{"id":32552,"date":"2022-03-08T13:00:43","date_gmt":"2022-03-08T13:00:43","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=32552"},"modified":"2025-03-10T02:01:14","modified_gmt":"2025-03-10T02:01:14","slug":"%e6%ac%a7%e5%87%a0%e9%87%8c%e5%be%97%e7%a9%ba%e9%97%b4-r%e2%81%bf","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/zh\/%e6%ac%a7%e5%87%a0%e9%87%8c%e5%be%97%e7%a9%ba%e9%97%b4-r%e2%81%bf\/","title":{"rendered":"\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4 R\u207f"},"content":{"rendered":"<style>\np, ul, ol{\ntext-align: justify;\n}\nh1{\ntext-align:center;\ntext-transform: uppercase;\n}\nh2{\ntext-align:center;\ntext-transform: uppercase;\nfont-size:24pt;\n}\nh3 { \n    text-align: center;\n    text-transform: uppercase;\n    font-size: 24px !important;\n}\n<\/style>\n<h1>\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">{\\mathbb{R}^n}<\/span><\/span><\/h1>\n<p style=\"text-align:center;\" dir=\"ltr\"><em>\u5728\u672c\u8bfe\u7a0b\u4e2d\uff0c\u6211\u4eec\u5c06\u63a2\u8ba8<strong>\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span><\/strong>\uff0c\u5176\u4ee3\u6570\u7ed3\u6784\u548c\u5ea6\u91cf\u6027\u8d28\u3002\u4f60\u5c06\u5b66\u4e60\u5411\u91cf\u8fd0\u7b97\u3001<strong>\u5185\u79ef<\/strong>\u3001<strong>\u8303\u6570<\/strong>\u548c<strong>\u6b27\u51e0\u91cc\u5f97\u8ddd\u79bb<\/strong>\uff0c\u8fd9\u4e9b\u6982\u5ff5\u5728\u51e0\u4f55\u548c\u5206\u6790\u4e2d\u81f3\u5173\u91cd\u8981\u3002\u672c\u8bfe\u7a0b\u63d0\u4f9b\u6e05\u6670\u7684\u89e3\u91ca\u548c\u76f4\u89c2\u7684\u793a\u4f8b\uff0c\u4f7f\u4f60\u80fd\u591f\u7406\u89e3\u5982\u4f55\u5728\u6570\u5b66\u4e0a\u5efa\u6a21\u591a\u7ef4\u7a7a\u95f4\u3002<\/em><\/p>\n<p style=\"text-align:center;\" dir=\"ltr\">\n<strong>\u5b66\u4e60\u76ee\u6807\uff1a<\/strong><br \/>\n\u5b8c\u6210\u672c\u8bfe\u7a0b\u540e\uff0c\u5b66\u751f\u5c06\u80fd\u591f\uff1a\n<\/p>\n<ol>\n<li><strong>\u5b9a\u4e49<\/strong>\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u53ca\u5176\u57fa\u672c\u6027\u8d28\u3002<\/li>\n<li><strong>\u89e3\u91ca<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u7684\u5411\u91cf\u7ed3\u6784\u53ca\u5176\u57fa\u672c\u8fd0\u7b97\u3002<\/li>\n<li><strong>\u5e94\u7528<\/strong> \u5185\u79ef\u8ba1\u7b97\u5411\u91cf\u4e4b\u95f4\u7684\u89d2\u5ea6\u548c\u6295\u5f71\u3002<\/li>\n<li><strong>\u8bc1\u660e<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u5185\u79ef\u7684\u4ee3\u6570\u548c\u5ea6\u91cf\u6027\u8d28\u3002<\/li>\n<li><strong>\u4f7f\u7528<\/strong> \u6b27\u51e0\u91cc\u5f97\u8303\u6570\u786e\u5b9a\u5411\u91cf\u7684\u5927\u5c0f\u3002<\/li>\n<li><strong>\u8ba1\u7b97<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u4e2d\u4e24\u70b9\u95f4\u7684\u6b27\u51e0\u91cc\u5f97\u8ddd\u79bb\uff0c\u5e76\u5206\u6790\u5176\u51e0\u4f55\u610f\u4e49\u3002<\/li>\n<li><strong>\u9a8c\u8bc1<\/strong> \u91cd\u8981\u7684\u4e0d\u7b49\u5f0f\uff0c\u5982\u67ef\u897f-\u65bd\u74e6\u8328\u4e0d\u7b49\u5f0f\u548c\u4e09\u89d2\u4e0d\u7b49\u5f0f\u3002<\/li>\n<\/ol>\n<p style=\"text-align:center;\" dir=\"ltr\"><strong>\u76ee\u5f55<\/strong><br \/>\n<a href=\"#1\">\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span><\/a><br \/>\n<a href=\"#2\">\u5185\u79ef<\/a><br \/>\n<a href=\"#3\">\u8303\u6570\u4e0e\u6b27\u51e0\u91cc\u5f97\u8ddd\u79bb<\/a><br \/>\n<a href=\"#4\">\u7ed3\u8bba<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/mV-G69l9LtI\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>\u5411\u91cf\u7a7a\u95f4 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span><\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=mV-G69l9LtI&#038;t=123s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u5728\u5b66\u4e60\u672c\u8282\u5185\u5bb9\u4e4b\u524d\uff0c\u4f60\u5e94\u8be5\u5df2\u7ecf\u719f\u6089\u4e86\u5b9e\u6570\u96c6 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R},<\/span><\/span> \u5e73\u9762 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^2,<\/span><\/span> \u6216\u8005\u4e09\u7ef4\u7a7a\u95f4 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3.<\/span><\/span><\/span><\/strong><\/a> \u8fd9\u4e9b\u6982\u5ff5\u6709\u52a9\u4e8e\u7406\u89e3\u66f4\u9ad8\u7ef4\u7684\u7a7a\u95f4 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n.<\/span><\/span> \u5177\u4f53\u6765\u8bf4\uff0c\u96c6\u5408 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n = \\{\\vec{x} = (x_1, \\cdots, x_n) | x_1, \\cdots, x_n \\in \\mathbb{R}\\},<\/span><\/span> \u5728\u5411\u91cf\u52a0\u6cd5\u548c\u6570\u4e58\u8fd0\u7b97\u7684\u5b9a\u4e49\u4e0b\u6784\u6210\u4e00\u4e2a\u5411\u91cf\u7a7a\u95f4\u3002\u8ba9\u6211\u4eec\u6df1\u5165\u63a2\u8ba8 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u7684\u57fa\u672c\u8fd0\u7b97\u3002<\/p>\n<h3><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u7684\u57fa\u672c\u8fd0\u7b97<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=mV-G69l9LtI&#038;t=232s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u8bbe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}=(x_1, \\cdots, x_n), \\vec{y}=(y_1, \\cdots, y_n)<\/span><\/span> \u662f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u4e2d\u7684\u5411\u91cf\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> \u662f\u4efb\u610f\u5b9e\u6570\u3002<\/span><\/strong><\/a> \u90a3\u4e48\uff0c\u5411\u91cf\u52a0\u6cd5\u548c\u6570\u4e58\u8fd0\u7b97\u5b9a\u4e49\u5982\u4e0b\uff1a<\/p>\n<p><strong>\u5411\u91cf\u52a0\u6cd5\uff1a<\/strong> \u901a\u8fc7\u4e0b\u8ff0\u51fd\u6570\u5b9a\u4e49\u5411\u91cf\u52a0\u6cd5\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\begin{array}{rcrl} +:&amp; \\mathbb{R}^n \\times \\mathbb{R}^n &amp; \\longrightarrow &amp; \\mathbb{R}^n \\\\ &amp; (\\vec{x},\\vec{y}) &amp; \\longmapsto &amp; \\vec{x}+\\vec{y} = (x_1+y_1, \\cdots, x_n + y_n) \\end{array} <\/span><\/span><\/p>\n<p><strong>\u6570\u4e58\u8fd0\u7b97\uff1a<\/strong> \u901a\u8fc7\u4e0b\u8ff0\u51fd\u6570\u5b9a\u4e49\u6570\u4e58\u8fd0\u7b97\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\begin{array}{rcrl} ():&amp; \\mathbb{R} \\times \\mathbb{R}^n &amp; \\longrightarrow &amp; \\mathbb{R}^n \\\\ &amp; (\\alpha,\\vec{x}) &amp; \\longmapsto &amp; (\\alpha\\vec{x}) = (\\alpha x_1, \\cdots, \\alpha x_n) \\end{array} <\/span>\n<h3><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u7684\u5411\u91cf\u7a7a\u95f4\u6027\u8d28<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=mV-G69l9LtI&#038;t=428s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u7531\u4e0a\u8ff0\u5b9a\u4e49\u7684\u8fd0\u7b97\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u662f\u4e00\u4e2a<strong>\u5411\u91cf\u7a7a\u95f4<\/strong>\u3002<\/span><\/strong><\/a><\/p>\n<p>\u56e0\u4e3a\u5176\u52a0\u6cd5\u548c\u6570\u4e58\u8fd0\u7b97\u6ee1\u8db3\u5982\u4e0b\u6027\u8d28\uff1a<\/p>\n<p>\u9996\u5148\uff0c\u6211\u4eec\u6709<strong>\u4ea4\u6362\u5f8b<\/strong>\u548c<strong>\u7ed3\u5408\u5f8b<\/strong>\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\vec{x} + \\vec{y} = \\vec{y} + \\vec{x}  \\\\ \\vec{x} + (\\vec{y}  + \\vec{z}) = (\\vec{x} + \\vec{y})  + \\vec{z}  \\\\ (\\alpha \\beta) \\vec{x}  = \\alpha (\\beta  \\vec{x}) = \\beta (\\alpha  \\vec{x}) = (\\beta\\alpha) \\vec{x}\n\n<\/span>\n<p><strong>\u6807\u91cf\u52a0\u6cd5\u5bf9\u6570\u4e58\u8fd0\u7b97\u5177\u6709\u5206\u914d\u6027\uff0c\u5411\u91cf\u52a0\u6cd5\u5bf9\u6570\u4e58\u8fd0\u7b97\u4e5f\u5177\u6709\u5206\u914d\u6027\uff1b<\/strong> \u5373\u6ee1\u8db3\u4ee5\u4e0b\u7b49\u5f0f\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> (\\alpha + \\beta) \\vec{x} = \\alpha\\vec{x} + \\beta\\vec{x} \\\\ \\alpha(\\vec{x} + \\vec{y}) = \\alpha\\vec{x} + \\alpha\\vec{y} <\/span>\n<p>\u5b58\u5728\u4e00\u4e2a<strong>\u52a0\u6cd5\u5355\u4f4d\u5143<\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{0}=(0,\\cdots, 0)<\/span>\uff0c\u4f7f\u5f97\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\vec{x} + \\vec{0} = \\vec{x} <\/span>\n<p>\u5b58\u5728<strong>\u6570\u4e58\u5355\u4f4d\u5143<\/strong>\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> 1 \\vec{x} = \\vec{x} <\/span>\n<p>\u5bf9\u4e8e\u4efb\u610f\u5411\u91cf <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\in\\mathbb{R}^n<\/span><\/span>\uff0c\u5b58\u5728<strong>\u52a0\u6cd5\u9006\u5143<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-\\vec{x},<\/span><\/span> \u4f7f\u5f97\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\vec{x} + -\\vec{x} = \\vec{0} <\/span><\/span><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/HL85aSpHdsI\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>\u5185\u79ef<\/h2>\n<p>\u5982\u679c\u6211\u4eec\u89c2\u5bdf <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u4f5c\u4e3a\u5411\u91cf\u7a7a\u95f4\u7684\u6784\u9020\uff0c\u6211\u4eec\u4f1a\u53d1\u73b0\u5b83\u672c\u8eab\u7f3a\u5c11\u5411\u91cf\u4e4b\u95f4\u7684\u4e58\u6cd5\u8fd0\u7b97\uff1b\u6362\u53e5\u8bdd\u8bf4\uff0c\u6211\u4eec\u4e0d\u80fd\u50cf\u5904\u7406\u5b9e\u6570\u90a3\u6837\u201c\u76f8\u4e58\u201d\u4e24\u4e2a\u5411\u91cf\u3002\u7136\u800c\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u5b9a\u4e49 <strong>\u5185\u79ef<\/strong> \u6765\u5f15\u5165\u8fd9\u79cd\u8fd0\u7b97\u3002<\/p>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=HL85aSpHdsI&#038;t=349s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u5185\u79ef\u4e0d\u5e94\u4e0e\u6570\u4e58\u6df7\u6dc6\uff0c<\/span><\/strong><\/a> \u524d\u8005\u662f\u4e24\u4e2a\u5411\u91cf\u4e4b\u95f4\u7684\u8fd0\u7b97\uff0c\u7ed3\u679c\u662f\u4e00\u4e2a\u6807\u91cf\uff0c\u800c\u540e\u8005\u662f\u4e00\u4e2a\u6807\u91cf\u4e0e\u5411\u91cf\u4e4b\u95f4\u7684\u8fd0\u7b97\uff0c\u7ed3\u679c\u662f\u53e6\u4e00\u4e2a\u5411\u91cf\u3002\u8003\u8651 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u4e2d\u7684\u4e24\u4e2a\u5411\u91cf <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}=(x_1, \\cdots, x_n)<\/span><\/span> \u548c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}=(y_1, \\cdots, y_n).<\/span><\/span> \u5b83\u4eec\u7684\u5185\u79ef\u5b9a\u4e49\u5982\u4e0b\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y} =\\displaystyle \\sum_{i=1}^n x_i y_i = x_1y_1 + \\cdots + x_ny_n<\/span>\n<p>\u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u4e2d\uff0c\u5185\u79ef\u6709\u591a\u79cd\u8868\u793a\u65b9\u6cd5\uff0c\u4e00\u79cd\u662f\u6211\u4eec\u521a\u521a\u4ecb\u7ecd\u7684\u516c\u5f0f\uff0c\u53e6\u4e00\u79cd\u5219\u57fa\u4e8e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u7684\u57fa\u5e95\u4ee5\u53ca<strong>\u7231\u56e0\u65af\u5766\u6c42\u548c\u7ea6\u5b9a\uff1a<\/strong> \u5982\u679c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\hat{e}_i\\}_{i=\\overline{1,n}}<\/span><\/span> \u662f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u7684\u4e00\u7ec4\u57fa\uff08\u901a\u5e38\u53d6\u6807\u51c6\u57fa\uff09\uff0c\u90a3\u4e48\u5411\u91cf <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span><\/span> \u548c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span><\/span> \u53ef\u4ee5\u8868\u793a\u4e3a\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}=\\displaystyle\\sum_{i=1}^n x_i\\hat{e}_i = x_1\\hat{e}_1 + \\cdots + x_n\\hat{e}_n<\/span>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}=\\displaystyle\\sum_{i=1}^n y_i\\hat{e}_i = y_1\\hat{e}_1 + \\cdots + y_n\\hat{e}_n<\/span>\n<p>\u8fd9\u91cc\u660e\u786e\u8868\u793a\u4e86\u5411\u91cf <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_i<\/span><\/span> \u548c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y_i<\/span><\/span> \u7684\u5750\u6807\u662f\u76f8\u5bf9\u4e8e\u57fa\u5e95\u7684\u3002<\/p>\n<h3>\u7231\u56e0\u65af\u5766\u6c42\u548c\u7ea6\u5b9a<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=HL85aSpHdsI&#038;t=518s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u7231\u56e0\u65af\u5766\u6c42\u548c\u7ea6\u5b9a<\/span><\/strong><\/a> \u4f7f\u5f97\u5411\u91cf\u8868\u793a\u53ca\u5185\u79ef\u8fd0\u7b97\u66f4\u52a0\u7b80\u6d01\u3002\u5982\u679c\u6211\u4eec\u89c2\u5bdf\u4e0a\u9762\u7684\u4e24\u4e2a\u8868\u8fbe\u5f0f\uff0c\u4f1a\u53d1\u73b0\u7d22\u5f15 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">i<\/span><\/span> \u5728\u5411\u91cf\u7684\u7cfb\u6570\u548c\u57fa\u5411\u91cf\u4e2d\u90fd\u51fa\u73b0\u4e86\u3002\u6839\u636e\u7231\u56e0\u65af\u5766\u7684\u7ea6\u5b9a\uff0c\u51fa\u73b0\u91cd\u590d\u7d22\u5f15\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u9ed8\u8ba4\u5b58\u5728\u6c42\u548c\u8fd0\u7b97\uff0c\u800c\u65e0\u9700\u663e\u5f0f\u5199\u51fa\u6c42\u548c\u7b26\u53f7\uff0c\u56e0\u6b64\u53ef\u4ee5\u7b80\u5316\u4e3a\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}=  x_i\\hat{e}_i<\/span>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}= y_i\\hat{e}_i <\/span>\n<p>\u4f7f\u7528\u8fd9\u79cd\u8bb0\u53f7\uff0c\u5185\u79ef\u53ef\u4ee5\u5199\u6210\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y} = x_i\\hat{e}_i \\cdot y_i\\hat{e}_i = x_iy_i \\underbrace{(\\hat{e}_i \\cdot \\hat{e}_i)}_{=1} = x_iy_i  <\/span>\n<p>\u5728\u6700\u540e\u7684\u7b49\u5f0f\u4e2d\uff0c\u6211\u4eec\u5047\u8bbe\u91c7\u7528\u7684\u662f\u6807\u51c6\u57fa\u3002<\/p>\n<h3>\u5185\u79ef\u7684\u5176\u4ed6\u8bb0\u53f7<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=HL85aSpHdsI&#038;t=825s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u5411\u91cf\u53ca\u5176\u8fd0\u7b97\u7684\u8bb0\u53f7\u5728\u4e0d\u540c\u4e0a\u4e0b\u6587\u4e2d\u53ef\u80fd\u6709\u6240\u4e0d\u540c\uff0c<\/span><\/strong><\/a> \u672c\u6587\u524d\u9762\u4f7f\u7528\u7684\u8bb0\u53f7\u662f\u5fae\u79ef\u5206\u4e2d\u6700\u5e38\u89c1\u7684\u3002\u5728\u7ebf\u6027\u4ee3\u6570\u4e2d\uff0c\u6211\u4eec\u6709\u65f6\u4f1a\u533a\u5206\u5411\u91cf\u548c\u5171\u53d8\u5411\u91cf\uff08covector\uff09\u3002<\/p>\n<p>\u5f53\u63d0\u5230\u5411\u91cf\u65f6\uff0c\u6211\u4eec\u901a\u5e38\u6307\u7684\u662f\u00bb\u5217\u5411\u91cf\u00bb\uff0c\u5176\u77e9\u9635\u5f62\u5f0f\u4e3a\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha^i = \\left( \\begin{array}{c}\\alpha_1 \\\\ \\vdots \\\\ \\alpha_n \\end{array} \\right)  <\/span>\n<p>\u800c\u5171\u53d8\u5411\u91cf\uff08covector\uff09\u901a\u5e38\u6307\u00bb\u884c\u5411\u91cf\u00bb\uff0c\u5176\u77e9\u9635\u8868\u793a\u5982\u4e0b\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta_i = \\left( \\beta_1 \\; \\cdots \\; \\beta_n  \\right)  <\/span>\n<p>\u56e0\u6b64\uff0c\u4e24\u4e2a\u5411\u91cf <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}=(x_1,\\cdots,x_n)<\/span><\/span> \u548c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}=(y_1,\\cdots,y_n)<\/span><\/span> \u7684\u5185\u79ef\u53ef\u4ee5\u89c6\u4e3a\u00bb\u5171\u53d8\u5411\u91cf\u00bb <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_i<\/span><\/span> \u4e0e\u5411\u91cf <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y^i<\/span><\/span> \u7684\u77e9\u9635\u4e58\u79ef\uff0c\u7ed3\u679c\u662f\u4e00\u4e2a\u5b9e\u6570\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left( x_1 \\; \\cdots \\; x_n  \\right) \\left( \\begin{array}{c}y_1 \\\\ \\vdots \\\\ y_n \\end{array} \\right)  = x_iy^i  <\/span>\n<p>\u6ce8\u610f\uff0c\u5728\u6700\u540e\u7684\u7b49\u5f0f\u4e2d\uff0c\u6211\u4eec\u518d\u6b21\u4f7f\u7528\u4e86\u7231\u56e0\u65af\u5766\u6c42\u548c\u7ea6\u5b9a\uff0c\u91cd\u590d\u7684\u7d22\u5f15\u8868\u793a\u6c42\u548c\u3002<\/p>\n<p>\u5728\u5e7f\u4e49\u76f8\u5bf9\u8bba\u548c\u7279\u6b8a\u76f8\u5bf9\u8bba\u4e2d\uff0c\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u00bb\u534f\u53d8\u8bb0\u53f7\u00bb\u6216\u00bb\u5f20\u91cf\u8bb0\u53f7\u00bb\u6765\u533a\u5206\u5411\u91cf\u548c\u5171\u53d8\u5411\u91cf\u3002\u8fd9\u79cd\u8bb0\u53f7\u7684\u4f18\u52bf\u5728\u4e8e\u5b83\u53ef\u4ee5\u81ea\u7136\u5730\u63a8\u5e7f\u5230\u5f20\u91cf\u7684\u8fd0\u7b97\uff0c\u800c\u5f20\u91cf\u6b63\u662f\u6211\u4eec\u7a0d\u540e\u4f1a\u8be6\u7ec6\u8ba8\u8bba\u7684\u6982\u5ff5\u3002\u5728\u91cf\u5b50\u529b\u5b66\u4e2d\uff0c\u901a\u5e38\u91c7\u7528 Bra-Ket \u8bb0\u53f7\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left&lt; x \\right| =\\left( x_1 \\; \\cdots \\; x_n  \\right) \\\\ \\\\ \\left|y\\right&gt; = \\left( \\begin{array}{c}y_1 \\\\ \\vdots \\\\ y_n \\end{array} \\right)\n\n <\/span>\n<p>\u8fd9\u6837\uff0c\u5185\u79ef\u53ef\u4ee5\u8868\u793a\u4e3a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left&lt;x|y\\right&gt;.<\/span><\/span><\/p>\n<h3>\u5185\u79ef\u7684\u6027\u8d28<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=HL85aSpHdsI&#038;t=1083s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u4ece\u5185\u79ef\u7684\u5b9a\u4e49\u51fa\u53d1\uff0c\u6211\u4eec\u53ef\u4ee5\u603b\u7ed3\u51fa\u4e00\u7cfb\u5217\u91cd\u8981\u6027\u8d28\uff0c<\/span><\/strong><\/a> \u8fd9\u4e9b\u6027\u8d28\u5728\u672a\u6765\u7684\u5b66\u4e60\u4e2d\u4f1a\u975e\u5e38\u91cd\u8981\u3002<\/p>\n<p>\u5982\u679c\u6211\u4eec\u7528\u5185\u79ef\u5b9a\u4e49\u51fd\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tilde{\\omega}(\\vec{x})=\\vec{\\omega} \\cdot \\vec{x} = \\omega_i x^i,<\/span><\/span> \u90a3\u4e48\u8be5\u51fd\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tilde{\\omega}<\/span><\/span> \u5177\u5907\u6240\u6709\u7ebf\u6027\u51fd\u6570\u7684\u6027\u8d28\u3002\u5b9e\u9645\u4e0a\uff0c\u6211\u4eec\u53ef\u4ee5\u8bc1\u660e\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl} \\tilde{\\omega}(\\alpha \\vec{x} + \\beta\\vec{y}) = \\alpha \\tilde{\\omega}(\\vec{x}) + \\beta\\tilde{\\omega}(\\vec{y}) \\end{array}<\/span>\n<p>\u56e0\u6b64\uff0c\u50cf <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tilde{\\omega}<\/span><\/span> \u8fd9\u6837\u7531\u5185\u79ef\u5b9a\u4e49\u7684\u5bf9\u8c61\u88ab\u79f0\u4e3a<strong>\u7ebf\u6027\u6cdb\u51fd<\/strong>\u3002\u6211\u4eec\u77e5\u9053 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span><\/span> \u662f<strong>\u5411\u91cf\u7a7a\u95f4<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u7684\u5143\u7d20\uff0c\u800c\u5728\u67d0\u4e9b\u60c5\u51b5\u4e0b\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tilde{\\omega}<\/span><\/span> \u5219\u662f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u7684<strong>\u5bf9\u5076\u7a7a\u95f4<\/strong>\u4e2d\u7684\u5bf9\u8c61\u3002<\/p>\n<p>\u56e0\u6b64\uff0c\u5185\u79ef\u4e0e\u7ebf\u6027\u51fd\u6570\u4e4b\u95f4\u5b58\u5728\u7740\u5bc6\u5207\u7684\u5173\u7cfb\u3002\u4e8b\u5b9e\u4e0a\uff0c\u5185\u79ef\u53ef\u4ee5\u6982\u62ec\u4e3a\u4e00\u53e5\u8bdd\uff1a<em><strong>\u00ab\u5185\u79ef\u662f\u4e00\u4e2a\u53cc\u7ebf\u6027\u3001\u5bf9\u79f0\u3001\u6b63\u5b9a\u4e14\u975e\u9000\u5316\u7684\u5f62\u5f0f\u00bb\u3002<\/strong><\/em> \u6211\u4eec\u6765\u9010\u4e00\u89e3\u6790\u8fd9\u4e9b\u7279\u6027\uff1a<\/p>\n<p>\u5f53\u6211\u4eec\u8bf4<strong>\u5185\u79ef\u662f\u53cc\u7ebf\u6027\u7684<\/strong>\uff0c\u6211\u4eec\u6307\u7684\u662f\u5982\u679c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x},\\vec{y}<\/span><\/span> \u548c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}<\/span><\/span> \u662f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u4e2d\u7684\u5411\u91cf\uff0c\u4e14 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha,\\beta \\in \\mathbb{R},<\/span><\/span> \u90a3\u4e48\u6ee1\u8db3\u5982\u4e0b\u7b49\u5f0f\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl} \\vec{x}\\cdot(\\alpha \\vec{y} + \\beta\\vec{z}) = \\alpha (\\vec{x}\\cdot\\vec{y}) + \\beta(\\vec{x}\\cdot\\vec{z}) \\\\ \\\\ (\\alpha \\vec{x} + \\beta\\vec{y})\\cdot\\vec{z} = \\alpha (\\vec{x} \\cdot \\vec{z}) + \\beta(\\vec{y}\\cdot\\vec{z}) \\end{array}<\/span>\n<p>\u5185\u79ef<strong>\u662f\u5bf9\u79f0\u7684<\/strong>\uff0c\u56e0\u4e3a\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\forall(\\vec{x},\\vec{y}\\in\\mathbb{R}^n)(\\vec{x}\\cdot\\vec{y} = \\vec{y}\\cdot\\vec{x})<\/span>\n<p>\u5b83\u662f<strong>\u6b63\u5b9a\u7684<\/strong>\uff0c\u56e0\u4e3a\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall\\vec{x}\\in\\mathbb{R}^n)(\\vec{x}\\cdot\\vec{x} \\geq 0)<\/span>\n<p>\u6700\u540e\uff0c\u5b83\u662f<strong>\u975e\u9000\u5316\u7684<\/strong>\uff0c\u56e0\u4e3a\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{x} = 0 \\leftrightarrow \\vec{x}=\\vec{0}<\/span>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/vTFqDBEyU4Y\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>\u8303\u6570\u4e0e\u6b27\u51e0\u91cc\u5f97\u8ddd\u79bb<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vTFqDBEyU4Y&#038;t=174s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u8303\u6570\u662f\u4e00\u79cd\u8861\u91cf\u5411\u91cf\u5927\u5c0f\u7684\u65b9\u5f0f\uff0c<\/span><\/strong><\/a> \u5f53\u4e00\u4e2a\u5411\u91cf\u7a7a\u95f4\u914d\u5907\u4e86\u4e00\u4e2a\u8303\u6570\u65f6\uff0c\u6211\u4eec\u79f0\u5176\u4e3a<strong>\u8d4b\u8303\u5411\u91cf\u7a7a\u95f4<\/strong>\u3002\u5982\u679c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x},\\vec{y}\\in\\mathbb{R}^n<\/span><\/span> \u4e14 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\lambda\\in\\mathbb{R},<\/span><\/span> \u5219\u51fd\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Norm( . )<\/span><\/span> \u662f\u4e00\u4e2a\u8303\u6570\uff0c\u5f53\u4e14\u4ec5\u5f53\u5b83\u6ee1\u8db3\u4ee5\u4e0b\u6027\u8d28\uff1a<\/p>\n<ol>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Norm(\\vec{x})\\geq 0<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Norm(\\vec{x}) = 0 \\leftrightarrow \\vec{x}=\\vec{0}<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Norm(\\lambda\\vec{x}) = |\\lambda| Norm(\\vec{x})<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Norm(\\vec{x} + \\vec{y}) \\leq Norm(\\vec{x}) + Norm(\\vec{y})<\/span><\/span><\/li>\n<\/ol>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vTFqDBEyU4Y&#038;t=350s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u5185\u79ef\u7684\u4e00\u4e2a\u91cd\u8981\u65b9\u9762<\/span><\/strong><\/a> \u662f\u5b83\u7279\u522b\u9002\u7528\u4e8e\u5b9a\u4e49\u6570\u5b66\u4e0a\u7684\u8ddd\u79bb\u6982\u5ff5\uff0c\u8fd9\u79cd\u5b9a\u4e49\u65b9\u5f0f\u4e0e\u6211\u4eec\u76f4\u89c2\u7406\u89e3\u4e24\u4e2a\u70b9\u4e4b\u95f4\u7684\u8ddd\u79bb\u76f8\u543b\u5408\u3002\u5bf9\u4e8e\u6bcf\u4e2a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\in\\mathbb{R}^n<\/span><\/span>\uff0c\u6211\u4eec\u5b9a\u4e49\u5176<strong>\u6b27\u51e0\u91cc\u5f97\u8303\u6570<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x}\\|<\/span><\/span> \u5982\u4e0b\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x}\\| = \\sqrt{\\vec{x}\\cdot\\vec{x}}<\/span>\n<p>\u56e0\u6b64\uff0c\u6211\u4eec\u79f0<strong>\u6b27\u51e0\u91cc\u5f97\u8303\u6570\u662f\u7531\u5185\u79ef\u8bf1\u5bfc\u51fa\u7684\u8303\u6570\u3002<\/strong><\/p>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vTFqDBEyU4Y&#038;t=846s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u8ddd\u79bb\uff0c\u6216\u79f0\u5ea6\u91cf\uff0c<\/span><\/strong><\/a> \u662f\u4e00\u4e2a\u7528\u4e8e\u63cf\u8ff0\u00bb\u96c6\u5408\u4e2d\u4e24\u4e2a\u5143\u7d20\u4e4b\u95f4\u5206\u79bb\u7a0b\u5ea6\u00bb\u7684\u51fd\u6570\u3002\u5982\u679c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}, \\vec{y}, \\vec{z}\\in\\mathbb{R}^n<\/span><\/span> \u4e14 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\lambda\\in\\mathbb{R},<\/span><\/span> \u5219\u51fd\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Dist( . )<\/span><\/span> \u662f\u4e00\u4e2a\u8ddd\u79bb\uff0c\u5f53\u4e14\u4ec5\u5f53\u5b83\u6ee1\u8db3\u4ee5\u4e0b\u6761\u4ef6\uff1a<\/p>\n<ol>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Dist(\\vec{x},\\vec{y})=0 \\leftrightarrow \\vec{x}=\\vec{y}<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Dist(\\vec{x},\\vec{y})=Dist(\\vec{y},\\vec{x})\\geq 0<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Dist(\\vec{x},\\vec{z})\\leq Dist(\\vec{x},\\vec{y}) + Dist(\\vec{y},\\vec{z})<\/span><\/span><\/li>\n<\/ol>\n<p>\u6700\u540e\u4e00\u4e2a\u8868\u8fbe\u5f0f\u79f0\u4e3a<strong>\u4e09\u89d2\u4e0d\u7b49\u5f0f<\/strong>\uff0c\u5982\u679c\u5b83\u4e0d\u6ee1\u8db3\uff0c\u5219\u51fd\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Dist(.)<\/span><\/span> \u4ec5\u662f\u00bb\u4f2a\u8ddd\u79bb\u00bb\u6216\u00bb\u4f2a\u5ea6\u91cf\u00bb\u3002\u4e00\u4e2a\u5e26\u6709\u8ddd\u79bb\u7684\u5411\u91cf\u7a7a\u95f4\u88ab\u79f0\u4e3a<strong>\u5ea6\u91cf\u7a7a\u95f4\u3002<\/strong><\/p>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vTFqDBEyU4Y&#038;t=1013s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u57fa\u4e8e\u6b27\u51e0\u91cc\u5f97\u8303\u6570<\/span><\/strong><\/a>\uff0c\u6211\u4eec\u5b9a\u4e49\u4e86\u4e24\u4e2a\u5411\u91cf\u4e4b\u95f4\u7684<strong>\u6b27\u51e0\u91cc\u5f97\u8ddd\u79bb<\/strong>\u3002\u5bf9\u4e8e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x},\\vec{y}\\in\\mathbb{R}^n,<\/span><\/span> \u5b83\u4eec\u4e4b\u95f4\u7684\u6b27\u51e0\u91cc\u5f97\u8ddd\u79bb <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">dist_e(\\vec{x},\\vec{y})<\/span><\/span> \u7531\u4e0b\u5f0f\u7ed9\u51fa\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">dist_e(\\vec{x},\\vec{y}) = \\|\\vec{x} - \\vec{y}\\|<\/span>\n<p>\u5982\u679c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}=(x_1,\\cdots,x_n)<\/span><\/span> \u4e14 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}=(y_1,\\cdots, y_n),<\/span><\/span> \u90a3\u4e48\u6839\u636e\u5185\u79ef\u548c\u8303\u6570\u7684\u6027\u8d28\uff0c\u6211\u4eec\u53ef\u4ee5\u8f7b\u677e\u8bc1\u660e\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">dist_e(\\vec{x},\\vec{y}) = \\sqrt{\\displaystyle \\sum_{i=1}^n (x_i - y_i)^2}<\/span>\n<p>\u5982\u679c\u6211\u4eec\u5728\u5411\u91cf\u7a7a\u95f4 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u4e2d\u5f15\u5165\u6b27\u51e0\u91cc\u5f97\u8ddd\u79bb\uff0c\u5219\u5f97\u5230\u7684\u7ed3\u6784\u79f0\u4e3a<strong>\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4<\/strong>\u3002<\/p>\n<p>\u56e0\u6b64\uff0c\u6211\u4eec\u79f0<strong>\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4\u7684\u5ea6\u91cf\u662f\u7531\u6b27\u51e0\u91cc\u5f97\u8303\u6570\u8bf1\u5bfc\u51fa\u7684\u5ea6\u91cf\u3002<\/strong><\/p>\n<h3>\u6b27\u51e0\u91cc\u5f97\u8303\u6570\u7684\u6027\u8d28<\/h3>\n<p>\u7531\u4e8e\u6211\u4eec\u7684\u7814\u7a76\u91cd\u70b9\u662f\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4\uff0c\u56e0\u6b64\u6709\u5fc5\u8981\u56de\u987e\u6b27\u51e0\u91cc\u5f97\u8303\u6570\u7684\u6027\u8d28\u3002<\/p>\n<h4>\u67ef\u897f-\u65bd\u74e6\u8328\u4e0d\u7b49\u5f0f<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vTFqDBEyU4Y&#038;t=1624s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u5982\u679c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x},\\vec{y}\\in\\mathbb{R}^n,<\/span><\/span><\/span><\/strong><\/a> \u5219\u6ee1\u8db3\u4ee5\u4e0b\u6027\u8d28\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|\\vec{x}\\cdot\\vec{y}|\\leq \\|\\vec{x}\\|\\|\\vec{y}\\|<\/span>\n<p><strong>\u8bc1\u660e\uff1a<\/strong><\/p>\n<p>\u4ee4 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\lambda = (\\vec{x}\\cdot\\vec{y})\/\\|\\vec{y}\\|^2,<\/span><\/span> \u5219\u6709\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl} 0\\leq \\|\\vec{x} - \\lambda \\vec{y}\\|^2 &amp;= (\\vec{x} - \\lambda\\vec{y}) \\cdot (\\vec{x} - \\lambda\\vec{y}) \\\\ \\\\\n\n\\displaystyle &amp;= \\vec{x}\\cdot\\vec{x} - \\lambda\\vec{x}\\cdot\\vec{y} + \\lambda\\vec{y}\\cdot\\vec{x} + \\lambda^2(\\vec{y}\\cdot\\vec{y})\\\\ \\\\\n\n&amp;= \\|\\vec{x}\\|^2 - 2\\lambda(\\vec{x}\\cdot\\vec{y}) + \\lambda^2 \\|\\vec{y}\\|^2 \\\\ \\\\\n\n\\displaystyle &amp;= \\|\\vec{x}\\|^2 - 2\\left(\\frac{\\vec{x}\\cdot\\vec{y}}{\\|\\vec{y}\\|^2}\\right)(\\vec{x}\\cdot\\vec{y}) + \\left(\\frac{\\vec{x}\\cdot\\vec{y}}{\\|\\vec{y}\\|^2}\\right)^2 {\\|\\vec{y}\\|^2}\\\\ \\\\\n\n\\displaystyle &amp;= \\|\\vec{x}\\|^2 - 2\\left(\\frac{(\\vec{x}\\cdot\\vec{y})^2}{\\|\\vec{y}\\|^2}\\right) + \\frac{\\left(\\vec{x}\\cdot\\vec{y}\\right)^2}{\\|\\vec{y}\\|^2}\\\\ \\\\\n\n&amp;= \\|\\vec{x}\\|^2 - \\frac{\\left(\\vec{x}\\cdot\\vec{y}\\right)^2}{\\|\\vec{y}\\|^2} \\end{array}<\/span>\n<p>\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u5f97\u51fa\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle 0 \\leq \\|\\vec{x}\\|^2 - \\frac{\\left(\\vec{x}\\cdot\\vec{y}\\right)^2}{\\|\\vec{y}\\|^2} <\/span>\n<p>\u56e0\u6b64\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\left(\\vec{x}\\cdot\\vec{y}\\right)^2 \\leq \\|\\vec{x}\\|^2 \\|\\vec{y}\\|^2 <\/span>\n<p>\u6700\u7ec8\uff0c\u53d6\u5e73\u65b9\u6839\u5f97\u5230\u6240\u8981\u8bc1\u660e\u7684\u7ed3\u8bba\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> |\\vec{x}\\cdot\\vec{y}| \\leq \\|\\vec{x}\\| \\|\\vec{y}\\|<\/span> \u2b1b<\/p>\n<h4>\u4e09\u89d2\u4e0d\u7b49\u5f0f<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vTFqDBEyU4Y&#038;t=2065s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u8bbe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x},\\vec{y}\\in\\mathbb{R}^n,<\/span><\/span><\/span><\/strong><\/a> \u5219\u8fd9\u4e9b\u5411\u91cf\u6ee1\u8db3\u4ee5\u4e0b\u5173\u7cfb\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x} + \\vec{y}\\| \\leq \\|\\vec{x}\\| + \\|\\vec{y}\\|<\/span>\n<p><strong>\u8bc1\u660e\uff1a<\/strong><\/p>\n<p>\u9996\u5148\u6211\u4eec\u6ce8\u610f\u5230\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl} \\|\\vec{x} + \\vec{y}\\|^2 &amp;= (\\vec{x} + \\vec{y})\\cdot(\\vec{x} + \\vec{y}) \\\\ \\\\ &amp;=\\|\\vec{x}\\|^2 + 2(\\vec{x}\\cdot\\vec{y}) + \\|\\vec{y}\\|^2 \\end{array}<\/span>\n<p>\u7531\u4e8e\u4ee5\u4e0b\u5173\u7cfb\u6210\u7acb\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y}\\leq |\\vec{x}\\cdot\\vec{y}| \\leq \\|\\vec{x}\\|\\|\\vec{y}\\|<\/span>\n<p>\u56e0\u6b64\u53ef\u4ee5\u5199\u6210\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n\\|\\vec{x} + \\vec{y}\\|^2 &amp;\\leq  \\|\\vec{x}\\|^2 + 2\\|\\vec{x}\\|\\|\\vec{y}\\| + \\|\\vec{y}\\|^2 \\\\ \\\\\n\n&amp;\\leq  \\left(\\|\\vec{x}\\|  + \\|\\vec{y}\\| \\right)^2\n\n\\end{array}<\/span>\n<p>\u6700\u7ec8\uff0c\u53d6\u5e73\u65b9\u6839\u540e\uff0c\u5f97\u5230\u8981\u8bc1\u660e\u7684\u7ed3\u8bba\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x} + \\vec{y}\\|\\leq  \\|\\vec{x}\\|  + \\|\\vec{y}\\|<\/span> \u2b1b <\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>\u7ed3\u8bba<\/h2>\n<p>\u5728\u672c\u8bfe\u7a0b\u4e2d\uff0c\u6211\u4eec\u63a2\u8ba8\u4e86\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u7684\u57fa\u672c\u6027\u8d28\uff0c\u5305\u62ec\u5176\u4ee3\u6570\u7ed3\u6784\u548c\u5ea6\u91cf\u7ed3\u6784\u3002\u6211\u4eec\u9996\u5148\u5b9a\u4e49\u4e86\u57fa\u672c\u8fd0\u7b97\uff0c\u5982\u5411\u91cf\u52a0\u6cd5\u548c\u6570\u4e58\uff0c\u786e\u7acb\u4e86\u5176\u4f5c\u4e3a\u5411\u91cf\u7a7a\u95f4\u7684\u7279\u6027\u3002\u968f\u540e\uff0c\u6211\u4eec\u6df1\u5165\u7814\u7a76\u4e86\u5185\u79ef\u7684\u6982\u5ff5\u53ca\u5176\u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u51e0\u4f55\u4e2d\u7684\u91cd\u8981\u6027\uff0c\u5f3a\u8c03\u4e86\u5176\u77e9\u9635\u8868\u793a\u53ca\u5176\u4e0e\u7ebf\u6027\u51fd\u6570\u7684\u5173\u7cfb\u3002<\/p>\n<p>\u63a5\u7740\uff0c\u6211\u4eec\u5206\u6790\u4e86\u6b27\u51e0\u91cc\u5f97\u8303\u6570\u53ca\u5176\u8bf1\u5bfc\u7684\u8ddd\u79bb\uff0c\u8ba8\u8bba\u4e86\u5982\u4f55\u5229\u7528\u8fd9\u4e9b\u5de5\u5177\u6765\u5ea6\u91cf\u8be5\u7a7a\u95f4\u4e2d\u7684\u957f\u5ea6\u548c\u8ddd\u79bb\u3002\u6b64\u5916\uff0c\u6211\u4eec\u56de\u987e\u4e86\u4e24\u4e2a\u91cd\u8981\u7684\u4e0d\u7b49\u5f0f\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> |\\vec{x}\\cdot\\vec{y}| \\leq \\|\\vec{x}\\| \\|\\vec{y}\\| <\/span>\n<p>\u67ef\u897f-\u65bd\u74e6\u8328\u4e0d\u7b49\u5f0f<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\|\\vec{x} + \\vec{y}\\|\\leq  \\|\\vec{x}\\|  + \\|\\vec{y}\\| <\/span>\n<p>\u4e09\u89d2\u4e0d\u7b49\u5f0f<\/p>\n<p>\u8fd9\u4e9b\u6027\u8d28\u5728\u66f4\u9ad8\u9636\u7684\u5206\u6790\u4e0e\u51e0\u4f55\u7406\u8bba\u4e2d\u5177\u6709\u5173\u952e\u4f5c\u7528\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4 \u5728\u672c\u8bfe\u7a0b\u4e2d\uff0c\u6211\u4eec\u5c06\u63a2\u8ba8\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4 \uff0c\u5176\u4ee3\u6570\u7ed3\u6784\u548c\u5ea6\u91cf\u6027\u8d28\u3002\u4f60\u5c06\u5b66\u4e60\u5411\u91cf\u8fd0\u7b97\u3001\u5185\u79ef\u3001\u8303\u6570\u548c\u6b27\u51e0\u91cc\u5f97\u8ddd\u79bb\uff0c\u8fd9\u4e9b\u6982\u5ff5\u5728\u51e0\u4f55\u548c\u5206\u6790\u4e2d\u81f3\u5173\u91cd\u8981\u3002\u672c\u8bfe\u7a0b\u63d0\u4f9b\u6e05\u6670\u7684\u89e3\u91ca\u548c\u76f4\u89c2\u7684\u793a\u4f8b\uff0c\u4f7f\u4f60\u80fd\u591f\u7406\u89e3\u5982\u4f55\u5728\u6570\u5b66\u4e0a\u5efa\u6a21\u591a\u7ef4\u7a7a\u95f4\u3002 \u5b66\u4e60\u76ee\u6807\uff1a 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