{"id":6988,"date":"2021-07-23T23:22:56","date_gmt":"2021-07-23T14:22:56","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=6988"},"modified":"2026-09-27T17:04:29","modified_gmt":"2026-09-27T08:04:29","slug":"euclidean-spaces","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-calculus\/euclidean-spaces\/","title":{"rendered":"\uc720\ud074\ub9ac\ub4dc \uacf5\uac04"},"content":{"rendered":"<div style=\"display: none; visibility: hidden;\">\n<style type=\"text\/css\">\n\timg.mfp-img { background-color: white; }\n<\/style>\n<p><!--\n\\[\n\\newcommand{\\vecf}{{\\mathbf{f}}}\n\\newcommand{\\vecL}{{\\mathbf{L}}}\n\\newcommand{\\vecR}{{\\mathbb{R}}}\n\\newcommand{\\imI}{\\boldsymbol{i}}\n\\]\n-->\n<\/div>\n<div class=\"box itc_intro\">\n<p>\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 0\uc7a5 8\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.&nbsp; <span class=\"itc_viewcontents\">(<a href=\"..\/\">\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30<\/a>)<\/span><\/p>\n<\/div>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"itc_h2\">\ubca1\ud130\uacf5\uac04<\/h2>\n<p>\\(d\\)\uac00 \uc591\uc758 \uc815\uc218\ub77c\uace0 \ud558\uc790. \uc2e4\uc218 \uc9d1\ud569 \\(\\mathbb{R}\\)\uc758 \\(d\\)\uc911 \ub370\uce74\ub974\ud2b8 \uacf1<br \/>\n\\[<br \/>\n\\mathbb{R}^d<br \/>\n=<br \/>\n\\underbrace{\\mathbb{R}\\times\\mathbb{R}\\times\\cdots\\times\\mathbb{R}}_{d\\text{\uac1c}}<br \/>\n\\]<br \/>\n\uc744 \uc0dd\uac01\ud558\uc790. \uadf8\ub9ac\uace0<br \/>\n\\[<br \/>\n\\mathbf{x}<br \/>\n=<br \/>\n(x_1,\\,x_2,\\,\\cdots,\\,x_d),<br \/>\n\\qquad<br \/>\n\\mathbf{y}<br \/>\n=<br \/>\n(y_1,\\,y_2,\\,\\cdots,\\,y_d)<br \/>\n\\]<br \/>\n\uac00 \\(\\mathbb{R}^d\\)\uc758 \uc6d0\uc18c\uc774\uace0 \\(k\\)\uac00 \uc2e4\uc218\ub77c\uace0 \ud558\uc790. \uc774\ub54c <span class=\"defined\">\ubca1\ud130\ud569<\/span> \\(\\mathbf{x}+\\mathbf{y}\\)\uc640 <span class=\"defined\">\uc2a4\uce7c\ub77c\ubc30<\/span> \\(k\\mathbf{x}\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\n\\mathbf{x}+\\mathbf{y}<br \/>\n&#038;=<br \/>\n(x_1+y_1,\\,x_2+y_2,\\,\\cdots,\\,x_d+y_d),<br \/>\n\\\\[6pt]<br \/>\nk\\mathbf{x}<br \/>\n&#038;=<br \/>\n(kx_1,\\,kx_2,\\,\\cdots,\\,kx_d).<br \/>\n\\end{align}<br \/>\n\\]<br \/>\n\uc774\ub7ec\ud55c \uc5f0\uc0b0\uc774 \uc8fc\uc5b4\uc9c4 \\(\\mathbb{R}^d\\)\ub97c \uc2e4\uc218 \uc704\uc758 \ubca1\ud130\uacf5\uac04\uc73c\ub85c \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4. \\(\\mathbb{R}^d\\)\uc758 \uc6d0\uc18c\ub97c <span class=\"defined\">\ubca1\ud130<\/span>(vector)\ub77c\uace0 \ubd80\ub974\uace0, \uc2e4\uc218\ub97c <span class=\"defined\">\uc2a4\uce7c\ub77c<\/span>(scalar)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\\(\\mathbf{v}\\in\\mathbb{R}^d\\)\ub77c\uace0 \ud558\uc790. \ud3b8\uc758\uc0c1 \\((-1)\\mathbf{v}\\)\ub97c \uac04\ub2e8\ud788 \\(-\\mathbf{v}\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ub610\ud55c \ubaa8\ub4e0 \uc131\ubd84\uc774 \\(0\\)\uc778 \ubca1\ud130<br \/>\n\\[<br \/>\n\\mathbf{0}<br \/>\n=<br \/>\n(0,\\,0,\\,\\cdots,\\,0)<br \/>\n\\]<br \/>\n\uc744 <span class=\"defined\">\uc601\ubca1\ud130<\/span>(zero vector)\ub77c\uace0 \ubd80\ub978\ub2e4. \\(\\mathbf{0}\\)\uc740 \ubca1\ud130\uc758 \ub367\uc148\uc5d0 \ub300\ud55c \ud56d\ub4f1\uc6d0\uc774\uace0, \\(-\\mathbf{v}\\)\ub294 \ub367\uc148\uc5d0 \ub300\ud55c \\(\\mathbf{v}\\)\uc758 \uc5ed\uc6d0\uc774\ub2e4.<\/p>\n<p>\\(\\mathbf{x},\\mathbf{y},\\mathbf{z}\\in\\mathbb{R}^d\\)\uc774\uace0 \\(a,b\\in\\mathbb{R}\\)\uc77c \ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ul>\n<li>\\(\\mathbf{x}+(\\mathbf{y}+\\mathbf{z})=(\\mathbf{x}+\\mathbf{y})+\\mathbf{z}\\).<\/li>\n<li>\\(\\mathbf{x}+\\mathbf{y}=\\mathbf{y}+\\mathbf{x}\\).<\/li>\n<li>\\(\\mathbf{x}+\\mathbf{0}=\\mathbf{x}\\).<\/li>\n<li>\\(\\mathbf{x}+(-\\mathbf{x})=\\mathbf{0}\\).<\/li>\n<li>\\(a(b\\mathbf{x})=(ab)\\mathbf{x}\\).<\/li>\n<li>\\(1\\mathbf{x}=\\mathbf{x}\\).<\/li>\n<li>\\(a(\\mathbf{x}+\\mathbf{y})=a\\mathbf{x}+a\\mathbf{y}\\).<\/li>\n<li>\\((a+b)\\mathbf{x}=a\\mathbf{x}+b\\mathbf{x}\\).<\/li>\n<\/ul>\n<p>\uac01 \ub4f1\uc2dd\uc740 \ubca1\ud130\uc758 \uac01 \uc131\ubd84\uc5d0 \uc2e4\uc218\uc758 \ub367\uc148\uacfc \uacf1\uc148\uc5d0 \uad00\ud55c \ubc95\uce59\uc744 \uc801\uc6a9\ud558\uba74 \uc9c1\uc811 \ud655\uc778\ud560 \uc218 \uc788\ub2e4.<\/p>\n<p>\uc774\ub97c \uc77c\ubc18\ud654\ud560 \uc218 \uc788\ub2e4. \\(K\\)\uac00 \uccb4(field)\ub77c\uace0 \ud558\uc790. \uc5ec\uae30\uc11c \uccb4\ub780 \ub367\uc148\uacfc \uacf1\uc148\uc774 \uc815\uc758\ub418\uc5b4 \uc788\uace0, \ub367\uc148\uacfc \uacf1\uc148\uc758 \ud56d\ub4f1\uc6d0\uacfc \uc5ed\uc6d0, \ubd84\ubc30\ubc95\uce59 \ub4f1 \uccb4\uc758 \uacf5\ub9ac\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc9d1\ud569\uc774\ub2e4. \uc9d1\ud569 \\(V\\)\uc5d0 \ub450 \uc5f0\uc0b0<br \/>\n\\[<br \/>\n+ : V\\times V\\rightarrow V,<br \/>\n\\qquad<br \/>\n\\cdot : K\\times V\\rightarrow V<br \/>\n\\]<br \/>\n\uac00 \uc8fc\uc5b4\uc838 \uc788\uace0 \uc704\uc758 \uc5ec\ub35f \uc131\uc9c8\uc744 \ub9cc\uc871\uc2dc\ud0a4\uba74 \\(V\\)\ub97c <span class=\"defined\">\\(\\boldsymbol{K}\\) \uc704\uc758 \ubca1\ud130\uacf5\uac04<\/span>(vector space over \\(K\\))\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774\ub54c \\(V\\)\uc758 \uc6d0\uc18c\ub97c \ubca1\ud130, \\(K\\)\uc758 \uc6d0\uc18c\ub97c \uc2a4\uce7c\ub77c\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"itc_h2\">\uae30\uc800\uc640 \ucc28\uc6d0<\/h2>\n<p>\\(V\\)\uac00 \\(K\\) \uc704\uc758 \ubca1\ud130\uacf5\uac04\uc774\uace0<br \/>\n\\[<br \/>\n\\mathbf{v}_1,\\,<br \/>\n\\mathbf{v}_2,\\,<br \/>\n\\ldots,\\,<br \/>\n\\mathbf{v}_d<br \/>\n\\]<br \/>\n\uac00 \\(V\\)\uc758 \ubca1\ud130\ub77c\uace0 \ud558\uc790. \\(V\\)\uc758 \ubaa8\ub4e0 \ubca1\ud130 \\(\\mathbf{v}\\)\uac00<br \/>\n\\[<br \/>\n\\mathbf{v}<br \/>\n=<br \/>\na_1\\mathbf{v}_1<br \/>\n+<br \/>\na_2\\mathbf{v}_2<br \/>\n+<br \/>\n\\cdots<br \/>\n+<br \/>\na_d\\mathbf{v}_d<br \/>\n\\]<br \/>\n\uc758 \uaf34\ub85c \uc720\uc77c\ud558\uac8c \ud45c\ud604\ub418\uba74<br \/>\n\\[<br \/>\n\\mathbf{v}_1,\\,<br \/>\n\\mathbf{v}_2,\\,<br \/>\n\\ldots,\\,<br \/>\n\\mathbf{v}_d<br \/>\n\\]<br \/>\n\ub97c \\(V\\)\uc758 <span class=\"defined\">\uae30\uc800<\/span>(basis)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\uc720\ud55c \uac1c\uc758 \ubca1\ud130\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uae30\uc800\ub97c \uac16\ub294 \ubca1\ud130\uacf5\uac04\uc744 <span class=\"defined\">\uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04<\/span>(finite-dimensional vector space)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc5d0\uc11c\ub294 \uc5b4\ub5a4 \uae30\uc800\ub97c \ud0dd\ud558\ub354\ub77c\ub3c4 \uae30\uc800\ub97c \uc774\ub8e8\ub294 \ubca1\ud130\uc758 \uac1c\uc218\uac00 \uac19\ub2e4. \uc774 \uc0ac\uc2e4\uc740 \uc120\ud615\ub300\uc218\ud559\uc758 \uae30\ubcf8\uc815\ub9ac \uac00\uc6b4\ub370 \ud558\ub098\uc774\uba70, \uc5ec\uae30\uc11c\ub294 \uc99d\uba85 \uc5c6\uc774 \uc0ac\uc6a9\ud55c\ub2e4. \ub530\ub77c\uc11c \uae30\uc800\ub97c \uc774\ub8e8\ub294 \ubca1\ud130\uc758 \uac1c\uc218\ub97c \\(V\\)\uc758 <span class=\"defined\">\ucc28\uc6d0<\/span>(dimension)\uc774\ub77c\uace0 \ubd80\ub974\uace0<br \/>\n\\[<br \/>\n\\dim V<br \/>\n\\]<br \/>\n\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<p>\uc601\ubca1\ud130\ub9cc\uc73c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \ubca1\ud130\uacf5\uac04 \\(\\{\\mathbf{0}\\}\\)\uc5d0\ub294 \uacf5\uc9d1\ud569\uc744 \uae30\uc800\ub85c \ub450\uace0 \uadf8 \ucc28\uc6d0\uc744 \\(0\\)\uc73c\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n<p>\\(\\mathbb{R}^d\\)\uc5d0\uc11c \\(i\\)\ubc88\uc9f8 \uc131\ubd84\ub9cc \\(1\\)\uc774\uace0 \ub098\uba38\uc9c0 \uc131\ubd84\uc740 \ubaa8\ub450 \\(0\\)\uc778 \ubca1\ud130\ub97c<br \/>\n\\[<br \/>\n\\mathbf{e}_i<br \/>\n=<br \/>\n(0,\\,\\ldots,\\,0,\\,1,\\,0,\\,\\ldots,\\,0)<br \/>\n\\]<br \/>\n\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74<br \/>\n\\[<br \/>\n\\mathbf{e}_1,\\,<br \/>\n\\mathbf{e}_2,\\,<br \/>\n\\ldots,\\,<br \/>\n\\mathbf{e}_d<br \/>\n\\]<br \/>\n\ub97c \\(\\mathbb{R}^d\\)\uc758 <span class=\"defined\">\ud45c\uc900\uae30\uc800<\/span>(standard basis)\ub77c\uace0 \ubd80\ub978\ub2e4. \uc2e4\uc81c\ub85c<br \/>\n\\[<br \/>\n\\mathbf{x}<br \/>\n=<br \/>\n(x_1,\\,x_2,\\,\\ldots,\\,x_d)<br \/>\n\\]<br \/>\n\uc774\uba74<br \/>\n\\[<br \/>\n\\mathbf{x}<br \/>\n=<br \/>\nx_1\\mathbf{e}_1<br \/>\n+<br \/>\nx_2\\mathbf{e}_2<br \/>\n+<br \/>\n\\cdots<br \/>\n+<br \/>\nx_d\\mathbf{e}_d<br \/>\n\\]<br \/>\n\uc774\uace0, \uc774\ub7ec\ud55c \ud45c\ud604\uc740 \uc720\uc77c\ud558\ub2e4. \ub530\ub77c\uc11c<br \/>\n\\[<br \/>\n\\dim\\mathbb{R}^d=d.<br \/>\n\\]<br \/>\n\ud45c\uc900\uae30\uc800\uc758 \uac01 \ubca1\ud130\ub97c <span class=\"defined\">\ud45c\uc900\ub2e8\uc704\ubca1\ud130<\/span>(standard unit vector)\ub77c\uace0 \ubd80\ub974\uae30\ub3c4 \ud55c\ub2e4.<\/p>\n<p>\uc608\ub97c \ub4e4\uc5b4<br \/>\n\\[<br \/>\n\\mathbb{R}^2:<br \/>\n\\qquad<br \/>\n\\mathbf{e}_1=(1,\\,0),<br \/>\n\\qquad<br \/>\n\\mathbf{e}_2=(0,\\,1),<br \/>\n\\]<br \/>\n\uc774\uace0<br \/>\n\\[<br \/>\n\\mathbb{R}^3:<br \/>\n\\qquad<br \/>\n\\mathbf{e}_1=(1,\\,0,\\,0),<br \/>\n\\qquad<br \/>\n\\mathbf{e}_2=(0,\\,1,\\,0),<br \/>\n\\qquad<br \/>\n\\mathbf{e}_3=(0,\\,0,\\,1)<br \/>\n\\]<br \/>\n\uc774\ub2e4.<\/p>\n<p>\ubb34\ud55c \uac1c\uc758 \ubca1\ud130\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uae30\uc800\ub3c4 \uc815\uc758\ud560 \uc218 \uc788\uc73c\uba70, \uc774\uc5d0 \ub530\ub77c \ubb34\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\ub3c4 \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4. \uadf8\ub7ec\ub098 \uadf8\uc640 \uad00\ub828\ub41c \ub17c\uc758\ub294 \uc774 \ucc45\uc758 \ubc94\uc704\ub97c \ubc97\uc5b4\ub098\ubbc0\ub85c \uc5ec\uae30\uc11c\ub294 \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\ub9cc \ub2e4\ub8ec\ub2e4.<\/p>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"itc_h2\">\uc120\ud615\ubcc0\ud658<\/h2>\n<p>\\(V\\)\uc640 \\(W\\)\uac00 \uac19\uc740 \uccb4 \\(K\\) \uc704\uc758 \ubca1\ud130\uacf5\uac04\uc774\uace0 \\(T:V\\rightarrow W\\)\uac00 \ud568\uc218\ub77c\uace0 \ud558\uc790. \\(V\\)\uc758 \uc784\uc758\uc758 \ubca1\ud130 \\(\\mathbf{x},\\mathbf{y}\\)\uc640 \uc2a4\uce7c\ub77c \\(k\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\nT(\\mathbf{x}+\\mathbf{y})<br \/>\n&#038;=<br \/>\nT(\\mathbf{x})+T(\\mathbf{y}),<br \/>\n\\\\[6pt]<br \/>\nT(k\\mathbf{x})<br \/>\n&#038;=<br \/>\nkT(\\mathbf{x})<br \/>\n\\end{align}<br \/>\n\\]<br \/>\n\uac00 \uc131\ub9bd\ud558\uba74 \\(T\\)\ub97c <span class=\"defined\">\uc120\ud615\ubcc0\ud658<\/span>(linear transformation)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\\(T:V\\rightarrow W\\)\uac00 \uc77c\ub300\uc77c \ub300\uc751\uc778 \uc120\ud615\ubcc0\ud658\uc774\uba74 \\(T\\)\ub97c <span class=\"defined\">\ub3d9\ud615\uc0ac\uc0c1<\/span>(isomorphism)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \\(V\\)\uc640 \\(W\\) \uc0ac\uc774\uc5d0 \ub3d9\ud615\uc0ac\uc0c1\uc774 \uc874\uc7ac\ud558\uba74 \\(V\\)\uc640 \\(W\\)\uac00 <span class=\"defined\">\ub3d9\ud615<\/span>(isomorphic)\uc774\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac. (\ub3d9\ud615\uc778 \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc758 \ucc28\uc6d0)<\/span><\/p>\n<p>\uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04 \\(V\\)\uc640 \\(W\\)\uac00 \ub3d9\ud615\uc774\uba74<br \/>\n\\[<br \/>\n\\dim V=\\dim W<br \/>\n\\]<br \/>\n\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p>\\(T:V\\rightarrow W\\)\uac00 \ub3d9\ud615\uc0ac\uc0c1\uc774\uace0<br \/>\n\\[<br \/>\n\\mathbf{v}_1,\\,<br \/>\n\\ldots,\\,<br \/>\n\\mathbf{v}_d<br \/>\n\\]<br \/>\n\uac00 \\(V\\)\uc758 \uae30\uc800\ub77c\uace0 \ud558\uc790. \\(T\\)\uac00 \uc704\ub85c\uc758 \ud568\uc218\uc774\ubbc0\ub85c \uc784\uc758\uc758 \\(\\mathbf{w}\\in W\\)\uc5d0 \ub300\ud558\uc5ec \\(T(\\mathbf{v})=\\mathbf{w}\\)\uc778 \\(\\mathbf{v}\\in V\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uc5b4\ub5a4 \uc2a4\uce7c\ub77c \\(a_1,\\ldots,a_d\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\n\\mathbf{v}<br \/>\n=<br \/>\na_1\\mathbf{v}_1+\\cdots+a_d\\mathbf{v}_d<br \/>\n\\]<br \/>\n\uc774\ubbc0\ub85c \\(T\\)\uc758 \uc120\ud615\uc131\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[<br \/>\n\\mathbf{w}<br \/>\n=<br \/>\na_1T(\\mathbf{v}_1)<br \/>\n+\\cdots+<br \/>\na_dT(\\mathbf{v}_d).<br \/>\n\\]\n<\/p>\n<p>\ub610\ud55c<br \/>\n\\[<br \/>\na_1T(\\mathbf{v}_1)+\\cdots+a_dT(\\mathbf{v}_d)<br \/>\n=<br \/>\nb_1T(\\mathbf{v}_1)+\\cdots+b_dT(\\mathbf{v}_d)<br \/>\n\\]<br \/>\n\ub77c\uace0 \ud558\uc790. \uc120\ud615\uc131\uc744 \uc774\uc6a9\ud558\uba74<br \/>\n\\[<br \/>\nT\\left(<br \/>\n(a_1-b_1)\\mathbf{v}_1<br \/>\n+\\cdots+<br \/>\n(a_d-b_d)\\mathbf{v}_d<br \/>\n\\right)<br \/>\n=<br \/>\nT(\\mathbf{0}).<br \/>\n\\]<br \/>\n\\(T\\)\uac00 \uc77c\ub300\uc77c \ud568\uc218\uc774\ubbc0\ub85c<br \/>\n\\[<br \/>\n(a_1-b_1)\\mathbf{v}_1<br \/>\n+\\cdots+<br \/>\n(a_d-b_d)\\mathbf{v}_d<br \/>\n=<br \/>\n\\mathbf{0}.<br \/>\n\\]<br \/>\n\uae30\uc800\uc5d0 \uc758\ud55c \ud45c\ud604\uc758 \uc720\uc77c\uc131\uc5d0 \ub530\ub77c<br \/>\n\\[<br \/>\na_i=b_i<br \/>\n\\qquad<br \/>\n(i=1,\\ldots,d)<br \/>\n\\]<br \/>\n\uc774\ub2e4. \ub530\ub77c\uc11c<br \/>\n\\[<br \/>\nT(\\mathbf{v}_1),\\ldots,T(\\mathbf{v}_d)<br \/>\n\\]<br \/>\n\ub294 \\(W\\)\uc758 \uae30\uc800\uc774\uace0<br \/>\n\\[<br \/>\n\\dim W=d=\\dim V<br \/>\n\\]<br \/>\n\uc774\ub2e4. <span class=\"qed\"><\/span><\/p>\n<\/div>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"itc_h2\">\uc810\uacf1\uacfc \uc720\ud074\ub9ac\ub4dc \uacf5\uac04<\/h2>\n<p>\\(\\mathbf{x},\\mathbf{y}\\in\\mathbb{R}^d\\)\uc774\uace0<br \/>\n\\[<br \/>\n\\mathbf{x}<br \/>\n=<br \/>\n(x_1,\\,x_2,\\,\\ldots,\\,x_d),<br \/>\n\\qquad<br \/>\n\\mathbf{y}<br \/>\n=<br \/>\n(y_1,\\,y_2,\\,\\ldots,\\,y_d)<br \/>\n\\]<br \/>\n\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(\\mathbf{x}\\)\uc640 \\(\\mathbf{y}\\)\uc758 <span class=\"defined\">\uc810\uacf1<\/span>(dot product) \ub610\ub294 <span class=\"defined\">\ud45c\uc900\ub0b4\uc801<\/span>(standard inner product)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[<br \/>\n\\mathbf{x}\\cdot\\mathbf{y}<br \/>\n=<br \/>\nx_1y_1+x_2y_2+\\cdots+x_dy_d.<br \/>\n\\]\n<\/p>\n<p>\ud45c\uc900\ub0b4\uc801\uc774 \uc8fc\uc5b4\uc9c4 \ubca1\ud130\uacf5\uac04 \\(\\mathbb{R}^d\\)\ub97c <span class=\"defined\">\\(\\boldsymbol{d}\\)\ucc28\uc6d0 \uc720\ud074\ub9ac\ub4dc \uacf5\uac04<\/span>(Euclidean \\(d\\)-space) \ub610\ub294 \uac04\ub2e8\ud788 <span class=\"defined\">\uc720\ud074\ub9ac\ub4dc \uacf5\uac04<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\ubca1\ud130 \\(\\mathbf{x}\\)\uc758 <span class=\"defined\">\ub178\ub984<\/span>(norm) \ub610\ub294 <span class=\"defined\">\uae38\uc774<\/span>(length)\ub97c<br \/>\n\\[<br \/>\n\\lVert\\mathbf{x}\\rVert<br \/>\n=<br \/>\n\\sqrt{\\mathbf{x}\\cdot\\mathbf{x}}<br \/>\n=<br \/>\n\\sqrt{x_1^2+x_2^2+\\cdots+x_d^2}<br \/>\n\\]<br \/>\n\ub85c \uc815\uc758\ud55c\ub2e4. \uae38\uc774\uac00 \\(1\\)\uc778 \ubca1\ud130\ub97c <span class=\"defined\">\ub2e8\uc704\ubca1\ud130<\/span>(unit vector)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\ub178\ub984\uc740 \uc784\uc758\uc758 \\(\\mathbf{x}\\in\\mathbb{R}^d\\)\uc640 \\(a\\in\\mathbb{R}\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\n\\lVert\\mathbf{x}\\rVert\\ge0,<br \/>\n\\qquad<br \/>\n\\lVert\\mathbf{x}\\rVert=0<br \/>\n\\Longleftrightarrow<br \/>\n\\mathbf{x}=\\mathbf{0},<br \/>\n\\qquad<br \/>\n\\lVert a\\mathbf{x}\\rVert<br \/>\n=<br \/>\n|a|\\,\\lVert\\mathbf{x}\\rVert<br \/>\n\\]<br \/>\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac. (Cauchy-Schwarz \ubd80\ub4f1\uc2dd\uacfc \uc0bc\uac01\ubd80\ub4f1\uc2dd)<\/span><\/p>\n<p>\\(\\mathbf{x},\\mathbf{y}\\in\\mathbb{R}^d\\)\uc774\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[<br \/>\n|\\mathbf{x}\\cdot\\mathbf{y}|<br \/>\n\\le<br \/>\n\\lVert\\mathbf{x}\\rVert<br \/>\n\\lVert\\mathbf{y}\\rVert<br \/>\n\\]<br \/>\n\uadf8\ub9ac\uace0<br \/>\n\\[<br \/>\n\\lVert\\mathbf{x}+\\mathbf{y}\\rVert<br \/>\n\\le<br \/>\n\\lVert\\mathbf{x}\\rVert+\\lVert\\mathbf{y}\\rVert.<br \/>\n\\]\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p>\uba3c\uc800 \\(\\mathbf{y}=\\mathbf{0}\\)\uc774\uba74 \uccab \ubc88\uc9f8 \ubd80\ub4f1\uc2dd\uc740 \uc790\uba85\ud558\ub2e4. \uc774\uc81c \\(\\mathbf{y}\\ne\\mathbf{0}\\)\uc774\ub77c\uace0 \ud558\uc790. \uc2e4\uc218<br \/>\n\\[<br \/>\nt<br \/>\n=<br \/>\n\\frac{\\mathbf{x}\\cdot\\mathbf{y}}<br \/>\n{\\mathbf{y}\\cdot\\mathbf{y}}<br \/>\n\\]<br \/>\n\ub97c \uc0dd\uac01\ud558\uba74<br \/>\n\\[<br \/>\n0<br \/>\n\\le<br \/>\n\\lVert\\mathbf{x}-t\\mathbf{y}\\rVert^2.<br \/>\n\\]<br \/>\n\uc624\ub978\ucabd\uc744 \uc804\uac1c\ud558\uba74<br \/>\n\\[<br \/>\n\\begin{aligned}<br \/>\n0<br \/>\n&#038;\\le<br \/>\n\\lVert\\mathbf{x}\\rVert^2<br \/>\n&#8211;<br \/>\n2t(\\mathbf{x}\\cdot\\mathbf{y})<br \/>\n+<br \/>\nt^2\\lVert\\mathbf{y}\\rVert^2<br \/>\n\\\\<br \/>\n&#038;=<br \/>\n\\lVert\\mathbf{x}\\rVert^2<br \/>\n&#8211;<br \/>\n\\frac{(\\mathbf{x}\\cdot\\mathbf{y})^2}<br \/>\n{\\lVert\\mathbf{y}\\rVert^2}.<br \/>\n\\end{aligned}<br \/>\n\\]<br \/>\n\ub530\ub77c\uc11c<br \/>\n\\[<br \/>\n(\\mathbf{x}\\cdot\\mathbf{y})^2<br \/>\n\\le<br \/>\n\\lVert\\mathbf{x}\\rVert^2<br \/>\n\\lVert\\mathbf{y}\\rVert^2,<br \/>\n\\]<br \/>\n\uc989<br \/>\n\\[<br \/>\n|\\mathbf{x}\\cdot\\mathbf{y}|<br \/>\n\\le<br \/>\n\\lVert\\mathbf{x}\\rVert<br \/>\n\\lVert\\mathbf{y}\\rVert<br \/>\n\\]<br \/>\n\uc774\ub2e4.<\/p>\n<p>\uc774\uc81c<br \/>\n\\[<br \/>\n\\begin{aligned}<br \/>\n\\lVert\\mathbf{x}+\\mathbf{y}\\rVert^2<br \/>\n&#038;=<br \/>\n\\lVert\\mathbf{x}\\rVert^2<br \/>\n+<br \/>\n2\\mathbf{x}\\cdot\\mathbf{y}<br \/>\n+<br \/>\n\\lVert\\mathbf{y}\\rVert^2<br \/>\n\\\\<br \/>\n&#038;\\le<br \/>\n\\lVert\\mathbf{x}\\rVert^2<br \/>\n+<br \/>\n2\\lVert\\mathbf{x}\\rVert<br \/>\n\\lVert\\mathbf{y}\\rVert<br \/>\n+<br \/>\n\\lVert\\mathbf{y}\\rVert^2<br \/>\n\\\\<br \/>\n&#038;=<br \/>\n\\left(<br \/>\n\\lVert\\mathbf{x}\\rVert<br \/>\n+<br \/>\n\\lVert\\mathbf{y}\\rVert<br \/>\n\\right)^2.<br \/>\n\\end{aligned}<br \/>\n\\]<br \/>\n\uc591\ubcc0\uc774 \uc74c\uc774 \uc544\ub2c8\ubbc0\ub85c \uc81c\uacf1\uadfc\uc744 \ucde8\ud558\uba74<br \/>\n\\[<br \/>\n\\lVert\\mathbf{x}+\\mathbf{y}\\rVert<br \/>\n\\le<br \/>\n\\lVert\\mathbf{x}\\rVert+\\lVert\\mathbf{y}\\rVert<br \/>\n\\]<br \/>\n\uc774\ub2e4. <span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\\(\\mathbf{x}\\)\uc640 \\(\\mathbf{y}\\)\uac00 \uc601\ubca1\ud130\uac00 \uc544\ub2c8\ub77c\uace0 \ud558\uc790. Cauchy-Schwarz \ubd80\ub4f1\uc2dd\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[<br \/>\n-1<br \/>\n\\le<br \/>\n\\frac{\\mathbf{x}\\cdot\\mathbf{y}}<br \/>\n{\\lVert\\mathbf{x}\\rVert\\lVert\\mathbf{y}\\rVert}<br \/>\n\\le<br \/>\n1.<br \/>\n\\]<br \/>\n\ub610\ud55c \ucf54\uc0ac\uc778\ud568\uc218\ub294 \uad6c\uac04 \\([0,\\pi]\\)\uc5d0\uc11c \\([-1,1]\\)\uc758 \uac01 \uac12\uc744 \uc815\ud655\ud788 \ud55c \ubc88\uc529 \uac16\ub294\ub2e4. \uc774 \uc0bc\uac01\ud568\uc218\uc758 \uc131\uc9c8\uc744 \uc774\uc6a9\ud558\uc5ec<br \/>\n\\[<br \/>\n\\cos\\theta<br \/>\n=<br \/>\n\\frac{\\mathbf{x}\\cdot\\mathbf{y}}<br \/>\n{\\lVert\\mathbf{x}\\rVert\\lVert\\mathbf{y}\\rVert},<br \/>\n\\qquad<br \/>\n0\\le\\theta\\le\\pi<br \/>\n\\]<br \/>\n\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc720\uc77c\ud55c \\(\\theta\\)\ub97c \\(\\mathbf{x}\\)\uc640 \\(\\mathbf{y}\\) \uc0ac\uc774\uc758 <span class=\"defined\">\uac01<\/span>\uc774\ub77c\uace0 \uc815\uc758\ud55c\ub2e4. \uc989<br \/>\n\\[<br \/>\n\\mathbf{x}\\cdot\\mathbf{y}<br \/>\n=<br \/>\n\\lVert\\mathbf{x}\\rVert<br \/>\n\\lVert\\mathbf{y}\\rVert<br \/>\n\\cos\\theta.<br \/>\n\\]<br \/>\n\\(d=2\\) \ub610\ub294 \\(d=3\\)\uc77c \ub54c \uc774 \uc815\uc758\ub294 \uae30\ud558\ud559\uc801\uc778 \ub450 \ubca1\ud130 \uc0ac\uc774\uc758 \uac01\uacfc \uc77c\uce58\ud55c\ub2e4.<\/p>\n<p>\\(\\mathbf{x}\\cdot\\mathbf{y}=0\\)\uc774\uba74 \\(\\mathbf{x}\\)\uc640 \\(\\mathbf{y}\\)\uac00 <span class=\"defined\">\uc9c1\uad50\ud55c\ub2e4<\/span>(orthogonal)\uace0 \ub9d0\ud558\uace0<br \/>\n\\[<br \/>\n\\mathbf{x}\\perp\\mathbf{y}<br \/>\n\\]<br \/>\n\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4. \uc774 \uc815\uc758\uc5d0 \ub530\ub974\uba74 \uc601\ubca1\ud130\ub294 \ubaa8\ub4e0 \ubca1\ud130\uc640 \uc9c1\uad50\ud55c\ub2e4.<\/p>\n<p>\uc5b4\ub5a4 \uc2a4\uce7c\ub77c \\(k\\)\uac00 \uc874\uc7ac\ud558\uc5ec<br \/>\n\\[<br \/>\n\\mathbf{x}=k\\mathbf{y}<br \/>\n\\]<br \/>\n\ub610\ub294<br \/>\n\\[<br \/>\n\\mathbf{y}=k\\mathbf{x}<br \/>\n\\]<br \/>\n\uac00 \uc131\ub9bd\ud558\uba74 \\(\\mathbf{x}\\)\uc640 \\(\\mathbf{y}\\)\uac00 <span class=\"defined\">\ud3c9\ud589\ud558\ub2e4<\/span>(parallel)\uace0 \ub9d0\ud55c\ub2e4. \uc774 \ucc45\uc5d0\uc11c\ub294 \uc601\ubca1\ud130\ub3c4 \ubaa8\ub4e0 \ubca1\ud130\uc640 \ud3c9\ud589\ud55c \uac83\uc73c\ub85c \uc57d\uc18d\ud55c\ub2e4. \ub450 \ubca1\ud130\uac00 \ud3c9\ud589\ud558\uba74<br \/>\n\\[<br \/>\n\\mathbf{x}\\cdot\\mathbf{y}<br \/>\n=<br \/>\n\\pm<br \/>\n\\lVert\\mathbf{x}\\rVert<br \/>\n\\lVert\\mathbf{y}\\rVert<br \/>\n\\]<br \/>\n\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"itc_h2\">\uac00\uc704\uacf1<\/h2>\n<p>\\(\\mathbf{x},\\mathbf{y}\\in\\mathbb{R}^3\\)\uc774\uace0<br \/>\n\\[<br \/>\n\\mathbf{x}<br \/>\n=<br \/>\n(x_1,\\,x_2,\\,x_3),<br \/>\n\\qquad<br \/>\n\\mathbf{y}<br \/>\n=<br \/>\n(y_1,\\,y_2,\\,y_3)<br \/>\n\\]<br \/>\n\ub77c\uace0 \ud558\uc790. \ub450 \ubca1\ud130\uc758 <span class=\"defined\">\uac00\uc704\uacf1<\/span>(cross product) \ub610\ub294 <span class=\"defined\">\uc678\uc801<\/span>\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[<br \/>\n\\mathbf{x}\\times\\mathbf{y}<br \/>\n=<br \/>\n\\left(<br \/>\nx_2y_3-x_3y_2,\\,<br \/>\nx_3y_1-x_1y_3,\\,<br \/>\nx_1y_2-x_2y_1<br \/>\n\\right).<br \/>\n\\]\n<\/p>\n<p>\uac00\uc704\uacf1\uc740 \ub450 \ubca1\ud130\uc5d0 \ubaa8\ub450 \uc9c1\uad50\ud55c\ub2e4. \uc2e4\uc81c\ub85c<br \/>\n\\[<br \/>\n\\begin{aligned}<br \/>\n(\\mathbf{x}\\times\\mathbf{y})\\cdot\\mathbf{x}<br \/>\n&#038;=<br \/>\nx_1(x_2y_3-x_3y_2)<br \/>\n+<br \/>\nx_2(x_3y_1-x_1y_3)<br \/>\n+<br \/>\nx_3(x_1y_2-x_2y_1)<br \/>\n\\\\<br \/>\n&#038;=0.<br \/>\n\\end{aligned}<br \/>\n\\]<br \/>\n\uac19\uc740 \ubc29\ubc95\uc73c\ub85c<br \/>\n\\[<br \/>\n(\\mathbf{x}\\times\\mathbf{y})\\cdot\\mathbf{y}=0<br \/>\n\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[<br \/>\n\\mathbf{x}\\times\\mathbf{y}\\perp\\mathbf{x},<br \/>\n\\qquad<br \/>\n\\mathbf{x}\\times\\mathbf{y}\\perp\\mathbf{y}.<br \/>\n\\]\n<\/p>\n<p>\\(\\mathbb{R}^3\\)\uc758 \ud45c\uc900\uae30\uc800 \ubca1\ud130\ub294<br \/>\n\\[<br \/>\n\\mathbf{e}_1\\times\\mathbf{e}_2=\\mathbf{e}_3,<br \/>\n\\qquad<br \/>\n\\mathbf{e}_2\\times\\mathbf{e}_3=\\mathbf{e}_1,<br \/>\n\\qquad<br \/>\n\\mathbf{e}_3\\times\\mathbf{e}_1=\\mathbf{e}_2<br \/>\n\\]<br \/>\n\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4.<\/p>\n<p>\uc601\ubca1\ud130\uac00 \uc544\ub2cc \ubca1\ud130\ub4e4\uc774 \uc11c\ub85c \ub2e4\ub978 \ub450 \ubca1\ud130\ub9c8\ub2e4 \uc9c1\uad50\ud558\uba74 \uadf8 \ubca1\ud130\ub4e4\uc744 <span class=\"defined\">\uc9c1\uad50\uc871<\/span>(orthogonal family)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc9c1\uad50\uc871\uc758 \uac01 \ubca1\ud130\uac00 \ubaa8\ub450 \ub2e8\uc704\ubca1\ud130\uc774\uba74 <span class=\"defined\">\uc815\uaddc\uc9c1\uad50\uc871<\/span>(orthonormal family)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub530\ub77c\uc11c<br \/>\n\\[<br \/>\n\\mathbf{e}_1,\\,<br \/>\n\\mathbf{e}_2,\\,<br \/>\n\\mathbf{e}_3<br \/>\n\\]<br \/>\n\ub294 \\(\\mathbb{R}^3\\)\uc758 \uc815\uaddc\uc9c1\uad50\uc871\uc774\uba70 \ub3d9\uc2dc\uc5d0 \ud45c\uc900\uae30\uc800\uc774\ub2e4.<\/p>\n<p><!-- ##################################################################### --><br \/>\n<!--\n\n\n<h2 class=\"itc_h2\">\uc81c\ubaa9<\/h2>\n\n\n--><\/p>\n<p><!--\n\n\n\n<div class=\"theorem margintop2\">\n\n\n<p><span class=\"theorem\">\uc815\ub9ac 1.<\/span>\n\n\n<\/p>\n\n<\/div>\n\n\n\n\n\n<div class=\"proof\">\n\n\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n\n\n\n\n<p>\n\n<span class=\"qed\"><\/span><\/p>\n\n<\/div>\n\n\n\n\n########\n\n\u201c\u201d\n\u2018\u2019\n\n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">\ubcf4\uae30 1.<\/span>\n\n<\/p>\n\n<\/div>\n\n\n\n\n\n<h2 class=\"itc_h2\"><\/h2>\n\n\n\n--><\/p>\n<p><!--\n\n\n<div style=\"display: none; visibility: hidden;\">\n\\[\n\\newcommand{\\Hom}{{\\operatorname{Hom}}}\n\\newcommand{\\Mat}{{\\operatorname{Mat}}}\n\\newcommand{\\proj}{{\\operatorname{proj}}}\n\\newcommand{\\adj}{{\\operatorname{adj}}}\n\\]\n<\/div>\n\n\n--><\/p>\n<div class=\"box itc_prev_next_box\">\n<ul class=\"itc_ul\">\n<li class=\"itc_li_prev\">\uc55e\uc758 \uae00 : <a href=\"..\/exponential-and-logarithmic-functions\">\uc9c0\uc218\ud568\uc218\uc640 \ub85c\uadf8\ud568\uc218<\/a><\/li>\n<li class=\"itc_li_next\">\ub2e4\uc74c \uae00 : <a href=\"..\/definition-of-a-sequence\">\uc218\uc5f4\uc758 \uc815\uc758<\/a><\/li>\n<\/ul>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 0\uc7a5 8\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.&nbsp; (\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30) \ubca1\ud130\uacf5\uac04 \\(d\\)\uac00 \uc591\uc758 \uc815\uc218\ub77c\uace0 \ud558\uc790. \uc2e4\uc218 \uc9d1\ud569 \\(\\mathbb{R}\\)\uc758 \\(d\\)\uc911 \ub370\uce74\ub974\ud2b8 \uacf1 \\( \\mathbb{R}^d = \\underbrace{\\mathbb{R}\\times\\mathbb{R}\\times\\cdots\\times\\mathbb{R}}_{d\\text{\uac1c}} \\) \uc744 \uc0dd\uac01\ud558\uc790. \uadf8\ub9ac\uace0 \\( \\mathbf{x} = (x_1,\\,x_2,\\,\\cdots,\\,x_d), \\qquad \\mathbf{y} = (y_1,\\,y_2,\\,\\cdots,\\,y_d) \\) \uac00 \\(\\mathbb{R}^d\\)\uc758 \uc6d0\uc18c\uc774\uace0 \\(k\\)\uac00 \uc2e4\uc218\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ubca1\ud130\ud569 \\(\\mathbf{x}+\\mathbf{y}\\)\uc640 \uc2a4\uce7c\ub77c\ubc30 \\(k\\mathbf{x}\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4. \\( \\begin{align} \\mathbf{x}+\\mathbf{y} &#038;= (x_1+y_1,\\,x_2+y_2,\\,\\cdots,\\,x_d+y_d),  k\\mathbf{x} &#038;= (kx_1,\\,kx_2,\\,\\cdots,\\,kx_d). \\end{align} \\)&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":6620,"menu_order":18,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"class_list":["post-6988","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6988","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=6988"}],"version-history":[{"count":22,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6988\/revisions"}],"predecessor-version":[{"id":10130,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6988\/revisions\/10130"}],"up":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6620"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=6988"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}