \uc0bc\uac01\ud615 \\(B\\)\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\ub2e4. \ub450 \uc0bc\uac01\ud615 \\(A\\)\uc640 \\(A+w\\)\uac00 \ub3d9\uc77c\ud55c \ub113\uc774\ub97c \uac00\uc9c0\ubbc0\ub85c
      \n\\[
      \n\\operatorname{Vol}(v, w)=\\operatorname{Vol}(A)+\\operatorname{Vol}(B)=\\operatorname{Vol}(A+w)+\\operatorname{Vol}(B)=\\operatorname{Vol}(v+w, w)
      \n\\]
      \n\ub97c \uc5bb\ub294\ub2e4.
      \n<\/span>\n<\/p>\n

      \n\uc774\uc81c \uc120\ud615\uc131\uc5d0 \uad00\ud55c \ub450 \ubc88\uc9f8 \uc870\uac74\uc744 \ubcf4\uc774\uc790. \ud3c9\uba74\uc5d0 \ub193\uc778 \ubca1\ud130 \\(w\\)\uac00 \\(0\\)\uc774 \uc544\ub2cc \ubca1\ud130\ub77c \ud558\uc790. \uadf8\ub9ac\uace0 \\(\\{v, w\\}\\)\uac00 \ub3d9\uc77c\ud55c \ud3c9\uba74\uc758 \uae30\uc800\uac00 \ub418\ub3c4\ub85d \\(v\\)\ub97c \ud0dd\ud558\uace0, \uc784\uc758\uc758 \uc2a4\uce7c\ub77c \\(c, d\\)\uc5d0 \ub300\ud558\uc5ec
      \n\\begin{equation}\\label{eq:01}
      \n\\operatorname{Vol}_{\\mathrm{O}}(cv+dw, w)=c\\cdot\\operatorname{Vol}_{\\mathrm{O}}(v, w) \\tag{1}
      \n\\end{equation}
      \n\uac00 \uc131\ub9bd\ud568\uc744 \ubcf4\uc774\uc790. \ub9cc\uc77c \\(d=0\\)\uc774\uba74 \ub531\ud788 \uc99d\uba85\ud560 \uac83\uc774 \uc5c6\ub2e4. \ub9cc\uc77c \\(d\\neq0\\)\uc774\uba74 \uc55e\uc11c \uc0b4\ud3b4\ubcf8 \uad00\ucc30\uc744 \ud65c\uc6a9\ud558\uc5ec
      \n\\[
      \nd\\cdot\\operatorname{Vol}_{\\mathrm{O}}(cv+dw, w)=\\operatorname{Vol}_{\\mathrm{O}}(cv+dw, dw)=c\\cdot\\operatorname{Vol}_{\\mathrm{O}}(v, dw)=cd\\cdot\\operatorname{Vol}_{\\mathrm{O}}(v, w)
      \n\\]
      \n\ub97c \uc5bb\uc73c\uba70 \\(d\\)\ub97c \uc18c\uac70\ud558\uc5ec \uad00\uacc4\uc2dd (1)\uc744 \uc5bb\ub294\ub2e4. \uad00\uacc4\uc2dd (1)\ub85c\ubd80\ud130 \uc9c0\uae08 \ubcf4\uc774\uace0\uc790 \ud558\ub294 \\(\\operatorname{Vol}_{\\mathrm{O}}\\)\uc758 \uc120\ud615\uc131\uc5d0 \uad00\ud55c \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4\ub294 \uac83\uc744 \uc5bb\uc744 \uc218 \uc788\ub2e4. \uc2e4\uc81c\ub85c
      \n\\[
      \nv_{1}=c_{1}v+d_{1}w,\\quad v_{2}=c_{2}v+d_{2}w
      \n\\]
      \n\ub85c \ub450\uba74,
      \n\\begin{align*}
      \n\\operatorname{Vol}_{\\mathrm{O}}(v_{1}+v_{2}, w)&=\\operatorname{Vol}_{\\mathrm{O}}\\left((c_{1}+c_{2})v+(d_{1}+d_{2})w, w\\right) \\\\
      \n&=(c_{1}+c_{2})\\operatorname{Vol}_{\\mathrm{O}}(v, w) \\\\
      \n&=c_{1}\\operatorname{Vol}_{\\mathrm{O}}(v, w)+c_{2}\\operatorname{Vol}_{\\mathrm{O}}(v, w) \\\\
      \n&=\\operatorname{Vol}_{\\mathrm{O}}(v_{1}, w)+\\operatorname{Vol}_{\\mathrm{O}}(v_{2}, w)
      \n\\end{align*}
      \n\uac00 \uc131\ub9bd\ud55c\ub2e4.\n<\/p>\n

      \n\uc9c0\uae08\uae4c\uc9c0 \ud568\uc218 \\(\\operatorname{Vol}_{\\mathrm{O}}\\)\uc774 \ud589\ub82c\uc2dd\uc744 \uaddc\uc815\ud558\ub294 \uc138 \uac00\uc9c0 \uc870\uac74\uc744 \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a8\ub2e4\ub294 \uac83\uc744 \ud655\uc778\ud558\uc600\ub2e4. \uc989
      \n\\[
      \n\\operatorname{Vol}_{\\mathrm{O}}(v, w)=\\det(v, w)
      \n\\]
      \n\uac00 \uc131\ub9bd\ud55c\ub2e4.\n<\/p>\n

      \t\t Remark.1.<\/span>
      \n\uc9c0\uae08\uae4c\uc9c0 \uc0b4\ud3b4\ubcf8 \uc99d\uba85\uc774 \ub2e4\uc18c \uae38\ub2e4\ub294 \ub290\ub08c\uc744 \ubc1b\uc558\uc744 \uc218 \uc788\ub2e4. \uadf8\ub7ec\ub098 \uac01 \uc99d\uba85 \ub2e8\uacc4\uc758 \ub0b4\uc6a9\uc740 \uaf64 \ub2e8\uc21c\ud558\ub2e4 \ud560 \uc218 \uc788\ub2e4. \ub354\uc6b1\uc774 \uac01 \ub2e8\uacc4\uc758 \ub17c\uc758\ub97c \uace0\ucc28\uc6d0\uc73c\ub85c \uc77c\ubc18\ud654\ud558\ub824 \ud560 \ub54c, \uc9c0\uae08\uae4c\uc9c0\uc758 \ub17c\uc99d\uacfc \uac70\uc758 \ub3d9\uc77c\ud55c \ubc29\uc2dd\uc73c\ub85c \uadf8 \ub0b4\uc6a9\uc744 \uc99d\uba85\ud560 \uc218 \uc788\ub2e4. \uc774\ub294 \ud589\ub82c\uc2dd\uc744 \uaddc\uc815\ud558\ub294 \uc870\uac74\uc744 \ubcfc \ub54c \ub450 \uac1c\uc758 \ubcc0\uc218\ub9cc\uc744 \ud3ec\ud568\ud558\ub294 \uc870\uac74\uc740 \ub298 \uc801\ub2f9\ud55c \\(2\\)-\ucc28\uc6d0 \ud3c9\uba74\uc5d0\uc11c \uc0dd\uac01\ud560 \uc218 \uc788\uae30 \ub54c\ubb38\uc774\ub2e4. \ub450 \uac1c\uc758 \ubcc0\uc218\ub97c \uc81c\uc678\ud55c \ubaa8\ub4e0 \ubcc0\uc218\ub97c \uace0\uc815\uc2dc\ud0a4\uba74 \uc9c0\uae08\uae4c\uc9c0\uc758 \uc99d\uba85\uc744 \ud55c \ubc88\uc5d0 \ud655\uc7a5\ud560 \uc218 \uc788\ub2e4.\n<\/p>\n

      \n\uc608\ub97c \ub4e4\uc5b4 \\(3\\)-\ucc28\uc6d0 \uacf5\uac04\uc5d0\uc11c \uc0dd\uac01\ud574\ubcf4\uc790. \uc138 \ubca1\ud130 \\(u, v, w\\)\ub85c \uc0dd\uc131\ub41c \ud3c9\ud589\uccb4(\uadf8\ub9bc 7), \uc989
      \n\\[
      \nt_{1}u+t_{2}v+t_{3}w,\\quad 0\\leq t_{i}\\leq 1
      \n\\]
      \n\uaf34\uc758 \ubaa8\ub4e0 \uc77c\ucc28\uacb0\ud569\uc758 \ubaa8\uc784\uc744 \\(P(u, v, w)\\)\ub85c \ub098\ud0c0\ub0b4\uc790.\n<\/p>\n

      \"\" <\/p>\n

      \n\uadf8\ub9ac\uace0 \uc774 \ud3c9\ud589\uccb4\uc758 \uccb4\uc801\uc740 \\(\\operatorname{Vol}(u, v, w)\\)\ub85c \ub098\ud0c0\ub0b4\uc790.\n<\/p>\n

      \n

      \n\t\t Theorem.8. <\/span>
      \n\uc138 \ubca1\ud130 \\(u, v, w\\)\ub85c \uc0dd\uc131\ub41c \ud3c9\ud589\uccb4\uc758 \uccb4\uc801\uc740 \ud589\ub82c\uc2dd \\(\\det(u, v, w)\\)\uc758 \uc808\ub313\uac12\uacfc \uac19\ub2e4. \uc989
      \n\\[
      \n\\operatorname{Vol}(u, v, w)=\\left\\vert\\det(u, v, w)\\right\\vert
      \n\\]
      \n\uc774 \uc131\ub9bd\ud55c\ub2e4.\n<\/p>\n<\/div>\n

      \n\uc774\uc758 \uc99d\uba85\uc740 \\(2\\)-\ucc28\uc6d0\uc778 \uacbd\uc6b0\uc640 \uc644\uc804\ud788 \uac19\uc740 \ubc29\uc2dd\uc73c\ub85c \uc5bb\ub294\ub2e4. \uc2e4\uc81c\ub85c \ud45c\uc900\uae30\uc800\uc758 \ub2e8\uc704\ubca1\ud130\ub4e4\ub85c \uc5bb\ub294 \uc815\uc721\uba74\uccb4\uc758 \uccb4\uc801\uc740 \\(1\\)\uc774\ub2e4. \ub9cc\uc77c \\(u, v, w\\) \uc911 \ub450 \ubca1\ud130\uac00 \uc77c\uce58\ud55c\ub2e4\uba74, \ud3c9\ud589\uccb4 \\(P(u, v, w)\\)\ub294 \\(2\\)-\ucc28\uc6d0 \ud3c9\ud589\uc0ac\ubcc0\ud615\uc774 \ub418\uc5b4 \uadf8 (3-\ucc28\uc6d0) \uccb4\uc801\uc740 \\(0\\)\uc784\uc744 \uc5bb\ub294\ub2e4. \ub9c8\uc9c0\ub9c9\uc73c\ub85c \uc120\ud615\uc131\uc5d0 \uad00\ud55c \uc99d\uba85\ub3c4 \ub3d9\uc77c\ud558\ub2e4. \uc65c\ub0d0\ud558\uba74 \uc120\ud615\uc131\uc744 \ub530\uc9c0\ub294 \uac83\uc740 \ud558\ub098 \ud639\uc740 \ub450 \uac1c\uc758 \ubcc0\uc218\uc5d0 \uad00\ud55c \uc870\uac74\uc774\uae30 \ub54c\ubb38\uc774\ub2e4. \ub2e4\ub978 \ubcc0\uc218\ub294 \ud45c\uae30\ub97c \uc704\ud574 \uc0ac\uc6a9\ub420 \ubfd0\uc774\uc9c0 \uc99d\uba85 \uacfc\uc815\uc5d0\uc11c \uc544\ubb34 \uc5ed\ud560\ub3c4 \ud558\uc9c0 \uc54a\ub294\ub2e4.\n<\/p>\n

      \n\ub9c8\ucc2c\uac00\uc9c0\ub85c \\(n\\)-\ucc28\uc6d0 \uccb4\uc801\uc744 \uc0dd\uac01\ud558\uace0 \uadf8\uc5d0 \ub300\uc751\ud558\ub294 \ub2e4\uc74c\uc758 \uc815\ub9ac\ub97c \uc5bb\uc744 \uc218 \uc788\ub2e4.\n<\/p>\n

      \n

      \n\t\t Theorem.9. <\/span>
      \n\ubca1\ud130\uacf5\uac04 \\(\\mathbb{R}^{n}\\)\uc758 \\(n\\)\uac1c\uc758 \ubca1\ud130 \\(v_{1}, \\ldots, v_{n}\\)\uc5d0 \ub300\ud558\uc5ec \\(v_{1}, \\ldots, v_{n}\\)\uc73c\ub85c \uc0dd\uc131\ub41c \\(n\\)-\ucc28\uc6d0 \ud3c9\ud589\uccb4\uc758 \\(n\\)-\ucc28\uc6d0 \uccb4\uc801\uc744 \\(\\operatorname{Vol}(v_{1}, \\ldots, v_{n})\\)\ub85c \ub098\ud0c0\ub0b4\uc790. \uadf8\ub7ec\uba74
      \n\\[
      \n\\operatorname{Vol}(v_{1}, \\ldots, v_{n})=\\left\\vert\\det(v_{1}, \\ldots, v_{n})\\right\\vert
      \n\\]
      \n\uac00 \uc131\ub9bd\ud55c\ub2e4.\n<\/p>\n<\/div>\n

      \n\ubb3c\ub860 \\(v_{1}, \\ldots, v_{n}\\)\uc73c\ub85c \uc0dd\uc131\ub41c \\(n\\)-\ucc28\uc6d0 \ud3c9\ud589\uccb4\ub780
      \n\\[
      \n\\sum_{i=1}^{n}t_{i}v_{i},\\quad 0\\leq t_{i}\\leq 1
      \n\\]
      \n\uaf34\uc758 \uc77c\ucc28\uacb0\ud569\uc744 \ubaa8\ub450 \ubaa8\uc544 \ub193\uc740 \uc9d1\ud569\uc744 \uc77c\uceeb\ub294\ub2e4.\n<\/p>\n

      \t\t Remark.2.<\/span>
      \n\uc9c0\uae08\uae4c\uc9c0\uc758 \uc99d\uba85\uc5d0\uc11c \ub113\uc774\uac00 \uac16\ub294 \uae30\ud558\ud559\uc801 \uc131\uc9c8\uc744 \uc0ac\uc6a9\ud558\uc600\ub2e4. \uc21c\uc218\ud558\uac8c \ud574\uc11d\ud559\uc801 \uad00\uc810\uc5d0\uc11c \uc774\ub7ec\ud55c \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcfc \uc218 \uc788\ub2e4. \ud765\ubbf8\uac00 \uc788\ub294 \ub3c5\uc790\ub294 Serge Lang\uc758 \uc800\uc11c [2]\ub97c \ucc38\uc870\ud558\uc5ec\ub77c.\n<\/p>\n

      \t\t Remark.3.<\/span>
      \n\uc0ac\uc2e4 \\(2\\)-\ucc28\uc6d0\uc758 \uacbd\uc6b0 \ud589\ub82c\uc2dd\uc758 \uc808\ub313\uac12\uc774 \ub113\uc774\uc640 \uc77c\uce58\ud55c\ub2e4\ub294 \uac83\uc744 (\uadf8\ub9bc \ud558\ub098\ub85c) \uc27d\uac8c \ubcf4\uc77c \uc218 \uc788\ub2e4. \uadf8\ub7ec\ub098 \uc774 \uae00\uc5d0\uc11c \ub2e4\uc18c \ubcf5\uc7a1\ud574\ubcf4\uc774\ub294 \ubc29\uc2dd\uc744 \uace0\uc9d1\ud55c \uc774\uc720\ub294 \\(3\\)-\ucc28\uc6d0 \ud639\uc740 \\(n\\)-\ucc28\uc6d0\uc73c\ub85c\uc758 \uc77c\ubc18\ud654\ub97c \uc5fc\ub450\uc5d0 \ub450\uc5c8\uae30 \ub54c\ubb38\uc774\ub2e4.\n<\/p>\n

      \n\uc774\uc81c \uc815\ub9ac 1\uc744 \uc120\ud615\uc0ac\uc0c1\uc758 \uad00\uc810\uc5d0\uc11c \uc0b4\ud3b4\ubcf4\uc790. \ud3c9\uba74 \uc0c1\uc758 \ub450 \ubca1\ud130 \\(v, w\\)\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \\(L(e_{1})=v\\)\uc640 \\(L(e_{2})=w\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc120\ud615\uc0ac\uc0c1
      \n\\[
      \nL: \\mathbb{R}^{2}\\to\\mathbb{R}^{2}
      \n\\]
      \n\uac00 \uc720\uc77c\ud558\uac8c \uc874\uc7ac\ud568\uc744 \uc54c\uace0 \uc788\ub2e4. \ub610\ud55c
      \n\\[
      \nv=ae_{1}+ce_{2},\\quad w=be_{1}+de_{2}
      \n\\]
      \n\ub85c \ub458 \ub54c, \uc120\ud615\uc0ac\uc0c1 \\(L\\)\uc758 \ud45c\uc900\uae30\uc800\uc5d0 \uad00\ud55c \ud589\ub82c\ud45c\ud604\uc740
      \n\\[
      \n\\begin{pmatrix}
      \na & b \\\\
      \nc & d
      \n\\end{pmatrix}
      \n\\]
      \n\uc774\ub2e4. \ub354\uc6b1\uc774 \\(e_{1}, e_{2}\\)\ub85c \uc0dd\uc131\ub41c \ub2e8\uc704 \uc815\uc0ac\uac01\ud615\uc744 \\(C\\)\ub85c \ub098\ud0c0\ub0b4\uace0 \\(v, w\\)\ub85c \uc0dd\uc131\ub41c \ud3c9\ud589\uc0ac\ubcc0\ud615\uc744 \\(P\\)\ub85c \ub098\ud0c0\ub0bc \ub54c, \\(P\\)\ub294 \\(C\\)\uc758 \\(L\\)\uc5d0 \uc758\ud55c \uc0c1\uc774\ub2e4. \uc989 \\(L(C)=P\\)\uc774\ub2e4. \uc774\ub294 \\(0\\leq t_{i}\\leq 1\\)\uc5d0 \ub300\ud558\uc5ec
      \n\\[
      \nL(t_{1}e_{1}+t_{2}e_{2})=t_{1}L(e_{1})+t_{2}L(e_{2})=t_{1}v+t_{2}w
      \n\\]
      \n\ub85c\ubd80\ud130 \ud655\uc778\ub41c\ub2e4. \ub9cc\uc77c \uc120\ud615\uc0ac\uc0c1\uc758 \ud589\ub82c\uc2dd\uc744 \uadf8 \uc120\ud615\uc0ac\uc0c1\uc758 \ud589\ub82c\ud45c\ud604\uc758 \ud589\ub82c\uc2dd\uc73c\ub85c \uc815\uc758\ud55c\ub2e4\uba74
      \n\\begin{equation}\\label{eq:star}
      \n(\\mbox{\\(P\\)\uc758 \ub113\uc774})=\\left\\vert\\det(L)\\right\\vert \\tag{2}
      \n\\end{equation}
      \n\ub97c \uc5bb\ub294\ub2e4.\n<\/p>\n

      \n\uad6c\uccb4\uc801\uc73c\ub85c \uacc4\uc0b0\ud558\ub294 \uc608\ub97c \ud558\ub098 \uc0b4\ud3b4\ubcf4\uc790. \ub450 \ubca1\ud130 \\((2, 1)\\)\uacfc \\((3, -1)\\)\ub85c \uc0dd\uc131\ub41c \ud3c9\ud589\uc0ac\ubcc0\ud615(\uadf8\ub9bc 8)\uc758 \ub113\uc774\ub294 \ud589\ub82c\uc2dd
      \n\\[
      \n\\begin{vmatrix}
      \n2 & 1 \\\\
      \n3 & -1
      \n\\end{vmatrix}=-5
      \n\\]
      \n\uc758 \uc808\ub313\uac12, \uc989 \\(5\\)\uc640 \uac19\ub2e4.\n<\/p>\n

      \"\" <\/p>\n

      \n

      \n\t\t Theorem.10. <\/span>
      \n\ub450 \ubca1\ud130\ub85c \uc0dd\uc131\ub41c \ud3c9\ud589\uc0ac\ubcc0\ud615 \\(P\\)\uc640 \uc120\ud615\uc0ac\uc0c1 \\(L: \\mathbb{R}^{2}\\to\\mathbb{R}^{2}\\)\uc5d0 \ub300\ud558\uc5ec
      \n\\[
      \n(\\mbox{\\(L(P)\\)\uc758 \ub113\uc774})=\\left\\vert\\det(L)\\right\\vert\\cdot(\\mbox{\\(P\\)\uc758 \ub113\uc774})
      \n\\]
      \n\uac00 \uc131\ub9bd\ud55c\ub2e4.\n<\/p>\n<\/div>\n

      \n\tProof.<\/span>
      \n\ud3c9\ud589\uc0ac\ubcc0\ud615 \\(P\\)\uac00 \ub450 \ubca1\ud130 \\(v, w\\)\ub85c \uc0dd\uc131\ub418\uc5c8\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(L(P)\\)\ub294 \\(L(v)\\)\uc640 \\(L(w)\\)\ub85c \uc0dd\uc131\ub41c\ub2e4(\uadf8\ub9bc 9). \uc870\uac74
      \n\\[
      \nL_{1}(e_{1})=v, \\quad L_{1}(e_{2})=w
      \n\\]
      \n\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc120\ud615\uc0ac\uc0c1 \\(L_{1}: \\mathbb{R}^{2}\\to\\mathbb{R}^{2}\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \ud45c\uc900\uae30\uc800\ub85c \uc0dd\uc131\ub41c \ub2e8\uc704 \uc815\uc0ac\uac01\ud615\uc744 \\(C\\)\ub85c \ub450\uba74 \\(P=L_{1}(C)\\)\uac00 \uc131\ub9bd\ud558\uba70 \uc55e\uc11c \uad00\ucc30\ud588\ub358 \\eqref{eq:star}\ub85c\ubd80\ud130
      \n\\[
      \n\\operatorname{Vol}\\left(L(P)\\right)=\\left\\vert\\det(L\\circ L_{1})\\right\\vert=\\left\\vert\\det(L)\\det(L_{1})\\right\\vert=\\left\\vert\\det(L)\\right\\vert\\cdot\\operatorname{Vol}(P)
      \n\\]
      \n\ub97c \uc5bb\ub294\ub2e4.
      \n<\/span>\n<\/p>\n

      \"\" <\/p>\n

      \n

      \n\t\t Corollary.11. <\/span>
      \n\uc88c\ud45c\ucd95\uacfc \ud3c9\ud589\ud55c \ubcc0\uc744 \uac16\ub294 \uc784\uc758\uc758 \uc9c1\uc0ac\uac01\ud615 \\(R\\)\uacfc \uc784\uc758\uc758 \uc120\ud615\uc0ac\uc0c1 \\(L: \\mathbb{R}^{2}\\to\\mathbb{R}^{2}\\)\uc5d0 \ub300\ud558\uc5ec
      \n\\[
      \n\\operatorname{Vol}\\left(L(R)\\right)=\\left\\vert\\det(L)\\right\\vert\\cdot\\operatorname{Vol}(R)
      \n\\]
      \n\uc774 \uc131\ub9bd\ud55c\ub2e4.\n<\/p>\n<\/div>\n

      \n\tProof.<\/span>
      \n\uc9c1\uc0ac\uac01\ud615 \\(R\\)\uc758 \uc774\uc6c3\ud55c \ub450 \ubcc0\uc758 \uae38\uc774\uac00 \\(c_{1}, c_{2}\\)\ub77c \ud558\uc790. \uadf8\ub9ac\uace0 \\(c_{1}e_{1}\\)\uacfc \\(c_{2}e_{2}\\)\ub85c \uc0dd\uc131\ub41c \uc9c1\uc0ac\uac01\ud615\uc744 \\(R_{1}\\)\uc73c\ub85c \ub450\uc790. \uadf8\ub7ec\uba74 \\(R\\)\uc740 \uc801\ub2f9\ud55c \ubca1\ud130 \\(u\\)\ub85c \\(R_{1}\\)\uc744 \ud3c9\ud589\uc774\ub3d9\ud558\uc5ec \uc5bb\uc740 \uc9c1\uc0ac\uac01\ud615\uc774 \ub41c\ub2e4. \uc774\ub54c
      \n\\[
      \nL(R)=L(R_{1}+u)=L(R_{1})+L(u)
      \n\\]
      \n\uc774\ubbc0\ub85c \\(L(R)\\)\uc740 \\(L(u)\\)\ub85c \\(L(R_{1})\\)\uc744 \ud3c9\ud589\uc774\ub3d9\ud558\uc5ec \uc5bb\uc740 \ub3c4\ud615\uc774\ub2e4(\uadf8\ub9bc 10). \ud3c9\ud589\uc774\ub3d9\uc740 \ub113\uc774\ub97c \ubcc0\ud654\uc2dc\ud0a4\uc9c0 \uc54a\uc73c\ubbc0\ub85c \uc815\ub9ac 1\ub85c\ubd80\ud130 \uc6d0\ud558\ub294 \uacb0\uacfc\ub97c \uc5bb\ub294\ub2e4.
      \n<\/span>\n<\/p>\n

      \"\" <\/p>\n

      \uc5f0\uc2b5\ubb38\uc81c <\/h2>\n
        \n
      1. \\(v, w\\)\uc5d0 \uad00\ud55c \ud568\uc218 \\(g(v, w)\\)\uac00 \ud589\ub82c\uc2dd\uc744 \uaddc\uc815\ud558\ub294 \uc138 \uc131\uc9c8 \uc911 \ucc98\uc74c\uc758 \ub450 \uc131\uc9c8\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4\uba74 \uc784\uc758\uc758 \\(v, w\\)\uc5d0 \ub300\ud558\uc5ec<\/li>\n

        \\[
        \ng(v, w)=-g(w, v)
        \n\\]
        \n\uac00 \uc131\ub9bd\ud568\uc744 \ubcf4\uc5ec\ub77c. <\/p>\n

      2. \ub2e4\uc74c \uc8fc\uc5b4\uc9c4 \ubca1\ud130\ub4e4\ub85c \uc0dd\uc131\ub41c \ud3c9\ud589\uc0ac\ubcc0\ud615\uc758 \ub113\uc774\ub97c \uad6c\ud558\uc5ec\ub77c.<\/li>\n
          \n
        1. \\((2, 1), (-4, 5)\\)<\/li>\n
        2. \\((3, 4), (-2, -3)\\)<\/li>\n<\/ol>\n
        3. \ub2e4\uc74c \uc8fc\uc5b4\uc9c4 \uc810\ub4e4\uc744 \uc138 \uaf2d\uc9d3\uc810\uc73c\ub85c \uac16\ub294 \ud3c9\ud589\uc0ac\ubcc0\ud615\uc758 \ub113\uc774\ub97c \uad6c\ud558\uc5ec\ub77c.<\/li>\n
            \n
          1. \\((1, 1), (2, -1), (4, 6)\\)<\/li>\n
          2. \\((-3, 2), (1, 4), (-2, -7)\\)<\/li>\n
          3. \\((2, 5), (-1, 4), (1, 2)\\)<\/li>\n
          4. \\((1, 1), (1, 0), (2, 3)\\)<\/li>\n<\/ol>\n
          5. \ub2e4\uc74c \uc8fc\uc5b4\uc9c4 \uc138 \ubca1\ud130\ub85c \uc0dd\uc131\ub41c \ud3c9\ud589\uccb4\uc758 \uccb4\uc801\uc744 \uad6c\ud558\uc5ec\ub77c.<\/li>\n
              \n
            1. \\((1, 1, 3), (1, 2, -1), (1, 4, 1)\\)<\/li>\n
            2. \\((1, -1, 4), (1, 1, 0), (-1, 2, 5)\\)<\/li>\n
            3. \\((-1, 2, 1,), (2, 0, 1), (1, 3, 0)\\)<\/li>\n
            4. \\((-2, 2, 1), (0, 1, 0), (-4, 3, 2)\\)<\/li>\n<\/ol>\n<\/ol>\n

              \ucc38\uace0\ubb38\ud5cc <\/h2>\n
                \n
              1. Lang, S. (1986). Introduction to Linear Algebra Second Edition.<\/i>, Springer <\/li>\n
              2. Lang, S. (2013). Undergraduate analysis<\/i>. Springer Science & Business Media <\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"

                \uc774 \uae00\uc740 \ud604\uc7ac\ub294 \uc808\ud310\ub41c Serge Lang\uc758 \uc120\ud615\ub300\uc218 \uad50\uc7ac \uc81c2\ud310([1], VII, \u00a76)\uc758 \ub0b4\uc6a9\uc744 \ud1a0\ub300\ub85c \uc4f4 \uac83\uc774\ub2e4. \ud37c\uac00\uc9c0 \ub9c8\uc2dc\ub77c. \uc774 \uae00\uc5d0\uc11c\ub294 \ud589\ub82c\uc2dd\uc744 \ud55c \ub3c4\ud615\uc758 \uccb4\uc801\uc73c\ub85c \uc774\ud574\ud558\ub294 \uc774\uc57c\uae30\ub97c \uc18c\uac1c\ud55c\ub2e4. \uba3c\uc800 2-\ucc28\uc6d0\uc758 \uacbd\uc6b0\ub97c \ub17c\ud560\ud150\ub370, ‘\uccb4\uc801(volume)’\uc774\ub77c\ub294 \uc6a9\uc5b4\ub97c 2-\ucc28\uc6d0 \ub3c4\ud615\uc758 \ub113\uc774\ub97c \uc77c\uceec\uc744 \ub54c\uc5d0\ub3c4 \uadf8\ub0e5 \uc0ac\uc6a9\ud558\uace0\uc790 \ud55c\ub2e4. \ub610\ud55c ‘\\(\\operatorname{Vol}\\)’\uc640 \uac19\uc740 \uae30\ud638\ub97c \uc774\uc6a9\ud558\uc5ec \ub113\uc774\ub97c \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud560 \uac83\uc774\ub2e4. \ubb3c\ub860 \uc774 \uae30\ud638\ub97c \uc77c\ubc18\uc801\uc778 \uace0\ucc28\uc6d0 \ub3c4\ud615\uc758 \uccb4\uc801\uc744 \ub098\ud0c0\ub0b4\ub294 \uae30\ud638\ub85c\ub3c4 \uc4f8 \uac83\uc774\ub2e4. \uba3c\uc800 \ub450 \ubca1\ud130 \\(v,…<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"categories":[55,57],"tags":[540,400,552,553,554,555,556,455],"class_list":["post-6516","post","type-post","status-publish","format-standard","hentry","category-classical-geometry","category-linear-algebra","tag-determinants","tag-linear-algebra","tag-volume","tag-553","tag-554","tag-555","tag-556","tag-455"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/6516","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=6516"}],"version-history":[{"count":11,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/6516\/revisions"}],"predecessor-version":[{"id":6527,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/6516\/revisions\/6527"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=6516"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=6516"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=6516"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}