{"id":5585,"date":"2020-10-18T15:21:08","date_gmt":"2020-10-18T06:21:08","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=5585"},"modified":"2021-05-30T23:52:03","modified_gmt":"2021-05-30T14:52:03","slug":"linear-algebra-inner-product-spaces","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/linear-algebra-inner-product-spaces\/","title":{"rendered":"\ub0b4\uc801\uacf5\uac04"},"content":{"rendered":"

\ubca1\ud130\uacf5\uac04\uc740 \ubb3c\uccb4\uc758 \uc704\uce58\ub97c \uae30\uc220\ud560 \uc218 \uc788\ub294 \ucd94\uc0c1\uc801\uc778 \ud615\ud0dc\uc758 \uacf5\uac04\uc774\ub2e4. \uadf8\ub7ec\ub098 \ubca1\ud130\uacf5\uac04\uc5d0\ub294 \ubca1\ud130\uc758 \ud569\uacfc \uc2a4\uce7c\ub77c \uacf1\uc774\ub77c\ub294 \ub450 \uac1c\uc758 \uc5f0\uc0b0\ub9cc \uc874\uc7ac\ud558\uae30 \ub54c\ubb38\uc5d0 \uc810 \uc0ac\uc774\uc758 \uac70\ub9ac\ub098 \ub450 \ubca1\ud130 \uc0ac\uc774\uc758 \uac01\uc758 \ud06c\uae30\ub97c \uc815\uc758\ud560 \uc218 \uc5c6\ub2e4. \ub300\uc2e0 \ubca1\ud130\uacf5\uac04\uc5d0 \ub0b4\uc801\uc774\ub77c\ub294 \uad6c\uc870\ub97c \ucd94\uac00\ud568\uc73c\ub85c\uc368 \uac70\ub9ac\uc640 \uac01\uc758 \ud06c\uae30\ub97c \uc815\uc758\ud560 \uc218 \uc788\ub2e4.<\/p>\n

\uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \ub0b4\uc801\uc758 \uac1c\ub150\uacfc \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf4\uace0, \uc774\ub85c\ubd80\ud130 \ud30c\uc0dd\ub418\ub294 \uae30\uc800\uc758 \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n

\n\\[
\n\\newcommand{\\Hom}{{\\operatorname{Hom}}}
\n\\newcommand{\\Mat}{{\\operatorname{Mat}}}
\n\\newcommand{\\proj}{{\\operatorname{proj}}}
\n\\]\n<\/div>\n

<\/p>\n

\uc2e4\ub0b4\uc801\uacf5\uac04<\/h2>\n

\\(V\\)\uac00 \uc2e4\ubca1\ud130\uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. \ud568\uc218
\n\\[ V\\times V \\,\\rightarrow\\,\\mathbb{R} ,\\,\\, (v,\\,w) \\,\\mapsto\\, \\langle v \\,\\vert\\, w \\rangle\\]
\n\uac00 \ub2e4\uc74c \uc138 \uc870\uac74\uc744 \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0ac \ub54c, \uc774 \ud568\uc218\ub97c \\(V\\) \uc704\uc5d0\uc11c\uc758 \uc2e4\ub0b4\uc801<\/span>(real inner product)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

    \n
  1. \uc784\uc758\uc758 \\(v\\in V\\)\uc5d0 \ub300\ud558\uc5ec \\(\\langle v\\,\\vert\\, v \\rangle \\ge 0\\)\uc774\uba70, \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(v = \\mathbf{0}\\)\uc778 \uac83\uc774\ub2e4. (\uc591\uc758 \uc815\ubd80\ud638 \uc131\uc9c8)<\/li>\n
  2. \uc784\uc758\uc758 \\(v,\\,w\\in V\\)\uc5d0 \ub300\ud558\uc5ec \\(\\langle v\\,\\vert\\, w\\rangle = \\langle w\\,\\vert\\,v\\rangle\\)\uc774\ub2e4. (\ub300\uce6d \uc131\uc9c8)<\/li>\n
  3. \uc784\uc758\uc758 \\(u,\\,v,\\,w\\in V\\)\uc640 \\(a\\in \\mathbb{R}\\)\uc5d0 \ub300\ud558\uc5ec \\(\\langle u+v \\,\\vert\\, w \\rangle = \\langle u\\,\\vert\\,w\\rangle + \\langle v\\,\\vert\\,w \\rangle\\) \uadf8\ub9ac\uace0 \\(\\langle av\\,\\vert\\,w\\rangle = a\\langle v\\,\\vert\\,w\\rangle\\)\uc774\ub2e4. (\uc30d\uc120\ud615 \uc131\uc9c8)<\/li>\n<\/ol>\n

    \uc2e4\ub0b4\uc801\uc744 \uac04\ub2e8\ud788 \ub0b4\uc801<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub0b4\uc801\uc774 \uc8fc\uc5b4\uc838 \uc788\ub294 \ubca1\ud130\uacf5\uac04\uc744 \ub0b4\uc801\uacf5\uac04<\/span>(inner product space)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ud2b9\ud788 \uccb4\uac00 \uc2e4\uc218\uacc4\uc778 \uacbd\uc6b0 \ub0b4\uc801\uacf5\uac04\uc744 \uc2e4\ub0b4\uc801\uacf5\uac04<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. [\ubcf5\uc18c\ub0b4\uc801\uacf5\uac04\uc740 \uc2e4\ub0b4\uc801\uacf5\uac04\uacfc\ub294 \uc870\uae08 \ub2e4\ub974\uac8c \uc815\uc758\ub41c\ub2e4.]<\/p>\n

    \ub0b4\uc801\uc744 \uc774\uc6a9\ud558\uc5ec \ubca1\ud130\uc758 \uae38\uc774\ub97c \uc815\uc758\ud560 \uc218 \uc788\ub2e4. \\(V\\)\uac00 \uc2e4\ub0b4\uc801\uacf5\uac04\uc774\uace0 \\(v\\in V\\)\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(v\\)\uc758 \uae38\uc774<\/span>(length)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
    \n\\[\\lvert v \\rvert = \\sqrt{\\langle v\\,\\vert\\, v\\rangle}.\\]
    \n\ubca1\ud130\uc758 \uae38\uc774\ub97c \ub178\ub984<\/span>(norm)\uc774\ub77c\uace0 \ubd80\ub974\uae30\ub3c4 \ud55c\ub2e4. \uae38\uc774\uac00 1\uc778 \ubca1\ud130\ub97c \ub2e8\uc704\ubca1\ud130<\/span>(unit vector)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

    \ubca1\ud130\uc758 \uae38\uc774\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n

      \n
    1. \uc784\uc758\uc758 \\(v\\in V\\)\uc5d0 \ub300\ud558\uc5ec \\(\\lvert v \\rvert = 0 \\,\\,\\Leftrightarrow\\,\\, v=\\mathbf{0}.\\)<\/li>\n
    2. \uc784\uc758\uc758 \\(v\\in V\\)\uc640 \\(a\\in\\mathbb{R}\\)\uc5d0 \ub300\ud558\uc5ec \\(\\lvert av \\rvert = \\lvert a \\rvert \\cdot \\lvert v \\rvert.\\) \uc5ec\uae30\uc11c \\(\\lvert a \\vert\\)\ub294 \uc2e4\uc218 \\(a\\)\uc758 \uc808\ub313\uac12\uc744 \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n<\/ol>\n

      \uc2e4\uc218\uc758 \uc808\ub313\uac12\uacfc \ubca1\ud130\uc758 \ud06c\uae30\ub97c \uad6c\ubd84\ud558\uae30 \uc704\ud558\uc5ec \\(v\\)\uc758 \ud06c\uae30\ub97c \\(\\lVert v \\rVert\\)\ub85c \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud55c\ub2e4.<\/p>\n

      \n

      \ubcf4\uae30 1.<\/span>
      \n\\(V=\\mathbb{R}^n\\)\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(V\\)\uc5d0\uc11c\uc758 \uc720\ud074\ub9ac\ub4dc \ub0b4\uc801<\/span>(Euclidean inner product)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
      \n\\[\\langle \\mathbf{x} \\,\\vert\\, \\mathbf{y} \\rangle = \\sum_{j=1}^n x_j y_j .\\]
      \n\\(\\mathbb{R}^n\\)\uc758 \ubca1\ud130\ub97c \uc5f4\ubca1\ud130\ub85c \ub098\ud0c0\ub0b4\uba74 \ubca1\ud130\uc758 \ub0b4\uc801\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \ud45c\ud604\ud560 \uc218 \uc788\ub2e4.
      \n\\[\\langle \\mathbf{x} \\,\\vert\\,\\mathbf{y}\\rangle = \\,^t\\! \\mathbf{x}\\mathbf{y}.\\]
      \n\ud55c\ud3b8 \uc720\ud074\ub9ac\ub4dc \ub0b4\uc801\uc73c\ub85c\ubd80\ud130 \uc720\ub3c4\ub418\ub294 \ubca1\ud130\uc758 \ud06c\uae30\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.
      \n\\[\\lvert\\mathbf{x}\\rvert = \\left( \\sum_{j=1}^n x_j^2 \\right)^{1\/2}.\\]\n<\/p>\n<\/div>\n

      \n

      \ubcf4\uae30 2.<\/span>
      \n\\(V = C^0([a,\\,b])\\)\ub77c\uace0 \ud558\uc790. \uc774\ub54c
      \n\\[\\langle f \\,\\vert\\, g\\rangle = \\int_a^b f(x)g(x)dx\\]
      \n\ub85c \uc815\uc758\ub41c \ud568\uc218 \\(\\langle \\cdot \\vert \\cdot \\rangle\\)\ub294 \ub0b4\uc801\uc758 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uc774 \ub0b4\uc801\uc73c\ub85c\ubd80\ud130 \uc720\ub3c4\ub418\ub294 \ubca1\ud130\uc758 \ud06c\uae30\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.
      \n\\[\\lvert f \\rvert = \\left( \\int_a^b (f(x))^2 dx \\right)^{1\/2}.\\]\n<\/p>\n<\/div>\n

      \uc2e4\uc218 \\(x_j,\\) \\(y_j\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc740 \ubd80\ub4f1\uc2dd\uc774 \uc131\ub9bd\ud55c\ub2e4.
      \n\\[ (x_1 y_1 + x_2 y_2 + \\cdots + x_n y_n)^2 \\le (x_1^2 +x_2^2 + \\cdots + x_n^2)(y_1^2 + y_2^2 + \\cdots + y_n^2).\\]
      \n\uc774\uc640 \uac19\uc740 \ubd80\ub4f1\uc2dd\uc744 \ucf54\uc2dc-\uc288\ubc14\ub974\uce20\ubd80\ub4f1\uc2dd\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774 \ubd80\ub4f1\uc2dd\uc740 \ub0b4\uc801\uacf5\uac04\uc5d0\uc11c\ub3c4 \uc131\ub9bd\ud55c\ub2e4. \ub2e4\uc74c \uc815\ub9ac\ub97c \ubcf4\uc790.<\/p>\n

      \n

      \uc815\ub9ac 1. (Cauchy-Schwarz \ubd80\ub4f1\uc2dd)<\/span><\/p>\n

      \\(V\\)\uac00 \ub0b4\uc801\uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(V\\)\uc758 \uc784\uc758\uc758 \uc6d0\uc18c \\(v,\\) \\(w\\)\uc5d0 \ub300\ud558\uc5ec
      \n\\[\\lvert\\langle v\\,\\vert\\,w\\rangle\\rvert \\le \\lvert v \\rvert \\cdot \\lvert w \\rvert\\]
      \n\uac00 \uc131\ub9bd\ud55c\ub2e4.\n<\/p>\n<\/div>\n

      \n

      \uc99d\uba85.<\/p>\n

      \\(w=\\mathbf{0}\\)\uc778 \uacbd\uc6b0\ub294 \uc815\ub9ac\uc758 \ubd80\ub4f1\uc2dd\uc774 \uc790\uba85\ud558\uac8c \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(w\\ne \\mathbf{0}\\)\uc774\ub77c\uace0 \ub450\uace0 \uc99d\uba85\ud558\uc790.<\/p>\n

      \uc784\uc758\uc758 \\(x\\in\\mathbb{R}\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
      \n\\[\\begin{gather}
      \n\\langle v+xw \\,\\vert\\,v+xw\\rangle \\ge 0,\\\\[5pt]
      \n\\langle v\\,\\vert\\,v\\rangle + 2 x\\langle v\\,\\vert\\,w\\rangle +x^2 \\langle w\\,\\vert\\,w\\rangle \\ge 0,\\\\[5pt]
      \n\\lvert w \\rvert^2 x^2 + 2 \\langle v\\,\\vert\\, w\\rangle x + \\lvert v \\rvert^2 \\ge 0.
      \n\\end{gather}\\]
      \n\ub9c8\uc9c0\ub9c9 \ubd80\ub4f1\uc2dd\uc740 \uc784\uc758\uc758 \\(x\\in\\mathbb{R}\\)\uc5d0 \ub300\ud558\uc5ec \uc131\ub9bd\ud574\uc57c \ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc88c\ubcc0\uc758 \ud310\ubcc4\uc2dd\uc758 \uac12\uc740 \\(0\\) \uc774\ud558\uc5ec\uc57c \ud55c\ub2e4. \uc989
      \n\\[4\\langle v\\,\\vert\\,w\\rangle^2 – 4\\lvert v \\rvert^2 \\lvert w \\rvert^2 \\le 0\\]
      \n\uc774\ubbc0\ub85c
      \n\\[\\langle v\\,\\vert\\,w\\rangle ^2 \\le \\lvert v \\rvert^2 \\lvert w \\rvert^2 \\]
      \n\uc774\ub2e4.
      \n<\/span><\/p>\n<\/div>\n

      \n

      \ub530\ub984\uc815\ub9ac 2. (\uc0bc\uac01\ubd80\ub4f1\uc2dd)<\/span><\/p>\n

      \\(V\\)\uac00 \ub0b4\uc801\uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(V\\)\uc758 \uc784\uc758\uc758 \uc6d0\uc18c \\(v,\\) \\(w\\)\uc5d0 \ub300\ud558\uc5ec
      \n\\[\\lvert v+w \\vert \\le \\lvert v \\rvert + \\lvert w \\rvert\\]
      \n\uac00 \uc131\ub9bd\ud55c\ub2e4.\n<\/p>\n<\/div>\n

      \n

      \uc99d\uba85.<\/p>\n

      \n\\[\\begin{align}
      \n\\lvert v+w \\rvert^2
      \n&= \\langle v+w \\,\\vert\\, v+w \\rangle \\\\[5pt]
      \n&= \\langle v\\,\\vert\\,v \\rangle + 2 \\langle v\\,\\vert\\,w \\rangle + \\langle w\\,\\vert\\,w\\rangle\\\\[5pt]
      \n&\\le \\lvert v \\rvert^2 + 2 \\lvert v \\rvert \\lvert w \\rvert + \\lvert w \\rvert^2\\\\[5pt]
      \n&\\le (\\lvert v \\rvert + \\lvert w \\rvert )^2.\\tag*{\\(\\blacksquare\\)}
      \n\\end{align}\\]\n<\/div>\n<\/div>\n

      \ub0b4\uc801\uc744 \uc774\uc6a9\ud558\uc5ec \ubca1\ud130\uc758 \ud06c\uae30\ub97c \uc815\uc758\ud55c \uac83\ucc98\ub7fc, \ub0b4\uc801\uc744 \uc774\uc6a9\ud558\uc5ec \ub450 \ubca1\ud130 \uc0ac\uc774\uc758 \uac01\uc758 \ud06c\uae30\ub97c \uc815\uc758\ud560 \uc218 \uc788\ub2e4. \\(V\\)\uac00 \ub0b4\uc801\uacf5\uac04\uc774\uace0 \\(v,\\) \\(w\\)\uac00 \\(V\\)\uc758 \ubca1\ud130\uc774\uba70 \uc601\ubca1\ud130\uac00 \uc544\ub2c8\ub77c\uace0 \ud558\uc790. \uc774\ub54c
      \n\\[\\cos\\theta = \\frac{\\langle v\\,\\vert\\,w\\rangle}{\\lvert v \\rvert \\lvert w \\rvert}\\]
      \n\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc218 \\(\\theta\\in [0,\\,\\pi ]\\)\uc758 \uac12\uc744 \ub450 \ubca1\ud130 \\(v\\)\uc640 \\(w\\) \uc0ac\uc774\uc758 \uac01<\/span>(angle)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \\(v\\)\uc640 \\(w\\)\uc758 \ub0b4\uc801\uc744 \\(v \\cdot w\\)\ub77c\uace0 \ub098\ud0c0\ub0b8\ub2e4\uba74 \uc774 \uc2dd\uc740
      \n\\[v\\cdot w = \\lvert v \\rvert \\lvert w \\rvert \\cos\\theta\\]
      \n\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.<\/p>\n

      \ub0b4\uc801\uacf5\uac04 \\(V\\)\uc758 \ub450 \ubca1\ud130 \\(v,\\) \\(w\\)\uc5d0 \ub300\ud558\uc5ec \\(\\langle v\\,\\vert\\,w\\rangle =0\\)\uc774 \uc131\ub9bd\ud560 \ub54c, \\(v\\)\ub294 \\(w\\)\uc640 \uc218\uc9c1\uc774\ub2e4<\/span>(orthogonal)\ub77c\uace0 \ub9d0\ud558\uace0, \uc774\uac83\uc744 \uae30\ud638\ub85c \\(v\\bot w\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n

      \uc601\ubca1\ud130\uac00 \uc544\ub2cc \\(m\\)\uac1c\uc758 \ubca1\ud130 \\(v_1 ,\\) \\(\\cdots ,\\) \\(v_m\\)\uc774 \uc788\uc744 \ub54c, \uc774 \ubca1\ud130\ub4e4\uc774 \uc9c1\uad50\ubca1\ud130\uc871<\/span>(orthogonal family)\uc774\ub77c \ud568\uc740 \\(i\\ne j\\)\uc77c \ub54c\ub9c8\ub2e4 \\(\\langle v_i \\,\\vert\\, v_j \\rangle =0\\)\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4. \ubaa8\ub4e0 \ubca1\ud130\uc758 \uae38\uc774\uac00 \\(1\\)\uc774\uace0, \uc774 \ubca1\ud130\ub4e4\uc774 \uc9c1\uad50\ubca1\ud130\uc871\uc77c \ub54c \uc774 \ubca1\ud130\ub4e4\uc744 \uc815\uaddc\uc9c1\uad50\uc871<\/span>(orthonormal family)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774 \uacbd\uc6b0 \\(\\langle v_i \\,\\vert\\, v_j \\rangle = \\delta_{ij}\\)\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

      \\(V\\)\uc758 \uae30\uc800\uac00 \uc9c1\uad50\ubca1\ud130\uc871\uc77c \ub54c, \uadf8\ub7ec\ud55c \uae30\uc800\ub97c \uc9c1\uad50\uae30\uc800<\/span>(orthogonal basis)\ub77c\uace0 \ubd80\ub978\ub2e4. \\(V\\)\uc758 \uae30\uc800\uac00 \uc815\uaddc\uc9c1\uad50\uc871\uc77c \ub54c, \uadf8\ub7ec\ud55c \uae30\uc800\ub97c \uc815\uaddc\uc9c1\uad50\uae30\uc800<\/span>(orthonormal basis)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

      \uc9c1\uad50\ubca1\ud130\uc871\uc740 \ubca1\ud130\uc758 \uc218\uac00 \ubb34\ud55c\uc77c \ub54c\uc5d0\ub3c4 \uc815\uc758\ud560 \uc218 \uc788\ub2e4. \uc9d1\ud569 \\(S\\)\uac00 \ubca1\ud130\uacf5\uac04 \\(V\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(S\\)\uc758 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc784\uc758\uc758 \uc720\ud55c\ubd80\ubd84\uc9d1\ud569\uc774 \uc9c1\uad50\ubca1\ud130\uc871\uc774\uba74 \\(S\\)\ub97c \uc9c1\uad50\ubca1\ud130\uc871\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc989 \\(S\\)\uac00 \uc9c1\uad50\ubca1\ud130\uc871\uc774\ub77c \ud568\uc740 \\(S\\)\uc758 \uc11c\ub85c \ub2e4\ub978 \uc784\uc758\uc758 \ub450 \ubca1\ud130 \\(v_i,\\) \\(v_j\\)\uc5d0 \ub300\ud558\uc5ec \\(\\langle v_i \\,\\vert\\, v_j \\rangle =0\\)\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4. \uc815\uaddc\uc9c1\uad50\uc871\ub3c4 \uac19\uc740 \ubc29\uc2dd\uc73c\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n

      \n

      \ubcf4\uae30 3.<\/span>
      \n\\(\\mathbb{R}^n\\)\uc5d0\uc11c \ubcf4\ud1b5\uae30\uc800 \\(\\mathbf{e}_1,\\) \\(\\cdots,\\) \\(\\mathbf{e}_n\\)\uc740 \uc815\uaddc\uc9c1\uad50\uae30\uc800\uc774\ub2e4.<\/p>\n

      [\\(\\mathbb{R}^n\\)\uc758 \uc815\uaddc\uc9c1\uad50\uae30\uc800\ub294 \ubcf4\ud1b5\uae30\uc800 \uc678\uc5d0\ub3c4 \ubb34\uc218\ud788 \ub9ce\uc774 \uc788\ub2e4.]\n<\/p>\n<\/div>\n

      \n

      \ubcf4\uae30 4.<\/span>
      \n\ubca1\ud130\uacf5\uac04 \\(C^0 ([-\\pi ,\\, \\pi ])\\)\ub97c \uc0dd\uac01\ud558\uc790. \uc774 \uacf5\uac04\uc5d0\uc11c \ub2e4\uc74c \ubca1\ud130\ub4e4\uc740 \uc9c1\uad50\ubca1\ud130\uc871\uc744 \uc774\ub8ec\ub2e4.
      \n\\[1,\\,\\, \\cos x ,\\,\\, \\sin x ,\\,\\, \\cos 2x ,\\,\\, \\sin 2x ,\\,\\, \\cos 3x ,\\,\\, \\sin 3x ,\\,\\, \\cdots .\\]
      \n\uc774 \ubca1\ud130\ub4e4\uc774 \uc9c1\uad50\ubca1\ud130\uc871\uc744 \uc774\ub8ec\ub2e4\ub294 \uc0ac\uc2e4\uc744 \uc99d\uba85\ud558\uae30 \uc704\ud574\uc11c\ub294 \uc0bc\uac01\ud568\uc218\uc758 \uacf5\uc2dd
      \n\\[\\begin{align}
      \n\\cos\\alpha \\,\\cos\\beta &= \\frac{1}{2} (\\cos(\\alpha + \\beta) + \\cos(\\alpha – \\beta)), \\\\[5pt]
      \n\\sin\\alpha \\,\\sin\\beta &= -\\frac{1}{2} (\\cos(\\alpha + \\beta) – \\cos(\\alpha – \\beta))
      \n\\end{align}\\]
      \n\uc744 \uc774\uc6a9\ud558\uba74 \ub41c\ub2e4. \uadf8\ub7ec\ub098 \uc774 \ubca1\ud130\ub4e4\uc774 \uc815\uaddc\uc9c1\uad50\uc871\uc744 \uc774\ub8e8\uc9c0\ub294 \uc54a\ub294\ub2e4.\n<\/p>\n<\/div>\n

      \uc9c1\uad50\ubca1\ud130\uc871\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uc7ac\ubbf8 \uc788\ub294 \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n

      \n

      \uc815\ub9ac 3.<\/span>
      \n\uc9c1\uad50\ubca1\ud130\uc871\uc758 \ubca1\ud130\ub4e4\uc740 \uc77c\ucc28\ub3c5\ub9bd\uc774\ub2e4.\n<\/p>\n<\/div>\n

      \n

      \uc99d\uba85.<\/p>\n

      \\(V\\)\uac00 \ub0b4\uc801\uacf5\uac04\uc774\uace0 \\(S\\)\uac00 \\(V\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uba70 \uc9c1\uad50\ubca1\ud130\uc871\uc774\ub77c\uace0 \ud558\uc790. \\(S\\)\uc5d0\uc11c \uc784\uc758\ub85c \\(m\\)\uac1c\uc758 \uc6d0\uc18c\ub97c \uaebc\ub0b4\uc5b4 \\(v_1 ,\\) \\(\\cdots ,\\) \\(v_m\\)\uc774\ub77c\uace0 \ud558\uc790. \uc774\uc81c
      \n\\[a_1 v_1 + a_2 v_2 + \\cdots + a_m v_m = \\mathbf{0}\\]
      \n\uc774\ub77c\uace0 \uac00\uc815\ud558\uc790. \uc774 \ub4f1\uc2dd\uc758 \uc88c\ubcc0\uacfc \\(v_k\\) \uc911 \ud558\ub098\uc640 \ub0b4\uc801\uc744 \ucde8\ud558\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
      \n\\[0 = \\langle \\mathbf{0} \\,\\vert\\, v_k \\rangle
      \n= \\left\\langle \\sum_{j=1}^m a_j v_j \\,\\bigg\\vert\\,v_k \\right\\rangle
      \n= \\sum _{j=1}^m a_j \\langle v_j \\,\\vert\\, v_k \\rangle
      \n= a_k \\langle v_k \\,\\vert\\,v_k \\rangle.\\]
      \n\uadf8\ub7f0\ub370 \\(\\langle v_k \\,\\vert\\,v_k \\rangle\\)\uac00 \\(0\\)\uc774 \uc544\ub2c8\ubbc0\ub85c \\(a_k =0\\)\uc774\ub2e4.
      \n<\/span><\/p>\n<\/div>\n

      \n

      \uc815\ub9ac 4. (\ud53c\ud0c0\uace0\ub77c\uc2a4 \uc815\ub9ac)<\/span><\/p>\n

      \\(v_1,\\) \\(v_2,\\) \\(\\cdots,\\) \\(v_m\\)\uc774 \ub0b4\uc801\uacf5\uac04 \\(V\\)\uc5d0\uc11c \uc9c1\uad50\ubca1\ud130\uc871\uc774\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
      \n\\[\\left\\lvert \\sum_{j=1}^{m} v_j \\right\\rvert^2 = \\sum_{j=1}^m \\lvert v_j \\rvert^2 .\\]\n<\/p>\n<\/div>\n

      \n

      \uc99d\uba85.<\/p>\n

      \n\\[\\begin{align}
      \n\\left\\lvert \\sum_{j=1}^m v_j \\right\\rvert^2
      \n&= \\left\\langle \\sum_{j=1} ^m v_j \\,\\bigg\\vert\\, \\sum_{j=1}^m v_j \\right\\rangle
      \n= \\sum_{j=1}^m \\sum_{k=1}^m \\langle v_j \\,\\vert\\, v_k \\rangle \\\\[5pt]
      \n&= \\sum_{j=1}^m \\langle v_j \\,\\vert\\, v_j \\rangle
      \n= \\sum_{j=1}^m \\lvert v_j \\rvert^2. \\tag*{\\(\\blacksquare\\)}
      \n\\end{align}\\]\n<\/div>\n<\/div>\n

      \ud589\ub82c \\(C\\in M_n (\\mathbb{R})\\)\ub97c \uc0dd\uac01\ud558\uc790. \ub9cc\uc57d \uc601\ubca1\ud130\uac00 \uc544\ub2cc \uc784\uc758\uc758 \\(\\mathbf{x}\\in\\mathbb{R}^n\\)\uc5d0 \ub300\ud558\uc5ec \\(^t \\! \\mathbf{x} C \\mathbf{x} > 0\\)\uc774 \uc131\ub9bd\ud558\uba74 \\(C\\)\ub97c \uc591\uc758 \uc815\ubd80\ud638 \ud589\ub82c<\/span>(positive definite)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

      \uc591\uc758 \uc815\ubd80\ud638 \ud589\ub82c\uc744 \uc67c\ucabd\uc5d0 \uacf1\ud558\ub294 \uc120\ud615\ubcc0\ud658\uc758 \ud575(kernel)\uc740 \uc790\uba85\ud55c \uacf5\uac04\uc774\ub2e4. \uc65c\ub0d0\ud558\uba74 \\(\\mathbf{x} \\ne \\mathbf{0}\\)\uc774\uba74 \\(C\\mathbf{x} \\ne \\mathbf{0}\\)\uc774\uae30 \ub54c\ubb38\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc591\uc758 \uc815\ubd80\ud638 \ud589\ub82c\uc740 \uc77c\ub300\uc77c\uc778 \uc120\ud615\ubcc0\ud658\uc744 \ub098\ud0c0\ub0b4\uba70, \uac00\uc5ed\uc778 \ud589\ub82c\uc774\ub2e4.<\/p>\n

      \ub2e4\uc74c \uc815\ub9ac\ub294 \uc591\uc758 \uc815\ubd80\ud638 \ud589\ub82c\uacfc \ubca1\ud130\uacf5\uac04\uc758 \ub0b4\uc801\uc758 \uad00\uacc4\ub97c \uc124\uba85\ud55c\ub2e4.<\/p>\n

      \n

      \uc815\ub9ac 5.<\/span>
      \n\\(V\\)\uac00 \ucc28\uc6d0\uc774 \\(n\\)\uc778 \uc720\ud55c\ucc28\uc6d0 \uc2e4\ubca1\ud130\uacf5\uac04\uc774\uace0 \uae30\uc800 \\(B\\)\ub97c \uac00\uc9c4\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(V\\) \uc704\uc5d0\uc11c \uc815\uc758\ub418\ub294 \uc784\uc758\uc758 \ub0b4\uc801\uc740 \uc801\ub2f9\ud55c \uc591\uc758 \uc815\ubd80\ud638 \ub300\uce6d\ud589\ub82c \\(C\\in\\operatorname{Mat}_{n\\times n}(\\mathbb{R})\\)\uc5d0 \ub300\ud558\uc5ec
      \n\\[\\langle v\\,\\vert\\,w\\rangle = \\,^t \\! \\gamma_B (v) C \\gamma_B (w) \\tag{\\(\\ast\\)}\\]
      \n\uc758 \uaf34\uc774 \ub41c\ub2e4.\n<\/p>\n<\/div>\n

      \n

      \uc99d\uba85.<\/p>\n

      \\(C\\)\uac00 \uc591\uc758 \uc815\ubd80\ud638 \ub300\uce6d\ud589\ub82c\uc774\uba74 \\((\\ast)\\)\ub294 \uba85\ubc31\ud788 \ub0b4\uc801\uc758 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ub0b4\uc801\uc774 \ud56d\uc0c1 \\((\\ast)\\)\uc758 \uaf34\ub85c \ub098\ud0c0\ub0a8\uc744 \ubcf4\uc774\uc790. \\(V\\)\uac00 \ub0b4\uc801\uc774 \uc8fc\uc5b4\uc9c4 \ubca1\ud130\uacf5\uac04\uc774\uace0
      \n\\[B = \\left\\{ v_1 ,\\, v_2 ,\\, \\cdots ,\\, v_n \\right\\}\\]
      \n\uc774 \\(V\\)\uc758 \uae30\uc800\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74
      \n\\[c_{ij} = \\langle v_i \\,\\vert\\, v_j \\rangle ,\\quad 1 \\le i \\le n ,\\, 1 \\le j \\le n\\]
      \n\uc774\ub77c\uace0 \uc815\uc758\ud568\uc73c\ub85c\uc368 \ub300\uce6d\ud589\ub82c \\(C = (c_{ij})\\)\ub97c \uc5bb\ub294\ub2e4. \ub354\uc6b1\uc774 \uc9c1\uc811 \uacc4\uc0b0\ud574\ubcf4\uba74 \\(C\\)\uac00 \ub4f1\uc2dd \\((\\ast)\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\uace0, \uc591\uc758 \uc815\ubd80\ud638 \ud589\ub82c\uc784\uc744 \ubcf4\uc77c \uc218 \uc788\ub2e4.
      \n<\/span><\/p>\n<\/div>\n

      <\/p>\n

      \uc9c1\uad50\uae30\uc800<\/h2>\n

      \\(V\\)\uc758 \uae30\uc800\uac00 \uc9c1\uad50\ubca1\ud130\uc871\uc77c \ub54c, \uadf8\ub7ec\ud55c \uae30\uc800\ub97c \uc9c1\uad50\uae30\uc800<\/span>(orthogonal basis)\ub77c\uace0 \ubd80\ub978\ub2e4. \\(V\\)\uc758 \uae30\uc800\uac00 \uc815\uaddc\uc9c1\uad50\uc871\uc77c \ub54c, \uadf8\ub7ec\ud55c \uae30\uc800\ub97c \uc815\uaddc\uc9c1\uad50\uae30\uc800<\/span>(orthonormal basis)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

      \ubca1\ud130\uacf5\uac04\uc5d0 \uc815\uaddc\uc9c1\uad50\uae30\uc800\uac00 \uc8fc\uc5b4\uc838 \uc788\uc73c\uba74 \uadf8 \uae30\uc800\ub97c \uae30\uc900\uc73c\ub85c \ubca1\ud130\uc758 \uc88c\ud45c\ub098 \uae38\uc774\ub97c \ub2e4\ub8f0 \ub54c \ud3b8\ub9ac\ud558\ub2e4.<\/p>\n

      \n

      \uc815\ub9ac 6.<\/span>
      \n\\(u_1,\\) \\(u_2,\\) \\(\\cdots,\\) \\(u_n\\)\uc774 \ub0b4\uc801\uacf5\uac04 \\(V\\)\uc758 \uc815\uaddc\uc9c1\uad50\uae30\uc800\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc784\uc758\uc758 \\(v\\in V\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

        \n
      1. \\(v=\\sum_{j=1}^{n} \\langle v \\,\\vert\\, u_j \\rangle u_j\\)<\/li>\n
      2. \\(\\lvert v \\rvert^2 = \\sum_{j=1}^n \\langle v \\,\\vert\\, u_j \\rangle ^2\\)<\/li>\n<\/ol>\n<\/div>\n
        \n

        \uc99d\uba85.<\/p>\n

        (i) \\(v = a_1 u_1 + a_2 u_2 + \\cdots + a_n u_n\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
        \n\\[\\langle v\\,\\vert\\, u_k \\rangle = \\left\\langle \\sum_{j=1}^n a_j u_j \\,\\bigg\\vert\\, u_k \\right\\rangle = a_k \\langle u_k \\,\\vert\\, u_k \\rangle = a_k.\\]<\/p>\n

        (ii)\ub294 \ud53c\ud0c0\uace0\ub77c\uc2a4 \uc815\ub9ac\ub85c\ubd80\ud130 \uace7\ubc14\ub85c \ub3c4\ucd9c\ub41c\ub2e4.
        \n<\/span><\/p>\n<\/div>\n

        \\(v\\)\uc640 \\(u\\)\uac00 \ub0b4\uc801\uacf5\uac04 \\(V\\)\uc758 \ubca1\ud130\uc774\uace0 \\(u\\)\uac00 \ub2e8\uc704\ubca1\ud130\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(u\\) \uc704\ub85c\uc758 \\(v\\)\uc758 \uc9c1\uad50\uc0ac\uc601<\/span>(orthogonal projection)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
        \n\\[\\proj_u (v) = \\langle v\\,\\vert\\, u \\rangle u.\\]
        \n\\(u\\)\uac00 \ub2e8\uc704\ubca1\ud130\uac00 \uc544\ub2c8\uace0 \uc601\ubca1\ud130\uace0 \uc544\ub2d0 \ub54c\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
        \n\\[\\proj_u (v) = \\frac{\\langle v\\,\\vert\\, u \\rangle}{\\langle u\\,\\vert\\, u \\rangle} u.\\]
        \n\\(u\\)\uac00 \uc601\ubca1\ud130\uc77c \ub54c\ub294 \\(u\\) \uc704\ub85c\uc758 \uc9c1\uad50\uc0ac\uc601\uc774 \uc815\uc758\ub418\uc9c0 \uc54a\ub294\ub2e4.<\/p>\n

        \ub354\uc6b1 \uc77c\ubc18\uc801\uc73c\ub85c, \\(W\\)\uac00 \\(V\\)\uc758 \ubd80\ubd84\uacf5\uac04\uc774\uace0 \\(u_1,\\) \\(\\cdots,\\) \\(u_m\\)\uc774 \\(W\\)\uc758 \uc815\uaddc\uc9c1\uad50\uae30\uc800\uc77c \ub54c \\(W\\) \uc704\ub85c\uc758 \\(v\\)\uc758 \uc9c1\uad50\uc0ac\uc601\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
        \n\\[\\proj_W (v) = \\sum_{j=1}^m \\langle v\\,\\vert\\, u_j \\rangle u_j.\\]\n<\/p>\n

        \n

        \uc815\ub9ac 7.<\/span>
        \n\\(W\\)\uac00 \ub0b4\uc801\uacf5\uac04 \\(V\\)\uc758 \ubd80\ubd84\uacf5\uac04\uc774\uace0 \\(u_1,\\) \\(\\cdots,\\) \\(u_m\\)\uc774 \\(W\\)\uc758 \uc815\uaddc\uc9c1\uad50\uae30\uc800\uc774\uba70 \\(v\\in V\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
        \n\\[(v-\\proj_W (v)) \\bot u_k \\quad ( k=1,\\,2,\\,\\cdots,\\,m )\\]
        \n\uc989 \\(v-\\proj_W (v)\\)\ub294 \\(W\\)\uc5d0 \uc18d\ud558\ub294 \uc784\uc758\uc758 \ubca1\ud130\uc640 \uc11c\ub85c \uc218\uc9c1\uc774\ub2e4.\n<\/p>\n<\/div>\n

        \n

        \uc99d\uba85.<\/p>\n

        \uc815\uc758\uc5d0 \ub530\ub77c \uacc4\uc0b0\ud558\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
        \n\\[\\begin{align}
        \n\\langle v-\\proj_W (v) \\,\\vert\\, u_k \\rangle
        \n&= \\langle v\\,\\vert\\, u_k \\rangle – \\langle \\proj_W (v) \\,\\vert\\, u_k \\rangle \\\\[5pt]
        \n&= \\langle v\\,\\vert\\, u_k \\rangle – \\left\\langle \\sum_{j=1}^m \\langle v\\,\\vert\\,u_j \\rangle u_j \\,\\bigg\\vert\\, u_k \\right\\rangle \\\\[5pt]
        \n&= \\langle v\\,\\vert\\,u_k \\rangle – \\sum_{j=1}^m \\langle v\\,\\vert\\,u_j \\rangle \\langle u_j \\,\\vert\\, u_k \\rangle \\\\[5pt]
        \n&=\\langle v\\,\\vert\\, u_k \\rangle – \\langle v\\,\\vert\\, u_k \\rangle \\langle u_k \\,\\vert\\, u_k \\rangle =0.
        \n\\end{align}\\]
        \n\ub354\uc6b1\uc774 \\(W\\)\uc5d0 \uc18d\ud558\ub294 \ubaa8\ub4e0 \ubca1\ud130\ub294 \\(W\\)\uc758 \uae30\uc800\uc758 \uc77c\ucc28\uacb0\ud569\uc73c\ub85c \ud45c\ud604\ub418\ubbc0\ub85c \\(v\\)\ub294 \\(W\\)\uc5d0 \uc18d\ud558\ub294 \ubaa8\ub4e0 \ubca1\ud130\uc640 \uc11c\ub85c \uc218\uc9c1\uc774\ub2e4.
        \n<\/span><\/p>\n<\/div>\n

        \uc774\ucc98\ub7fc \ubca1\ud130\uacf5\uac04\uc5d0 \uc815\uaddc\uc9c1\uad50\uae30\uc800\uac00 \uc8fc\uc5b4\uc838 \uc788\uc73c\uba74 \ubb34\ucc99 \ud3b8\ub9ac\ud558\ub2e4. \uadf8\ub807\ub2e4\uba74 \ubaa8\ub4e0 \ubca1\ud130 \uacf5\uac04\uc5d0 \uc815\uaddc\uc9c1\uad50\uae30\uc800\uac00 \uc874\uc7ac\ud560\uae4c? \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc758 \uacbd\uc6b0\uc5d0\ub294 \uc815\uaddc\uc9c1\uad50\uae30\uc800\ub97c \uc9c1\uc811 \uad6c\uc131\ud558\ub294 \uc54c\uace0\ub9ac\ub4ec\uc774 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n

        \n

        \uc815\ub9ac 8. (Gram-Schmidt \uc815\uaddc\uc9c1\uad50\ud654 \uacfc\uc815)<\/span><\/p>\n

        \\(V\\)\uac00 \uc720\ud55c\ucc28\uc6d0 \ub0b4\uc801\uacf5\uac04\uc774\uba74 \\(V\\)\uc758 \uc815\uaddc\uc9c1\uad50\uae30\uc800\uac00 \uc874\uc7ac\ud55c\ub2e4.\n<\/p>\n<\/div>\n

        \n

        \uc99d\uba85.<\/p>\n

        \ub9cc\uc57d \\(V\\)\uac00 0\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc774\ub77c\uba74 \uacf5\uc9d1\ud569\uc740 \\(V\\)\uc758 \uc815\uaddc\uc9c1\uad50\uae30\uc800\uc774\ub2e4.<\/p>\n

        \ub9cc\uc57d \\(V\\)\uc758 \ucc28\uc6d0\uc774 \\(1\\)\uc774\ub77c\uba74 \\(V\\)\uc758 \uc601 \uc544\ub2cc \ubca1\ud130 \\(v\\)\ub97c \ud0dd\ud558\uba74 \\(v\/\\lvert v \\rvert\\)\ub294 \\(V\\)\uc758 \uc815\uaddc\uc9c1\uad50\uae30\uc800\uac00 \ub41c\ub2e4.<\/p>\n

        \uc774\uc81c \\(V\\)\uc758 \ucc28\uc6d0\uc774 \\(n\\)\uc774\uace0 \\(n\\ge 2\\)\ub77c\uace0 \ud558\uc790. \uc784\uc758\uc758 \ubca1\ud130\uacf5\uac04\uc740 \uae30\uc800\ub97c \uac00\uc9c0\ubbc0\ub85c, \\(V\\)\uc758 \uae30\uc800\ub97c \\(v_1,\\) \\(\\cdots,\\) \\(v_n\\)\uc73c\ub85c \ub450\uc790. \uadf8\ub9ac\uace0 \\[W_j = \\operatorname{Span}(v_1 ,\\, \\cdots ,\\, v_j)\\]\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74
        \n\\[W_1 \\subseteq W_2 \\subseteq \\cdots \\subseteq W_n = V\\]
        \n\uc774\ub2e4. \uc774\uc81c \\(k = 2 ,\\, \\cdots ,\\, n\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud558\uc790.
        \n\\[\\begin{align}
        \nu_1 &= \\frac{v_1}{\\lvert v_1 \\rvert} , \\\\[5pt]
        \nu_2 &= \\frac{v_2 – \\proj_{W_1}(v_2)}{\\lvert v_2 – \\proj_{W_1}(v_2) \\rvert} , \\\\[5pt]
        \n&\\,\\,\\vdots \\\\[5pt]
        \nu_k &= \\frac{v_k – \\proj_{W_{k-1}} (v_k)}{\\lvert v_k – \\proj_{W_{k-1}} (v_k) \\rvert} .
        \n\\end{align}\\]
        \n\uadf8\ub7ec\uba74 \\(u_1,\\) \\(\\cdots,\\) \\(u_n\\)\uc740 \\(W_n = V\\)\uc758 \uc815\uaddc\uc9c1\uad50\uae30\uc800\uc774\ub2e4.
        \n<\/span><\/p>\n<\/div>\n

        \n

        \ubcf4\uae30 5.<\/span>
        \n\\(\\mathbb{R}^3\\)\uc5d0 \ub2e4\uc74c\uacfc \uac19\uc740 \uc138 \ubca1\ud130\uac00 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790.
        \n\\[v_1 = (2,\\,0,\\,0) ,\\,\\, v_2 = (1,\\,5,\\,0) ,\\,\\, v_3 = (1,\\,2,\\,2).\\]
        \n\uc774\ub4e4 \ubca1\ud130\ub294 \uc77c\ucc28\ub3c5\ub9bd\uc774\ubbc0\ub85c \\(\\mathbb{R}^3\\)\uc758 \uae30\uc800\uac00 \ub41c\ub2e4. \uc774 \uae30\uc800\uc5d0 \uadf8\ub78c-\uc288\ubbf8\uce20 \uc9c1\uad50\ud654\ub97c \uc801\uc6a9\ud558\uc5ec \uc815\uaddc\uc9c1\uad50\uae30\uc800\ub97c \ub9cc\ub4e4\uc5b4 \ubcf4\uc790. \uba3c\uc800
        \n\\[u_1 = \\frac{v_1}{\\lvert v_1 \\rvert} = (1,\\,0,\\,0)\\]
        \n\uc774\ub2e4. \ub2e4\uc74c\uc73c\ub85c
        \n\\[(1,\\,5,\\,0) – 1\\cdot (1,\\,0,\\,0) = (0,\\,5,\\,0)\\]
        \n\uc774\ubbc0\ub85c, \uc774 \ubca1\ud130\ub97c \uc815\uaddc\ud654(\ud06c\uae30\uac00 1\uc778 \ubca1\ud130\ub85c \ub9cc\ub4e6)\ud558\uc5ec
        \n\\[u_2 = (0,\\,1,\\,0)\\]
        \n\uc744 \uc5bb\ub294\ub2e4. \ub05d\uc73c\ub85c
        \n\\[(1,\\,2,\\,2) – 1\\cdot (1,\\,0,\\,0) – 2\\cdot (0,\\,1,\\,0) = (0,\\,0,\\,2)\\]
        \n\uc774\ubbc0\ub85c, \uc774 \ubca1\ud130\ub97c \uc815\uaddc\ud654\ud558\uc5ec
        \n\\[u_3 = (0,\\,0,\\,1)\\]
        \n\uc744 \uc5bb\ub294\ub2e4. \uc774\ub807\uac8c \uc5bb\uc740 \uc138 \ubca1\ud130
        \n\\[u_1 = (1,\\,0,\\,0) ,\\,\\, u_2 = (0,\\,1,\\,0) ,\\,\\, u_3 = (0,\\,0,\\,1)\\]
        \n\uc740 \\(\\mathbb{R}^3\\)\uc758 \uc815\uaddc\uc9c1\uad50\uae30\uc800\uc774\ub2e4. [\uc774 \uae30\uc800\uac00 \ubcf4\ud1b5\uae30\uc800\uc640 \uc77c\uce58\ud558\ub294 \uac83\uc740 \uc6b0\uc5f0\uc77c \ubfd0\uc774\ub2e4.]\n<\/p>\n<\/div>\n

        \n

        \ubcf4\uae30 6.<\/span>
        \n\\(V\\)\uac00 \\(C^0 ([-1,\\,1])\\)\uc758 \ubd80\ubd84\uacf5\uac04\uc774\uace0 \ub2e4\uc74c \uc138 \ubc85\ud130\ub97c \uae30\uc800\ub85c \uac00\uc9c4\ub2e4\uace0 \ud558\uc790.
        \n\\[1,\\,\\,x,\\,\\,x^2 .\\]
        \n\uc774 \uae30\uc800\uc5d0 \uadf8\ub78c-\uc288\ubbf8\uce20 \uc9c1\uad50\ud654\ub97c \uc801\uc6a9\ud558\uc5ec \uace0\ub41c \uacc4\uc0b0\uc744 \ud558\uba74 \ub2e4\uc74c\uacfc \uac19\uc740 \uc138 \ubca1\ud130\ub97c \uc5bb\ub294\ub2e4.
        \n\\[\\frac{\\sqrt{2}}{2} ,\\,\\, \\frac{\\sqrt{6}}{2} x ,\\,\\, \\frac{\\sqrt{10}}{4} (3x^2 – 1).\\]
        \n\uc0ac\uc2e4 \ubaa8\ub4e0 \ub2e8\ud56d\uc2dd\uc744 \uc774\uc640 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \uc815\uaddc\uc9c1\uad50\ud654\ud560 \uc218 \uc788\ub294\ub370, \uadf8\ub807\uac8c \uc5bb\uc5b4\uc9c4 \ub2e4\ud56d\ud568\uc218\uacf5\uac04\uc758 \uae30\uc800\uc758 \uc6d0\uc18c\ub97c \ub974\uc7a5\ub4dc\ub974 \ub2e4\ud56d\uc2dd<\/span>(Legendre polynomial)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.\n<\/p>\n<\/div>\n

        \\(V\\)\uac00 \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc774\uace0 \\(W\\)\uac00 \\(V\\)\uc758 \ubd80\ubd84\uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c
        \n\\[W^\\bot = \\left\\{ v\\in V \\,\\vert\\, \\langle v \\,\\vert\\, w \\rangle = 0 \\text{ for all }w\\in W\\right\\}\\]
        \n\ub294 \\(V\\)\uc758 \ubd80\ubd84\uacf5\uac04\uc774 \ub418\ub294\ub370, \uc774\ub7ec\ud55c \uacf5\uac04\uc744 \\(W\\)\uc758 \uc9c1\uad50\uc5ec\uacf5\uac04<\/span>(orthogonal complement)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

        \n

        \uc815\ub9ac 9.<\/span>
        \n\\(W\\)\uac00 \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04 \\(V\\)\uc758 \ubd80\ubd84\uacf5\uac04\uc77c \ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

          \n
        1. \\(W^\\bot\\)\uc740 \\(V\\)\uc758 \ubd80\ubd84\uacf5\uac04\uc774\ub2e4.<\/li>\n
        2. \\(V = W \\oplus W^\\bot\\)<\/li>\n<\/ol>\n<\/div>\n
          \n

          \uc99d\uba85.<\/p>\n

          (i)\uc740 \uc790\uba85\ud558\ub2e4. \uc65c\ub0d0\ud558\uba74 \\(W^\\bot\\)\ub294 \uc2a4\uce7c\ub77c \uacf1\uacfc \ubca1\ud130 \ud569\uc5d0 \ub300\ud558\uc5ec \ub2eb\ud600\uc788\uae30 \ub54c\ubb38\uc774\ub2e4.<\/p>\n

          (ii) \\(u\\in W \\cap W^\\bot\\)\uc774\uba74 \\(u\\)\ub294 \uc790\uae30 \uc790\uc2e0\uacfc \uc218\uc9c1\uc774\ubbc0\ub85c \\(u=\\mathbf{0}\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(W\\cap W^\\bot = \\left\\{ \\mathbf{0} \\right\\}\\)\uc774\ub2e4. \ub354\uc6b1\uc774 \uc784\uc758\uc758 \\(v\\in V\\)\uc5d0 \ub300\ud558\uc5ec
          \n\\[w = \\proj_W (v) ,\\,\\, w^\\bot = v-w\\]
          \n\ub77c\uace0 \ud558\uba74 \\[v = w + w^\\bot ,\\,\\, w\\in W ,\\,\\, w^\\bot \\in W^\\bot\\]
          \n\uc774\ubbc0\ub85c \\(V = W + W^\\bot\\)\uc774\ub2e4.
          \n<\/span><\/p>\n<\/div>\n

          \n

          \ub530\ub984\uc815\ub9ac 10.<\/span>
          \n\\(V\\)\uac00 \uc720\ud55c\ucc28\uc6d0 \ub0b4\uc801\uacf5\uac04\uc774\uace0 \\(v\\in V\\)\uc774\uba70 \\(W\\)\uac00 \\(V\\)\uc758 \ubd80\ubd84\uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790.
          \n\uc9c1\uad50\uc0ac\uc601 \\(\\proj_W (v)\\)\ub294 \\(W\\)\uc758 \uae30\uc800\uc5d0 \uc0c1\uad00 \uc5c6\uc774 \uc720\uc77c\ud558\uac8c \uc815\ud574\uc9c4\ub2e4.\n<\/p>\n<\/div>\n

          \n

          \uc99d\uba85.<\/p>\n

          \\(w = \\proj_W (v),\\) \\(w^\\bot = v-w\\)\ub77c\uace0 \ud558\uba74
          \n\\[v=w+w^\\bot \\tag{*}\\]
          \n\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(V = W \\oplus W^\\bot\\)\uc774\uace0 \\(w\\in W,\\) \\(w^\\bot \\in W^\\bot\\)\uc774\ubbc0\ub85c (*)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ubca1\ud130 \\(w\\)\ub294 \uc720\uc77c\ud558\uac8c \uc815\ud574\uc9c4\ub2e4.
          \n<\/span><\/p>\n<\/div>\n

          \n

          \ub530\ub984\uc815\ub9ac 11.<\/span>
          \n\\(V\\)\uac00 \uc720\ud55c\ucc28\uc6d0 \ub0b4\uc801\uacf5\uac04\uc774\uace0 \\(v\\in V\\)\uc774\uba70 \\(W\\)\uac00 \\(V\\)\uc758 \ubd80\ubd84\uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. \\(W\\)\uc5d0 \uc18d\ud558\ub294 \ubca1\ud130 \uc911 \\(v\\)\uc5d0 \uac00\uc7a5 \uac00\uae4c\uc6b4 \uac83\uc740 \\(\\proj_W (v)\\)\uc774\ub2e4.\n<\/p>\n<\/div>\n

          \n

          \uc99d\uba85.<\/p>\n

          \\(w = \\proj_W (v),\\) \\(w^\\bot = v-w\\)\ub77c\uace0 \ud558\uba74 \\(v = w+w^\\bot\\)\uc774\ub2e4. \ub354\uc6b1\uc774 \uc774 \ub4f1\uc2dd\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \\(w\\)\uc640 \\(w^\\bot\\)\ub294 \uac01\uac01 \uc720\uc77c\ud558\uac8c \uc815\ud574\uc9c4\ub2e4.
          \n\uc784\uc758\uc758 \\(w ‘ \\in W\\)\uc5d0 \ub300\ud558\uc5ec, \ud53c\ud0c0\uace0\ub77c\uc2a4 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec,
          \n\\[\\lvert w ‘ – v \\rvert^2 = \\lvert (w ‘ – w) – w^\\bot \\rvert^2 = \\lvert w ‘ – w \\rvert^2 + \\lvert w^\\bot \\rvert ^2\\]
          \n\uc778\ub370, \uc774 \uac12\uc740 \\(w ‘ = w\\)\uc77c \ub54c \uac00\uc7a5 \uc791\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(w=\\proj_W (v)\\)\ub294 \\(W\\)\uc758 \ubca1\ud130 \uc911 \\(v\\)\uc5d0 \uac00\uc7a5 \uac00\uae4c\uc6b4 \uc810\uc774\ub2e4.
          \n<\/span><\/p>\n<\/div>\n

          <\/p>\n

          \ubcf5\uc18c\ub0b4\uc801\uacf5\uac04<\/h2>\n

          \ubcf5\uc18c\ub0b4\uc801\uacf5\uac04\uc740 \uc2e4\ub0b4\uc801\uacf5\uac04\uacfc \uc57d\uac04 \ub2e4\ub974\uac8c \uc815\uc758\ub41c\ub2e4. \\(V\\)\uac00 \ubcf5\uc18c\ubca1\ud130\uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. \ud568\uc218
          \n\\[V\\times V \\,\\rightarrow\\,\\mathbb{C} ,\\,\\, (v,\\,w) \\,\\mapsto\\, \\langle v \\,\\vert\\, w \\rangle\\]
          \n\uac00 \ub2e4\uc74c \uc138 \uc870\uac74\uc744 \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0ac \ub54c, \uc774 \ud568\uc218\ub97c \\(V\\) \uc704\uc5d0\uc11c\uc758 \ubcf5\uc18c\ub0b4\uc801<\/span>(complex inner product)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

            \n
          1. \uc784\uc758\uc758 \\(v\\in V\\)\uc5d0 \ub300\ud558\uc5ec \\(\\langle v\\,\\vert\\, v \\rangle \\ge 0\\)\uc774\uba70, \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(v = \\mathbf{0}\\)\uc778 \uac83\uc774\ub2e4.<\/li>\n
          2. \uc784\uc758\uc758 \\(v,\\,w\\in V\\)\uc5d0 \ub300\ud558\uc5ec \\(\\langle v\\,\\vert\\, w\\rangle = \\overline{\\langle w\\,\\vert\\,v\\rangle}\\)\uc774\ub2e4.<\/li>\n
          3. \uc784\uc758\uc758 \\(u,\\,v,\\,w\\in V\\)\uc640 \\(a\\in\\mathbb{C}\\)\uc5d0 \ub300\ud558\uc5ec \\(\\langle u+v \\,\\vert\\, w \\rangle = \\langle u\\,\\vert\\,w\\rangle + \\langle v\\,\\vert\\,w \\rangle\\) \uadf8\ub9ac\uace0 \\(\\langle av \\,\\vert\\, w \\rangle = a\\langle v\\,\\vert\\,w \\rangle\\)\uc774\ub2e4.<\/li>\n<\/ol>\n

            (ii)\uc5d0 \uc758\ud558\uc5ec, \uc784\uc758\uc758 \\(v\\)\uc5d0 \ub300\ud558\uc5ec \\(\\langle v\\,\\vert\\,v\\rangle\\)\uc740 \uc2e4\uc218\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c (i)\uc758 \uc870\uac74\uc774 \uc758\ubbf8\ub97c \uac00\uc9c4\ub2e4.<\/p>\n

            (ii)\uc640 (iii)\uc73c\ub85c\ubd80\ud130 \ub2e4\uc74c\uacfc \uac19\uc740 \ubc18\uc120\ud615 \uc131\uc9c8<\/span>(antilinearity)\uc774 \uc720\ub3c4\ub41c\ub2e4:<\/p>\n

            \n

            \n\uc784\uc758\uc758 \\(u,\\,v,\\,w\\in V\\)\uc5d0 \ub300\ud558\uc5ec \\(\\langle u \\,\\vert\\, v+w \\rangle = \\langle u\\,\\vert\\, v\\rangle + \\langle u \\,\\vert\\,w\\rangle\\)\uc774\uba70,
            \n\uc784\uc758\uc758 \\(v,\\,w\\in V\\)\uc640 \\(a\\in\\mathbb{C}\\)\uc5d0 \ub300\ud558\uc5ec \\(\\langle v\\,\\vert\\, aw \\rangle = \\overline{a} \\langle v\\,\\vert\\,w\\rangle\\)\uc774\ub2e4.\n<\/p>\n<\/div>\n

            \n

            \ubcf4\uae30 7.<\/span>
            \n\\(V = \\mathbb{C}^n\\)\uc77c \ub54c \\(V\\)\uc758 \ubcf4\ud1b5\ub0b4\uc801<\/span>\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
            \n\\[\\langle \\mathbf{x} \\,\\vert\\, \\mathbf{y} \\rangle = \\sum_{j=1}^n x_j \\overline{y_j}.\\]\n<\/p>\n<\/div>\n

            \n

            \ubcf4\uae30 8.<\/span>
            \n\\(V\\)\uac00 \ub2eb\ud78c\uad6c\uac04 \\([a,\\,b]\\)\uc5d0\uc11c \uc815\uc758\ub41c \uc5f0\uc18d\uc778 \ubcf5\uc18c\uc22b\uac12 \ud568\uc218\ub4e4\uc758 \uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. \uc774 \uacf5\uac04\uc5d0\uc11c \ub0b4\uc801\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
            \n\\[\\langle f \\,\\vert\\, g\\rangle = \\int_a^b f(x) \\overline{g(x)} dx .\\]\n<\/p>\n<\/div>\n

            \uc2e4\ub0b4\uc801\uacf5\uac04\uc5d0\uc11c\uc640 \ub9c8\ucc2c\uac00\uc9c0\ub85c \ubcf5\uc18c\ub0b4\uc801\uacf5\uac04\uc5d0\uc11c\ub3c4 \ubca1\ud130\uc758 \uae38\uc774\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
            \n\\[\\lvert v \\rvert = \\sqrt{\\langle v \\,\\vert\\, v \\rangle} .\\]<\/p>\n

            \ubcf5\uc18c\ub0b4\uc801\uacf5\uac04\uc5d0\uc11c\ub3c4 Cauchy-Schwarz \ubd80\ub4f1\uc2dd\uacfc \ud53c\ud0c0\uace0\ub77c\uc2a4 \uc815\ub9ac\uac00 \uc131\ub9bd\ud55c\ub2e4. \uadf8 \uc99d\uba85\uc740 \uc2e4\ub0b4\uc801\uacf5\uac04\uc77c \ub54c\ubcf4\ub2e4 \uc870\uae08 \ub354 \ubcf5\uc7a1\ud558\ub2e4.<\/p>\n

            \n

            \uc815\ub9ac 12. (Cauchy-Schwarz \ubd80\ub4f1\uc2dd)<\/span><\/p>\n

            \\(V\\)\uac00 \ub0b4\uc801\uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(V\\)\uc758 \uc784\uc758\uc758 \uc6d0\uc18c \\(v,\\) \\(w\\)\uc5d0 \ub300\ud558\uc5ec
            \n\\[\\lvert \\langle v \\,\\vert\\, w \\rangle \\rvert \\le \\lvert v \\rvert \\cdot \\lvert w \\rvert\\]
            \n\uac00 \uc131\ub9bd\ud55c\ub2e4.\n<\/p>\n<\/div>\n

            \n

            \uc99d\uba85.<\/p>\n

            \\(w=\\mathbf{0}\\)\uc77c \ub54c\ub294 \uc790\uba85\ud558\uac8c \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(w\\ne\\mathbf{0}\\)\uc774\ub77c\uace0 \ud558\uc790.
            \n\\[\\lambda = \\frac{\\langle v\\,\\vert\\,w\\rangle}{\\lvert w \\rvert^2} ,\\,\\, v ‘ = v – \\lambda w\\tag{*}\\]
            \n\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\lambda w\\)\ub294 \\(v\\)\ub97c \\(w\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub41c \uacf5\uac04\uc5d0 \uc9c1\uad50\uc0ac\uc601\ud55c \uac83\uc774\ubbc0\ub85c \\(\\langle v ‘ \\,\\vert\\, w \\rangle = 0\\)\uc774\ub2e4.
            \n\\(v ‘ \\)\uc758 \uae38\uc774\uc758 \uc81c\uacf1\uc744 \uc804\uac1c\ud558\uba74
            \n\\[\\begin{align}
            \n0 \\le \\lvert v ‘ \\rvert^2
            \n&= \\langle v ‘ \\,\\vert\\, v – \\lambda w \\rangle \\\\[5pt]
            \n&= \\langle v ‘ \\,\\vert\\, v \\rangle \\\\[5pt]
            \n&= \\langle v – \\lambda w \\,\\vert\\, v \\rangle \\\\[5pt]
            \n&= \\lvert v \\rvert ^2 – \\langle \\lambda w \\,\\vert\\, v \\rangle
            \n\\end{align}\\]
            \n\uc774\ubbc0\ub85c
            \n\\[\\lambda \\langle w\\,\\vert\\, v \\rangle \\le \\lvert v \\rvert ^2\\]
            \n\uc774\ub2e4. \uc5ec\uae30\uc5d0\uc11c \\(\\lambda\\)\ub97c (*)\uc758 \uc2dd\uc73c\ub85c \uce58\ud658\ud558\uba74
            \n\\[\\frac{\\langle v\\,\\vert\\,w \\rangle \\langle w\\,\\vert\\,v \\rangle}{\\lvert w \\rvert^2} \\le \\lvert v \\rvert^2\\]
            \n\uc774\uba70, \uc774 \uc2dd\uc744 \ubcc0\ud615\ud558\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
            \n\\[\\langle v \\,\\vert\\, w \\rangle \\langle w \\,\\vert\\,v \\rangle \\le \\lvert v \\rvert^2 \\lvert w \\rvert^2\\tag{**}\\]
            \n\uadf8\ub7f0\ub370 \\(\\langle v \\,\\vert\\,w \\rangle\\)\uc640 \\(\\langle w\\,\\vert\\,v \\rangle\\)\uc740 \uc11c\ub85c \ucf24\ub808\ubcf5\uc18c\uc218\uc774\ubbc0\ub85c (**)\uc758 \uc88c\ubcc0\uc740 \\(\\lvert \\langle v\\,\\vert\\,w \\rangle \\rvert^2\\)\uacfc \uac19\ub2e4. \uc774\ub85c\uc368 (**)\uc758 \uc591\ubcc0\uc758 \uc81c\uacf1\uadfc\uc744 \ucde8\ud558\uba74 \uc6d0\ud558\ub294 \ubd80\ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.
            \n<\/span><\/p>\n<\/div>\n

            \n

            \ub530\ub984\uc815\ub9ac 13. (\uc0bc\uac01\ubd80\ub4f1\uc2dd)<\/span><\/p>\n

            \\(V\\)\uac00 \ub0b4\uc801\uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(V\\)\uc758 \uc784\uc758\uc758 \uc6d0\uc18c \\(v,\\) \\(w\\)\uc5d0 \ub300\ud558\uc5ec
            \n\\[\\lvert v+w \\vert \\le \\lvert v \\rvert + \\lvert w \\rvert\\]
            \n\uac00 \uc131\ub9bd\ud55c\ub2e4.\n<\/p>\n<\/div>\n

            \n

            \uc99d\uba85.<\/p>\n

            \uc784\uc758\uc758 \\(v,\\) \\(w\\)\uc5d0 \ub300\ud558\uc5ec
            \n\\[\\lvert \\operatorname{Re}(\\langle v \\,\\vert\\,w\\rangle)\\rvert \\le \\lvert \\langle v \\,\\vert\\,w \\rangle \\rvert \\le \\lvert v \\rvert \\cdot \\lvert w \\rvert \\]
            \n\uc774\ubbc0\ub85c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
            \n\\[\\begin{align}
            \n\\lvert v+w \\rvert^2
            \n&= \\langle v+w \\,\\vert\\, v+w\\rangle \\\\[5pt]
            \n&= \\langle v\\,\\vert\\,v \\rangle + \\langle v\\,\\vert\\,w \\rangle + \\langle w\\,\\vert\\,v \\rangle + \\langle w\\,\\vert\\,w\\rangle \\\\[5pt]
            \n&= \\langle v\\,\\vert\\,v \\rangle + 2\\operatorname{Re}(\\langle v \\,\\vert\\, w\\rangle ) + \\langle w\\,\\vert\\,w\\rangle \\\\[5pt]
            \n&\\le \\lvert v \\rvert^2 + 2 \\lvert v \\rvert \\lvert w \\rvert + \\lvert w \\rvert^2 \\\\[5pt]
            \n&= ( \\lvert v \\rvert + \\lvert w \\rvert )^2 . \\tag*{\\(\\blacksquare\\)}
            \n\\end{align}\\]\n<\/p>\n<\/div>\n

            \\(v,\\) \\(w\\)\uac00 \ubcf5\uc18c\ub0b4\uc801\uacf5\uac04\uc758 \uc601 \uc544\ub2cc \ubca1\ud130\uc77c \ub54c
            \n\\[\\cos \\theta = \\frac{\\operatorname{Re}(\\langle v \\,\\vert\\,w \\rangle)}{\\lvert v \\rvert \\lvert w \\rvert}\\]
            \n\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc218 \\(\\theta \\in [ 0 ,\\, \\pi ]\\)\ub97c \\(v\\)\uc640 \\(w\\) \uc0ac\uc774\uc758 \uac01<\/span>\uc73c\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n

            \ubcf5\uc18c\ub0b4\uc801\uacf5\uac04\uc5d0\uc11c \ubca1\ud130\uc758 \uc9c1\uad50\uc131\uacfc \uc815\uaddc\uc9c1\uad50\uc131\uc740 \uc2e4\ub0b4\uc801\uacf5\uac04\uc5d0\uc11c\uc640 \ub9c8\ucc2c\uac00\uc9c0\ub85c \uc815\uc758\ub418\uba70, \ud53c\ud0c0\uace0\ub77c\uc2a4 \uc815\ub9ac, \uadf8\ub78c-\uc288\ubbf8\uce20 \uc9c1\uad50\ud654 \uacfc\uc815 \ub4f1 \uc5ec\ub7ec \uac00\uc9c0 \uc131\uc9c8\ub3c4 \uc2e4\ub0b4\uc801\uacf5\uac04\uc5d0\uc11c\uc640 \ub9c8\ucc2c\uac00\uc9c0\ub85c \uc720\ub3c4\ub41c\ub2e4.<\/p>\n

            <\/p>\n

            <\/p>\n","protected":false},"excerpt":{"rendered":"

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