{"id":32556,"date":"2022-03-08T13:00:42","date_gmt":"2022-03-08T13:00:42","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=32556"},"modified":"2025-03-10T02:13:22","modified_gmt":"2025-03-10T02:13:22","slug":"%d8%a7%d9%84%d9%81%d8%b6%d8%a7%d8%a1-%d8%a7%d9%84%d8%a5%d9%82%d9%84%d9%8a%d8%af%d9%8a-%e2%84%9d%e2%81%bf","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/ar\/%d8%a7%d9%84%d9%81%d8%b6%d8%a7%d8%a1-%d8%a7%d9%84%d8%a5%d9%82%d9%84%d9%8a%d8%af%d9%8a-%e2%84%9d%e2%81%bf\/","title":{"rendered":"\u0627\u0644\u0641\u0636\u0627\u0621 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a \u211d\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>\u0627\u0644\u0641\u0636\u0627\u0621 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a <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>\u0641\u064a \u0647\u0630\u0647 \u0627\u0644\u062f\u0631\u0633\u060c \u0646\u0633\u062a\u0643\u0634\u0641 <strong>\u0627\u0644\u0641\u0636\u0627\u0621 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span><\/strong>\u060c \u0647\u064a\u0643\u0644\u0647 \u0627\u0644\u062c\u0628\u0631\u064a \u0648\u062e\u0635\u0627\u0626\u0635\u0647 \u0627\u0644\u0645\u062a\u0631\u064a\u0629. \u0633\u062a\u062a\u0639\u0644\u0645 \u0639\u0646 \u0627\u0644\u0639\u0645\u0644\u064a\u0627\u062a \u0627\u0644\u0645\u062a\u062c\u0647\u064a\u0629\u060c <strong>\u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0642\u064a\u0627\u0633\u064a<\/strong>\u060c <strong>\u0627\u0644\u0645\u0639\u064a\u0627\u0631<\/strong> \u0648<strong>\u0627\u0644\u0645\u0633\u0627\u0641\u0629 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a\u0629<\/strong>\u060c \u0648\u0647\u064a \u0645\u0641\u0627\u0647\u064a\u0645 \u0623\u0633\u0627\u0633\u064a\u0629 \u0641\u064a \u0627\u0644\u0647\u0646\u062f\u0633\u0629 \u0648\u0627\u0644\u062a\u062d\u0644\u064a\u0644. \u0645\u0639 \u0634\u0631\u0648\u062d\u0627\u062a \u0648\u0627\u0636\u062d\u0629 \u0648\u0623\u0645\u062b\u0644\u0629 \u062a\u0648\u0636\u064a\u062d\u064a\u0629\u060c \u0633\u064a\u0645\u0643\u0646\u0643 \u0647\u0630\u0627 \u0627\u0644\u0645\u062d\u062a\u0648\u0649 \u0645\u0646 \u0641\u0647\u0645 \u0643\u064a\u0641\u064a\u0629 \u062a\u0645\u062b\u064a\u0644 \u0627\u0644\u0641\u0636\u0627\u0621 \u0631\u064a\u0627\u0636\u064a\u064b\u0627 \u0641\u064a \u0623\u0628\u0639\u0627\u062f \u0645\u062a\u0639\u062f\u062f\u0629.<\/em><\/p>\n<p style=\"text-align:center;\" dir=\"ltr\">\n<strong>\u0623\u0647\u062f\u0627\u0641 \u0627\u0644\u062a\u0639\u0644\u0645:<\/strong><br \/>\n\u0628\u0639\u062f \u0627\u0644\u0627\u0646\u062a\u0647\u0627\u0621 \u0645\u0646 \u0647\u0630\u0627 \u0627\u0644\u062f\u0631\u0633\u060c \u0633\u064a\u0643\u0648\u0646 \u0627\u0644\u0637\u0627\u0644\u0628 \u0642\u0627\u062f\u0631\u064b\u0627 \u0639\u0644\u0649:\n<\/p>\n<ol>\n<li><strong>\u062a\u0639\u0631\u064a\u0641<\/strong> \u0627\u0644\u0641\u0636\u0627\u0621 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u0648\u062e\u0635\u0627\u0626\u0635\u0647 \u0627\u0644\u0623\u0633\u0627\u0633\u064a\u0629.<\/li>\n<li><strong>\u0634\u0631\u062d<\/strong> \u0627\u0644\u0628\u0646\u064a\u0629 \u0627\u0644\u0645\u062a\u062c\u0647\u064a\u0629 \u0644\u0640 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u0645\u0646 \u062e\u0644\u0627\u0644 \u0639\u0645\u0644\u064a\u0627\u062a\u0647 \u0627\u0644\u0623\u0633\u0627\u0633\u064a\u0629.<\/li>\n<li><strong>\u062a\u0637\u0628\u064a\u0642<\/strong> \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0642\u064a\u0627\u0633\u064a \u0644\u062d\u0633\u0627\u0628 \u0627\u0644\u0632\u0648\u0627\u064a\u0627 \u0648\u0627\u0644\u0625\u0633\u0642\u0627\u0637\u0627\u062a \u0628\u064a\u0646 \u0627\u0644\u0645\u062a\u062c\u0647\u0627\u062a.<\/li>\n<li><strong>\u0625\u062b\u0628\u0627\u062a<\/strong> \u0627\u0644\u062e\u0635\u0627\u0626\u0635 \u0627\u0644\u062c\u0628\u0631\u064a\u0629 \u0648\u0627\u0644\u0645\u062a\u0631\u064a\u0629 \u0644\u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0642\u064a\u0627\u0633\u064a \u0641\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span>.<\/li>\n<li><strong>\u0627\u0633\u062a\u062e\u062f\u0627\u0645<\/strong> \u0627\u0644\u0645\u0639\u064a\u0627\u0631 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a \u0644\u062a\u062d\u062f\u064a\u062f \u0645\u0642\u062f\u0627\u0631 \u0627\u0644\u0645\u062a\u062c\u0647.<\/li>\n<li><strong>\u062d\u0633\u0627\u0628<\/strong> \u0627\u0644\u0645\u0633\u0627\u0641\u0629 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a\u0629 \u0628\u064a\u0646 \u0646\u0642\u0637\u062a\u064a\u0646 \u0641\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u0648\u062a\u062d\u0644\u064a\u0644 \u0645\u0639\u0646\u0627\u0647\u0627 \u0627\u0644\u0647\u0646\u062f\u0633\u064a.<\/li>\n<li><strong>\u0627\u0644\u062a\u062d\u0642\u0642<\/strong> \u0645\u0646 \u0635\u062d\u0629 \u0627\u0644\u0645\u062a\u0628\u0627\u064a\u0646\u0627\u062a \u0627\u0644\u0623\u0633\u0627\u0633\u064a\u0629 \u0645\u062b\u0644 \u0645\u062a\u0628\u0627\u064a\u0646\u0629 \u0643\u0648\u0634\u064a-\u0634\u0648\u0627\u0631\u062a\u0632 \u0648\u0645\u062a\u0628\u0627\u064a\u0646\u0629 \u0627\u0644\u0645\u062b\u0644\u062b.<\/li>\n<\/ol>\n<p style=\"text-align:center;\" dir=\"ltr\"><strong>\u0627\u0644\u0641\u0647\u0631\u0633<\/strong><br \/>\n<a href=\"#1\">\u0627\u0644\u0641\u0636\u0627\u0621 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span><\/a><br \/>\n<a href=\"#2\">\u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0642\u064a\u0627\u0633\u064a<\/a><br \/>\n<a href=\"#3\">\u0627\u0644\u0645\u0639\u064a\u0627\u0631 \u0648\u0627\u0644\u0645\u0633\u0627\u0641\u0629 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a\u0629<\/a><br \/>\n<a href=\"#4\">\u0627\u0644\u062e\u0627\u062a\u0645\u0629<\/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>\u0627\u0644\u0641\u0636\u0627\u0621 \u0627\u0644\u0645\u062a\u062c\u0647\u064a <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;\">\u0645\u0646 \u0627\u0644\u0645\u062d\u062a\u0645\u0644 \u0623\u0646\u0643 \u0643\u0646\u062a \u0639\u0644\u0649 \u062f\u0631\u0627\u064a\u0629 \u0645\u0633\u0628\u0642\u0629 \u0628\u062e\u0635\u0627\u0626\u0635 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R},<\/span><\/span> \u0623\u0648 \u0627\u0644\u0645\u0633\u062a\u0648\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^2,<\/span><\/span> \u0623\u0648 \u0627\u0644\u0641\u0636\u0627\u0621 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3.<\/span><\/span><\/span><\/strong><\/a> \u0643\u0644 \u0647\u0630\u0647 \u0627\u0644\u0623\u0641\u0643\u0627\u0631 \u0645\u0641\u064a\u062f\u0629 \u0644\u0641\u0647\u0645 \u0627\u0644\u0641\u0636\u0627\u0621 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n.<\/span><\/span> \u0628\u0634\u0643\u0644 \u0639\u0627\u0645\u060c \u0627\u0644\u0645\u062c\u0645\u0648\u0639\u0629 <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> \u0645\u0639 \u0627\u0644\u0639\u0645\u0644\u064a\u0627\u062a \u0627\u0644\u0645\u0639\u062a\u0627\u062f\u0629 \u0644\u062c\u0645\u0639 \u0627\u0644\u0645\u062a\u062c\u0647\u0627\u062a \u0648\u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0639\u062f\u062f\u064a \u0647\u0648 \u0641\u0636\u0627\u0621 \u0645\u062a\u062c\u0647\u064a. \u062f\u0639\u0648\u0646\u0627 \u0646\u0633\u062a\u0643\u0634\u0641 \u0647\u0630\u0627 \u0628\u0645\u0632\u064a\u062f \u0645\u0646 \u0627\u0644\u062a\u0641\u0627\u0635\u064a\u0644 \u0645\u0646 \u062e\u0644\u0627\u0644 \u0645\u0631\u0627\u062c\u0639\u0629 \u0627\u0644\u0639\u0645\u0644\u064a\u0627\u062a \u0627\u0644\u0623\u0633\u0627\u0633\u064a\u0629 \u0641\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n.<\/span><\/span> <\/p>\n<h3>\u0627\u0644\u0639\u0645\u0644\u064a\u0627\u062a \u0627\u0644\u0623\u0633\u0627\u0633\u064a\u0629 \u0641\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span><\/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;\">\u0625\u0630\u0627 \u0643\u0627\u0646 <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> \u0645\u062a\u062c\u0647\u064a\u0646 \u0641\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u0648 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> \u0639\u062f\u062f \u062d\u0642\u064a\u0642\u064a\u060c<\/span><\/strong> <\/a>\u0641\u0625\u0646 \u0639\u0645\u0644\u064a\u0627\u062a <strong>\u062c\u0645\u0639 \u0627\u0644\u0645\u062a\u062c\u0647\u0627\u062a<\/strong> \u0648<strong>\u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0639\u062f\u062f\u064a<\/strong> \u062a\u064f\u0639\u0631\u0651\u0641 \u0643\u0645\u0627 \u064a\u0644\u064a:<\/p>\n<p><strong>\u062c\u0645\u0639 \u0627\u0644\u0645\u062a\u062c\u0647\u0627\u062a:<\/strong> \u064a\u064f\u0648\u0635\u0641 \u062c\u0645\u0639 \u0627\u0644\u0645\u062a\u062c\u0647\u0627\u062a \u0628\u0627\u0644\u062f\u0627\u0644\u0629 \u0627\u0644\u062a\u0627\u0644\u064a\u0629:<\/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>\u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0639\u062f\u062f\u064a:<\/strong> \u064a\u064f\u0648\u0635\u0641 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0639\u062f\u062f\u064a \u0628\u0627\u0644\u062f\u0627\u0644\u0629 \u0627\u0644\u062a\u0627\u0644\u064a\u0629:<\/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>\u062e\u0635\u0627\u0626\u0635 \u0627\u0644\u0641\u0636\u0627\u0621 \u0627\u0644\u0645\u062a\u062c\u0647\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span><\/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;\">\u0627\u0644\u0641\u0636\u0627\u0621 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u0645\u0639 \u0627\u0644\u0639\u0645\u0644\u064a\u0627\u062a \u0627\u0644\u0645\u0630\u0643\u0648\u0631\u0629 \u0623\u0639\u0644\u0627\u0647<\/span><\/strong><\/a> \u0647\u0648 <strong>\u0641\u0636\u0627\u0621 \u0645\u062a\u062c\u0647\u064a\u060c<\/strong> \u0644\u0623\u0646 \u0639\u0645\u0644\u064a\u0627\u062a \u0627\u0644\u062c\u0645\u0639 \u0648\u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0639\u062f\u062f\u064a \u062a\u062d\u0642\u0642 \u0627\u0644\u062e\u0635\u0627\u0626\u0635 \u0627\u0644\u062a\u0627\u0644\u064a\u0629:<\/p>\n<p>\u0623\u0648\u0644\u064b\u0627\u060c \u0644\u062f\u064a\u0646\u0627 \u062e\u0627\u0635\u064a\u062a\u0627 <strong>\u0627\u0644\u062a\u0628\u0627\u062f\u0644<\/strong> \u0648<strong>\u0627\u0644\u062a\u062c\u0645\u064a\u0639.<\/strong><\/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>\u064a\u062a\u0645 \u062a\u0648\u0632\u064a\u0639 \u062c\u0645\u0639 \u0627\u0644\u0623\u0639\u062f\u0627\u062f \u0627\u0644\u062d\u0642\u064a\u0642\u064a\u0629 \u0628\u0627\u0644\u0646\u0633\u0628\u0629 \u0644\u0644\u0636\u0631\u0628 \u0627\u0644\u0639\u062f\u062f\u064a\u060c \u0648\u064a\u062a\u0645 \u062a\u0648\u0632\u064a\u0639 \u062c\u0645\u0639 \u0627\u0644\u0645\u062a\u062c\u0647\u0627\u062a \u0628\u0627\u0644\u0646\u0633\u0628\u0629 \u0644\u0644\u0636\u0631\u0628 \u0627\u0644\u0639\u062f\u062f\u064a\u061b<\/strong> \u0628\u0645\u0639\u0646\u0649 \u0623\u0646 \u0627\u0644\u0645\u0639\u0627\u062f\u0644\u0627\u062a \u0627\u0644\u062a\u0627\u0644\u064a\u0629 \u062a\u062a\u062d\u0642\u0642:<\/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>\u064a\u0648\u062c\u062f <strong>\u0639\u0646\u0635\u0631 \u0645\u062d\u0627\u064a\u062f \u062c\u0645\u0639\u064a<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{0}=(0,\\cdots, 0)<\/span><\/span> \u064a\u062d\u0642\u0642 \u0627\u0644\u062e\u0627\u0635\u064a\u0629:<\/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>\u064a\u0648\u062c\u062f \u0639\u0646\u0635\u0631 <strong>\u0645\u062d\u0627\u064a\u062f \u0636\u0631\u0628\u064a<\/strong> \u0644\u0639\u0645\u0644\u064a\u0629 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0639\u062f\u062f\u064a:<\/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>\u0648\u0643\u0644 \u0645\u062a\u062c\u0647 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\in\\mathbb{R}^n<\/span><\/span> \u0644\u062f\u064a\u0647 <strong>\u0645\u0639\u0643\u0648\u0633 \u062c\u0645\u0639\u064a<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-\\vec{x},<\/span><\/span> \u064a\u062d\u0642\u0642 \u0627\u0644\u062e\u0627\u0635\u064a\u0629:<\/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>\u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0642\u064a\u0627\u0633\u064a<\/h2>\n<p>\u0625\u0630\u0627 \u0646\u0638\u0631\u0646\u0627 \u0625\u0644\u0649 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u0643\u0641\u0636\u0627\u0621 \u0645\u062a\u062c\u0647\u064a\u060c \u0633\u0646\u0644\u0627\u062d\u0638 \u0623\u0646\u0647 \u064a\u0641\u062a\u0642\u0631 \u0625\u0644\u0649 \u0639\u0645\u0644\u064a\u0629 \u0636\u0631\u0628 \u0628\u064a\u0646 \u0627\u0644\u0645\u062a\u062c\u0647\u0627\u062a\u061b \u0641\u064a \u0627\u0644\u0628\u062f\u0627\u064a\u0629\u060c \u0644\u0627 \u064a\u0645\u0643\u0646\u0646\u0627 \u00ab\u0636\u0631\u0628\u00bb \u0627\u0644\u0645\u062a\u062c\u0647\u0627\u062a \u0628\u0628\u0639\u0636\u0647\u0627 \u0643\u0645\u0627 \u0646\u0641\u0639\u0644 \u0645\u0639 \u0627\u0644\u0623\u0639\u062f\u0627\u062f \u0627\u0644\u062d\u0642\u064a\u0642\u064a\u0629. \u0648\u0645\u0639 \u0630\u0644\u0643\u060c \u064a\u0645\u0643\u0646 \u062a\u0639\u0631\u064a\u0641 \u0639\u0645\u0644\u064a\u0629 \u0627\u0644\u0636\u0631\u0628 \u0628\u064a\u0646 \u0627\u0644\u0645\u062a\u062c\u0647\u0627\u062a \u0628\u0637\u0631\u064a\u0642\u0629 \u062a\u0639\u0631\u0641 \u0628\u0640 <strong>\u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0642\u064a\u0627\u0633\u064a.<\/strong><\/p>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=HL85aSpHdsI&#038;t=349s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u0644\u0627 \u064a\u062c\u0628 \u0627\u0644\u062e\u0644\u0637 \u0628\u064a\u0646 \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0642\u064a\u0627\u0633\u064a \u0648\u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0639\u062f\u062f\u064a\u060c<\/span><\/strong><\/a> \u062d\u064a\u062b \u0623\u0646 \u0627\u0644\u0623\u0648\u0644 \u0647\u0648 \u062d\u0627\u0635\u0644 \u0636\u0631\u0628 \u0628\u064a\u0646 \u0645\u062a\u062c\u0647\u064a\u0646 \u064a\u0639\u0637\u064a \u0639\u062f\u062f\u064b\u0627 \u062d\u0642\u064a\u0642\u064a\u064b\u0627\u060c \u0628\u064a\u0646\u0645\u0627 \u0627\u0644\u062b\u0627\u0646\u064a \u0647\u0648 \u0636\u0631\u0628 \u0639\u062f\u062f \u062d\u0642\u064a\u0642\u064a \u0641\u064a \u0645\u062a\u062c\u0647 \u064a\u0646\u062a\u062c \u0639\u0646\u0647 \u0645\u062a\u062c\u0647 \u0622\u062e\u0631. \u0644\u0646\u0623\u062e\u0630 \u0645\u062a\u062c\u0647\u064a\u0646 \u0641\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n:<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}=(x_1, \\cdots, x_n)<\/span><\/span> \u0648 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}=(y_1, \\cdots, y_n).<\/span><\/span> \u0628\u0646\u0627\u0621\u064b \u0639\u0644\u0649 \u0630\u0644\u0643\u060c \u064a\u062a\u0645 \u062a\u0639\u0631\u064a\u0641 \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0642\u064a\u0627\u0633\u064a \u0644\u0640 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span><\/span> \u0645\u0639 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y},<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y},<\/span><\/span> \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0627\u0644\u0635\u064a\u063a\u0629 \u0627\u0644\u062a\u0627\u0644\u064a\u0629:<\/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>\u0647\u0646\u0627\u0643 \u0639\u062f\u0629 \u0637\u0631\u0642 \u0644\u062a\u0645\u062b\u064a\u0644 \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0642\u064a\u0627\u0633\u064a \u0628\u064a\u0646 \u0627\u0644\u0645\u062a\u062c\u0647\u0627\u062a \u0641\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n,<\/span><\/span> \u0625\u062d\u062f\u0627\u0647\u0627 \u0647\u064a \u0627\u0644\u0635\u064a\u063a\u0629 \u0627\u0644\u062a\u064a \u0630\u0643\u0631\u0646\u0627\u0647\u0627 \u0644\u0644\u062a\u0648\u060c \u0648\u0623\u062e\u0631\u0649 \u062a\u0639\u062a\u0645\u062f \u0639\u0644\u0649 \u0642\u0627\u0639\u062f\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u0648<strong>\u0627\u062a\u0641\u0627\u0642\u064a\u0629 \u062c\u0645\u0639 \u0623\u064a\u0646\u0634\u062a\u0627\u064a\u0646:<\/strong> \u0625\u0630\u0627 \u0643\u0627\u0646\u062a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\hat{e}_i\\}_{i=\\overline{1,n}}<\/span><\/span> \u0642\u0627\u0639\u062f\u0629 \u0644\u0640 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> (\u0639\u0627\u062f\u0629\u064b \u0627\u0644\u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0642\u064a\u0627\u0633\u064a\u0629)\u060c \u0641\u0625\u0646\u0647 \u064a\u0645\u0643\u0646 \u0627\u0644\u062a\u0639\u0628\u064a\u0631 \u0639\u0646 \u0627\u0644\u0645\u062a\u062c\u0647\u064a\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span><\/span> \u0648 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span><\/span> \u0628\u0627\u0644\u0634\u0643\u0644 \u0627\u0644\u062a\u0627\u0644\u064a:<\/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>\u062a\u0648\u0636\u062d \u0647\u0630\u0647 \u0627\u0644\u0635\u064a\u063a\u0629 \u0623\u0646 \u0627\u0644\u0645\u0639\u0627\u0645\u0644\u0627\u062a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_i<\/span><\/span> \u0648 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y_i<\/span><\/span> \u0627\u0644\u062e\u0627\u0635\u0629 \u0628\u0627\u0644\u0645\u062a\u062c\u0647\u0627\u062a \u062a\u064f\u062d\u062f\u062f \u0628\u0627\u0644\u0646\u0633\u0628\u0629 \u0625\u0644\u0649 \u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0641\u0636\u0627\u0621.<\/p>\n<h3>\u0627\u062a\u0641\u0627\u0642\u064a\u0629 \u062c\u0645\u0639 \u0623\u064a\u0646\u0634\u062a\u0627\u064a\u0646<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=HL85aSpHdsI&#038;t=518s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u0627\u062a\u0641\u0627\u0642\u064a\u0629 \u062c\u0645\u0639 \u0623\u064a\u0646\u0634\u062a\u0627\u064a\u0646<\/span><\/strong><\/a> \u062a\u0633\u0647\u0644 \u062a\u0645\u062b\u064a\u0644 \u0627\u0644\u0645\u062a\u062c\u0647\u0627\u062a \u0628\u0634\u0643\u0644 \u0639\u0627\u0645 \u0648\u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0642\u064a\u0627\u0633\u064a \u0628\u0634\u0643\u0644 \u062e\u0627\u0635. \u0625\u0630\u0627 \u0644\u0627\u062d\u0638\u0646\u0627 \u0627\u0644\u062a\u0639\u0628\u064a\u0631\u064a\u0646 \u0627\u0644\u0633\u0627\u0628\u0642\u064a\u0646\u060c \u0633\u0646\u062c\u062f \u0623\u0646 \u0627\u0644\u0641\u0647\u0631\u0633 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">i<\/span><\/span> \u064a\u062a\u0643\u0631\u0631 \u0641\u064a \u0645\u0639\u0627\u0645\u0644 \u0627\u0644\u0645\u062a\u062c\u0647 \u0648\u0643\u0630\u0644\u0643 \u0641\u064a \u0639\u0646\u0635\u0631 \u0627\u0644\u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0645\u062a\u062c\u0647\u064a\u0629\u061b \u0628\u0627\u0644\u0646\u0633\u0628\u0629 \u0644\u0623\u064a\u0646\u0634\u062a\u0627\u064a\u0646\u060c \u0648\u062c\u0648\u062f \u0627\u0644\u0641\u0647\u0627\u0631\u0633 \u0627\u0644\u0645\u062a\u0643\u0631\u0631\u0629 \u0643\u0627\u0641\u064d \u0644\u0627\u0641\u062a\u0631\u0627\u0636 \u0648\u062c\u0648\u062f \u0639\u0645\u0644\u064a\u0629 \u0627\u0644\u062c\u0645\u0639 \u0641\u064a \u0627\u0644\u062a\u0639\u0628\u064a\u0631\u060c \u0645\u0645\u0627 \u064a\u0633\u0645\u062d \u0644\u0646\u0627 \u0628\u0643\u062a\u0627\u0628\u0629:<\/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>\u0648\u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0647\u0630\u0647 \u0627\u0644\u0627\u062a\u0641\u0627\u0642\u064a\u0629\u060c \u064a\u0635\u0628\u062d \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0642\u064a\u0627\u0633\u064a \u0628\u0627\u0644\u0635\u0648\u0631\u0629:<\/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>\u0641\u064a \u0647\u0630\u0647 \u0627\u0644\u0645\u0633\u0627\u0648\u0627\u0629 \u0627\u0644\u0623\u062e\u064a\u0631\u0629\u060c \u062a\u0645 \u0627\u0641\u062a\u0631\u0627\u0636 \u0623\u0646\u0646\u0627 \u0646\u0639\u0645\u0644 \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0627\u0644\u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0642\u064a\u0627\u0633\u064a\u0629.<\/p>\n<h3>\u062a\u0631\u0645\u064a\u0632 \u0622\u062e\u0631 \u0644\u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0642\u064a\u0627\u0633\u064a<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=HL85aSpHdsI&#038;t=825s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u0644\u0627 \u064a\u0643\u0648\u0646 \u0627\u0644\u062a\u0631\u0645\u064a\u0632 \u0627\u0644\u0645\u0633\u062a\u062e\u062f\u0645 \u0644\u0644\u0645\u062a\u062c\u0647\u0627\u062a \u0648\u0639\u0645\u0644\u064a\u0627\u062a\u0647\u0627 \u0645\u0648\u062d\u062f\u064b\u0627 \u0641\u064a \u062c\u0645\u064a\u0639 \u0627\u0644\u0633\u064a\u0627\u0642\u0627\u062a\u060c<\/span><\/strong><\/a> \u0627\u0644\u062a\u0631\u0645\u064a\u0632 \u0627\u0644\u0630\u064a \u0627\u0633\u062a\u062e\u062f\u0645\u062a\u0647 \u0641\u064a \u0627\u0644\u0641\u0642\u0631\u0627\u062a \u0627\u0644\u0623\u0648\u0644\u0649 \u0647\u0648 \u0627\u0644\u0623\u0643\u062b\u0631 \u0634\u064a\u0648\u0639\u064b\u0627 \u0641\u064a \u062d\u0633\u0627\u0628 \u0627\u0644\u062a\u0641\u0627\u0636\u0644 \u0648\u0627\u0644\u062a\u0643\u0627\u0645\u0644. \u0641\u064a \u0627\u0644\u062c\u0628\u0631 \u0627\u0644\u062e\u0637\u064a\u060c \u064a\u062a\u0645 \u0623\u062d\u064a\u0627\u0646\u064b\u0627 \u0627\u0644\u062a\u0645\u064a\u064a\u0632 \u0628\u064a\u0646 \u0627\u0644\u0645\u062a\u062c\u0647\u0627\u062a \u0648\u0627\u0644\u0645\u0634\u0627\u0647\u062f\u0627\u062a \u0627\u0644\u0645\u0634\u062a\u0631\u0643\u0629:<\/p>\n<p>\u0639\u0646\u062f\u0645\u0627 \u0646\u062a\u062d\u062f\u062b \u0639\u0646 \u0627\u0644\u0645\u062a\u062c\u0647\u0627\u062a\u060c \u0641\u0625\u0646\u0646\u0627 \u0646\u0634\u064a\u0631 \u0625\u0644\u0649 \u0645\u0627 \u064a\u064f\u0639\u0631\u0641 \u0628\u0640 \u00ab\u0627\u0644\u0645\u062a\u062c\u0647 \u0627\u0644\u0639\u0645\u0648\u062f\u064a\u00bb\u060c \u0648\u0627\u0644\u0630\u064a \u064a\u064f\u0645\u062b\u0644 \u0628\u0627\u0644\u0634\u0643\u0644 \u0627\u0644\u0645\u0635\u0641\u0648\u0641\u064a \u0627\u0644\u062a\u0627\u0644\u064a:<\/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>\u0623\u0645\u0627 \u0639\u0646\u062f\u0645\u0627 \u0646\u062a\u062d\u062f\u062b \u0639\u0646 \u0627\u0644\u0645\u0634\u0627\u0647\u062f\u0627\u062a \u0627\u0644\u0645\u0634\u062a\u0631\u0643\u0629 (covectors)\u060c \u0641\u0625\u0646\u0646\u0627 \u0646\u0634\u064a\u0631 \u0625\u0644\u0649 \u00ab\u0627\u0644\u0645\u062a\u062c\u0647 \u0627\u0644\u0635\u0641\u0651\u064a\u00bb\u060c \u0648\u0627\u0644\u0630\u064a \u064a\u064f\u0645\u062b\u0644 \u0628\u0627\u0644\u0634\u0643\u0644:<\/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>\u0628\u0647\u0630\u0647 \u0627\u0644\u0637\u0631\u064a\u0642\u0629\u060c \u064a\u0645\u0643\u0646 \u062a\u0641\u0633\u064a\u0631 \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0642\u064a\u0627\u0633\u064a \u0628\u064a\u0646 \u0645\u062a\u062c\u0647\u064a\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}=(x_1,\\cdots,x_n)<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}=(y_1,\\cdots,y_n)<\/span><\/span> \u0639\u0644\u0649 \u0623\u0646\u0647 \u0636\u0631\u0628 \u0645\u0635\u0641\u0648\u0641\u064a \u0628\u064a\u0646 \u00ab\u0627\u0644\u0645\u0634\u0627\u0647\u062f\u0629 \u0627\u0644\u0645\u0634\u062a\u0631\u0643\u0629\u00bb <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_i<\/span><\/span> \u0648\u0627\u0644\u0645\u062a\u062c\u0647 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y^i,<\/span><\/span> \u0645\u0645\u0627 \u064a\u0624\u062f\u064a \u0625\u0644\u0649 \u0627\u0644\u0639\u062f\u062f \u0627\u0644\u062d\u0642\u064a\u0642\u064a \u0627\u0644\u062a\u0627\u0644\u064a:<\/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>\u0644\u0627\u062d\u0638 \u0623\u0646 \u0647\u0630\u0647 \u0627\u0644\u0645\u0633\u0627\u0648\u0627\u0629 \u0627\u0644\u0623\u062e\u064a\u0631\u0629 \u062a\u062a\u0636\u0645\u0646 \u0645\u0631\u0629 \u0623\u062e\u0631\u0649 \u0627\u062a\u0641\u0627\u0642\u064a\u0629 \u062c\u0645\u0639 \u0623\u064a\u0646\u0634\u062a\u0627\u064a\u0646\u060c \u062d\u064a\u062b \u064a\u0634\u064a\u0631 \u062a\u0643\u0631\u0627\u0631 \u0627\u0644\u0645\u0624\u0634\u0631\u0627\u062a \u0625\u0644\u0649 \u0623\u0646 \u0627\u0644\u0646\u062a\u064a\u062c\u0629 \u0627\u0644\u0646\u0647\u0627\u0626\u064a\u0629 \u0647\u064a \u0645\u062c\u0645\u0648\u0639.<\/p>\n<p>\u0627\u0644\u062a\u0631\u0645\u064a\u0632 \u0627\u0644\u0630\u064a \u064a\u0645\u064a\u0632 \u0628\u064a\u0646 \u0627\u0644\u0645\u062a\u062c\u0647\u0627\u062a \u0648\u0627\u0644\u0645\u0634\u0627\u0647\u062f\u0629 \u0627\u0644\u0645\u0634\u062a\u0631\u0643\u0629 \u0645\u0646 \u062e\u0644\u0627\u0644 \u0627\u0644\u0641\u0647\u0627\u0631\u0633 \u0627\u0644\u0641\u0648\u0642\u064a\u0629 \u0648\u0627\u0644\u062a\u062d\u062a\u064a\u0629 \u064a\u064f\u0639\u0631\u0641 \u0628\u0640 \u00ab\u0627\u0644\u062a\u0631\u0645\u064a\u0632 \u0627\u0644\u062a\u0648\u0627\u0641\u0642\u064a\u00bb \u0623\u0648 \u00ab\u062a\u0631\u0645\u064a\u0632 \u0627\u0644\u062a\u0646\u0633\u0648\u0631\u00bb\u060c \u0648\u0647\u0648 \u0634\u0627\u0626\u0639 \u0641\u064a \u062f\u0631\u0627\u0633\u0629 \u0646\u0638\u0631\u064a\u0629 \u0627\u0644\u0646\u0633\u0628\u064a\u0629 \u0627\u0644\u062e\u0627\u0635\u0629 \u0648\u0627\u0644\u0639\u0627\u0645\u0629. \u0643\u0645\u0627 \u0623\u0646 \u0644\u0647\u0630\u0627 \u0627\u0644\u062a\u0631\u0645\u064a\u0632 \u0641\u0627\u0626\u062f\u0629 \u0641\u064a \u062a\u0633\u0647\u064a\u0644 \u0627\u0644\u062a\u0639\u0627\u0645\u0644 \u0645\u0639 \u0627\u0644\u062a\u0646\u0633\u0648\u0631\u0627\u062a\u060c \u0648\u0647\u0648 \u0645\u0641\u0647\u0648\u0645 \u0639\u0627\u0645 \u064a\u0634\u0645\u0644 \u062c\u0645\u064a\u0639 \u0645\u0627 \u0646\u0627\u0642\u0634\u0646\u0627\u0647 \u0648\u0633\u0646\u062a\u0637\u0631\u0642 \u0625\u0644\u064a\u0647 \u0628\u0645\u0632\u064a\u062f \u0645\u0646 \u0627\u0644\u062a\u0641\u0627\u0635\u064a\u0644 \u0641\u064a \u0648\u0642\u062a \u0644\u0627\u062d\u0642. \u0641\u064a \u0645\u062c\u0627\u0644\u0627\u062a \u0623\u062e\u0631\u0649 \u0645\u062b\u0644 \u0645\u064a\u0643\u0627\u0646\u064a\u0643\u0627 \u0627\u0644\u0643\u0645\u060c \u064a\u062a\u0645 \u062a\u0641\u0636\u064a\u0644 \u062a\u0631\u0645\u064a\u0632 \u0628\u0631\u0627-\u0643\u064a\u062a\u060c \u062d\u064a\u062b:<\/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>\u0648\u0628\u0647\u0630\u0647 \u0627\u0644\u0637\u0631\u064a\u0642\u0629\u060c \u064a\u062a\u0645 \u062a\u0645\u062b\u064a\u0644 \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0642\u064a\u0627\u0633\u064a \u0628\u0627\u0644\u0634\u0643\u0644 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left&lt;x|y\\right&gt;.<\/span><\/span><\/p>\n<h3>\u062e\u0635\u0627\u0626\u0635 \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0642\u064a\u0627\u0633\u064a<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=HL85aSpHdsI&#038;t=1083s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u0627\u0646\u0637\u0644\u0627\u0642\u064b\u0627 \u0645\u0646 \u062a\u0639\u0631\u064a\u0641 \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0642\u064a\u0627\u0633\u064a\u060c \u064a\u0645\u0643\u0646\u0646\u0627 \u0627\u0633\u062a\u062e\u0644\u0627\u0635 \u0645\u062c\u0645\u0648\u0639\u0629 \u0645\u0646 \u0627\u0644\u062e\u0635\u0627\u0626\u0635<\/span><\/strong><\/a> \u0627\u0644\u062a\u064a \u0633\u062a\u0643\u0648\u0646 \u0630\u0627\u062a \u0623\u0647\u0645\u064a\u0629 \u0643\u0628\u064a\u0631\u0629 \u0641\u064a \u0627\u0644\u0645\u0633\u062a\u0642\u0628\u0644.<\/p>\n<p>\u0625\u0630\u0627 \u0627\u0633\u062a\u062e\u062f\u0645\u0646\u0627 \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0642\u064a\u0627\u0633\u064a \u0644\u062a\u0639\u0631\u064a\u0641 \u0627\u0644\u062f\u0627\u0644\u0629 <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> \u0641\u0633\u0646\u0631\u0649 \u0623\u0646 \u0627\u0644\u062f\u0627\u0644\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tilde{\\omega}<\/span><\/span> \u0627\u0644\u0645\u0639\u0631\u0641\u0629 \u0628\u0647\u0630\u0647 \u0627\u0644\u0637\u0631\u064a\u0642\u0629 \u062a\u0645\u062a\u0644\u0643 \u062c\u0645\u064a\u0639 \u062e\u0635\u0627\u0626\u0635 \u0627\u0644\u062f\u0648\u0627\u0644 \u0627\u0644\u062e\u0637\u064a\u0629\u060c \u062d\u064a\u062b \u064a\u0645\u0643\u0646 \u0628\u0633\u0647\u0648\u0644\u0629 \u0625\u062b\u0628\u0627\u062a \u0623\u0646:<\/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>\u0648\u0644\u0647\u0630\u0627 \u0627\u0644\u0633\u0628\u0628\u060c \u0641\u0625\u0646 \u0627\u0644\u0643\u064a\u0627\u0646\u0627\u062a \u0645\u062b\u0644 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tilde{\\omega}<\/span><\/span> \u0627\u0644\u062a\u064a \u062a\u064f\u0639\u0631\u0641 \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0642\u064a\u0627\u0633\u064a \u062a\u064f\u0633\u0645\u0649 <strong>\u062f\u0648\u0627\u0644 \u062e\u0637\u064a\u0629.<\/strong> \u0643\u0645\u0627 \u0646\u0639\u0644\u0645\u060c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span><\/span> \u0647\u0648 \u0645\u062a\u062c\u0647 \u064a\u0646\u062a\u0645\u064a \u0625\u0644\u0649 <strong>\u0627\u0644\u0641\u0636\u0627\u0621 \u0627\u0644\u0645\u062a\u062c\u0647\u064a<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n,<\/span><\/span> \u0648\u0643\u0645\u0627 \u0633\u0646\u0631\u0649 \u0644\u0627\u062d\u0642\u064b\u0627\u060c \u0641\u0625\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tilde{\\omega}<\/span><\/span> \u0647\u0648 \u0643\u064a\u0627\u0646 \u064a\u0646\u062a\u0645\u064a \u0625\u0644\u0649 <strong>\u0627\u0644\u0641\u0636\u0627\u0621 \u0627\u0644\u0645\u0632\u062f\u0648\u062c<\/strong> \u0644\u0640<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n.<\/span><\/span><\/p>\n<p>\u0628\u0646\u0627\u0621\u064b \u0639\u0644\u0649 \u0630\u0644\u0643\u060c \u0647\u0646\u0627\u0643 \u0639\u0644\u0627\u0642\u0629 \u0648\u062b\u064a\u0642\u0629 \u0628\u064a\u0646 \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0642\u064a\u0627\u0633\u064a \u0648\u0627\u0644\u062f\u0648\u0627\u0644 \u0627\u0644\u062e\u0637\u064a\u0629\u061b \u0641\u064a \u0627\u0644\u0648\u0627\u0642\u0639\u060c \u062a\u0639\u0628\u064a\u0631 \u0648\u0627\u062d\u062f \u064a\u0645\u0643\u0646\u0647 \u062a\u0644\u062e\u064a\u0635 \u062c\u0645\u064a\u0639 \u0627\u0644\u062e\u0635\u0627\u0626\u0635 \u0627\u0644\u0645\u0647\u0645\u0629 \u0644\u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0642\u064a\u0627\u0633\u064a \u0647\u0648: <em><strong>\u00ab\u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0642\u064a\u0627\u0633\u064a \u0647\u0648 \u0634\u0643\u0644 \u062e\u0637\u064a \u0645\u0632\u062f\u0648\u062c\u060c \u0645\u062a\u0645\u0627\u062b\u0644\u060c \u0645\u0648\u062c\u0628 \u0648\u063a\u064a\u0631 \u0645\u0646\u062d\u0644\u00bb.<\/strong><\/em> \u062f\u0639\u0648\u0646\u0627 \u0646\u0633\u062a\u0643\u0634\u0641 \u0645\u0639\u0646\u0649 \u0643\u0644 \u062c\u0632\u0621 \u0645\u0646 \u0647\u0630\u0627 \u0627\u0644\u062a\u0639\u0631\u064a\u0641:<\/p>\n<p>\u0639\u0646\u062f\u0645\u0627 \u0646\u0642\u0648\u0644 <strong>\u0625\u0646 \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0642\u064a\u0627\u0633\u064a \u0647\u0648 \u0634\u0643\u0644 \u062e\u0637\u064a \u0645\u0632\u062f\u0648\u062c\u060c<\/strong> \u0641\u0625\u0646\u0646\u0627 \u0646\u0639\u0646\u064a \u0623\u0646\u0647 \u0625\u0630\u0627 \u0643\u0627\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x},\\vec{y}<\/span><\/span> \u0648 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}<\/span><\/span> \u0645\u062a\u062c\u0647\u0627\u062a \u0641\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha,\\beta \\in \\mathbb{R},<\/span><\/span> \u0641\u0625\u0646 \u0627\u0644\u0645\u0633\u0627\u0648\u0627\u062a\u064a\u0646 \u0627\u0644\u062a\u0627\u0644\u064a\u062a\u064a\u0646 \u062a\u062a\u062d\u0642\u0642\u0627\u0646:<\/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>\u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0642\u064a\u0627\u0633\u064a <strong>\u0645\u062a\u0645\u0627\u062b\u0644<\/strong> \u0644\u0623\u0646\u0647:<\/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>\u064a\u0643\u0648\u0646 <strong>\u0645\u064f\u0639\u0631\u0641\u064b\u0627 \u0625\u064a\u062c\u0627\u0628\u064a\u064b\u0627<\/strong> \u0644\u0623\u0646\u0647:<\/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>\u0648\u0623\u062e\u064a\u0631\u064b\u0627\u060c \u064a\u0643\u0648\u0646 <strong>\u063a\u064a\u0631 \u0645\u062a\u062d\u0644\u0644<\/strong> \u0644\u0623\u0646\u0647:<\/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>\u0627\u0644\u0645\u0639\u064a\u0627\u0631 \u0648\u0627\u0644\u0645\u0633\u0627\u0641\u0629 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a\u0629<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vTFqDBEyU4Y&#038;t=174s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u0627\u0644\u0645\u0639\u064a\u0627\u0631 \u0647\u0648 \u0637\u0631\u064a\u0642\u0629 \u0644\u0642\u064a\u0627\u0633 \u0645\u0642\u062f\u0627\u0631 \u0627\u0644\u0645\u062a\u062c\u0647\u060c<\/span><\/strong><\/a> \u0639\u0646\u062f\u0645\u0627 \u064a\u0643\u0648\u0646 \u0627\u0644\u0641\u0636\u0627\u0621 \u0627\u0644\u0645\u062a\u062c\u0647\u064a \u0645\u0632\u0648\u062f\u064b\u0627 \u0628\u0645\u0639\u064a\u0627\u0631\u060c \u0641\u0625\u0646\u0647 \u064a\u064f\u0633\u0645\u0649 <strong>\u0641\u0636\u0627\u0621 \u0645\u062a\u062c\u0647\u064a \u0645\u0639\u064a\u0627\u0631\u064a.<\/strong> \u0625\u0630\u0627 \u0643\u0627\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x},\\vec{y}\\in\\mathbb{R}^n<\/span><\/span> \u0648 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\lambda\\in\\mathbb{R},<\/span><\/span> \u0641\u0625\u0646 \u0627\u0644\u062f\u0627\u0644\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Norm( . )<\/span><\/span> \u062a\u064f\u0639\u062a\u0628\u0631 \u0645\u0639\u064a\u0627\u0631\u064b\u0627 \u0625\u0630\u0627 \u062d\u0642\u0642\u062a \u0627\u0644\u062e\u0635\u0627\u0626\u0635 \u0627\u0644\u062a\u0627\u0644\u064a\u0629:<\/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;\">\u0645\u0646 \u0627\u0644\u062e\u0635\u0627\u0626\u0635 \u0627\u0644\u0645\u0647\u0645\u0629 \u0644\u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0642\u064a\u0627\u0633\u064a<\/span><\/strong><\/a> \u0623\u0646\u0647 \u0645\u0641\u064a\u062f \u0628\u0634\u0643\u0644 \u062e\u0627\u0635 \u0644\u062a\u0639\u0631\u064a\u0641 \u0645\u0641\u0647\u0648\u0645 \u0631\u064a\u0627\u0636\u064a \u0644\u0644\u0645\u0633\u0627\u0641\u0629 \u064a\u062a\u0648\u0627\u0641\u0642 \u0645\u0639 \u0641\u0647\u0645\u0646\u0627 \u0627\u0644\u0637\u0628\u064a\u0639\u064a \u0644\u0644\u0645\u0633\u0627\u0641\u0627\u062a \u0628\u064a\u0646 \u0646\u0642\u0637\u062a\u064a\u0646. \u0644\u0643\u0644 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\in\\mathbb{R}^n<\/span><\/span>\u060c \u064a\u062a\u0645 \u062a\u0639\u0631\u064a\u0641 <strong>\u0627\u0644\u0645\u0639\u064a\u0627\u0631 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a\u060c<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x}\\|<\/span><\/span> \u0648\u0641\u0642\u064b\u0627 \u0644\u0644\u0645\u0639\u0627\u062f\u0644\u0629:<\/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>\u0645\u0646 \u0647\u0646\u0627\u060c \u0646\u0642\u0648\u0644 \u0625\u0646 <strong>\u0627\u0644\u0645\u0639\u064a\u0627\u0631 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a \u0647\u0648 \u0627\u0644\u0645\u0639\u064a\u0627\u0631 \u0627\u0644\u0645\u0633\u062a\u062d\u062b \u0628\u0648\u0627\u0633\u0637\u0629 \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0642\u064a\u0627\u0633\u064a.<\/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;\">\u0627\u0644\u0645\u0633\u0627\u0641\u0629\u060c \u0623\u0648 \u0627\u0644\u0645\u062a\u0631\u064a\u0629\u060c<\/span><\/strong><\/a> \u0647\u064a \u062f\u0627\u0644\u0629 \u062a\u062d\u062f\u062f \u00ab\u0645\u062f\u0649 \u0627\u0644\u0641\u0635\u0644 \u0628\u064a\u0646 \u0639\u0646\u0635\u0631\u064a\u0646 \u0641\u064a \u0645\u062c\u0645\u0648\u0639\u0629\u00bb. \u0625\u0630\u0627 \u0643\u0627\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}, \\vec{y}, \\vec{z}\\in\\mathbb{R}^n<\/span><\/span> \u0648 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\lambda\\in\\mathbb{R},<\/span><\/span> \u0641\u0625\u0646 \u0627\u0644\u062f\u0627\u0644\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Dist( . )<\/span><\/span> \u062a\u064f\u0639\u062a\u0628\u0631 \u0645\u0633\u0627\u0641\u0629 \u0625\u0630\u0627 \u0627\u0633\u062a\u0648\u0641\u062a \u0627\u0644\u062e\u0635\u0627\u0626\u0635 \u0627\u0644\u062a\u0627\u0644\u064a\u0629:<\/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>\u0627\u0644\u0645\u0639\u0627\u062f\u0644\u0629 \u0627\u0644\u0623\u062e\u064a\u0631\u0629 \u062a\u064f\u0639\u0631\u0641 \u0628\u0640 <strong>\u0639\u062f\u0645 \u0627\u0644\u0645\u0633\u0627\u0648\u0627\u0629 \u0627\u0644\u0645\u062b\u0644\u062b\u064a\u0629\u060c<\/strong> \u0648\u0625\u0630\u0627 \u0644\u0645 \u062a\u062a\u062d\u0642\u0642\u060c \u0641\u0625\u0646 \u0627\u0644\u062f\u0627\u0644\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Dist(.)<\/span><\/span> \u062a\u064f\u0639\u0631\u0641 \u0628\u0623\u0646\u0647\u0627 \u00ab\u0634\u0628\u0647 \u0645\u0633\u0627\u0641\u0629\u00bb \u0623\u0648 \u00ab\u0634\u0628\u0647 \u0645\u062a\u0631\u064a\u0629\u00bb. \u0627\u0644\u0641\u0636\u0627\u0621 \u0627\u0644\u0645\u062a\u062c\u0647\u064a \u0627\u0644\u0630\u064a \u064a\u062a\u0636\u0645\u0646 \u0645\u0641\u0647\u0648\u0645 \u0627\u0644\u0645\u0633\u0627\u0641\u0629 \u064a\u064f\u0633\u0645\u0649 <strong>\u0641\u0636\u0627\u0621 \u0645\u062a\u0631\u064a.<\/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;\">\u0627\u0633\u062a\u0646\u0627\u062f\u064b\u0627 \u0625\u0644\u0649 \u0627\u0644\u0645\u0639\u064a\u0627\u0631 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a\u060c<\/span><\/strong><\/a> \u064a\u062a\u0645 \u062a\u0639\u0631\u064a\u0641 <strong>\u0627\u0644\u0645\u0633\u0627\u0641\u0629 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a\u0629<\/strong> \u0628\u064a\u0646 \u0645\u062a\u062c\u0647\u064a\u0646. \u0625\u0630\u0627 \u0643\u0627\u0646 \u0644\u062f\u064a\u0646\u0627 \u0645\u062a\u062c\u0647\u0627\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x},\\vec{y}\\in\\mathbb{R}^n,<\/span><\/span> \u0641\u0625\u0646 \u0627\u0644\u0645\u0633\u0627\u0641\u0629 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a\u0629 \u0628\u064a\u0646\u0647\u0645\u0627\u060c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">dist_e(\\vec{x},\\vec{y})<\/span><\/span> \u062a\u064f\u0639\u0637\u0649 \u0628\u0648\u0627\u0633\u0637\u0629:<\/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>\u0625\u0630\u0627 \u0643\u0627\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}=(x_1,\\cdots,x_n)<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}=(y_1,\\cdots, y_n),<\/span><\/span> \u0641\u0645\u0646 \u0627\u0644\u0633\u0647\u0644 \u0625\u062b\u0628\u0627\u062a\u060c \u0627\u0633\u062a\u0646\u0627\u062f\u064b\u0627 \u0625\u0644\u0649 \u062e\u0635\u0627\u0626\u0635 \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0642\u064a\u0627\u0633\u064a \u0648\u0627\u0644\u0645\u0642\u062f\u0627\u0631\u060c \u0623\u0646:<\/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>\u0625\u0630\u0627 \u0632\u0648\u062f\u0646\u0627 \u0627\u0644\u0641\u0636\u0627\u0621 \u0627\u0644\u0645\u062a\u062c\u0647\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u0628\u0627\u0644\u0645\u0633\u0627\u0641\u0629 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a\u0629\u060c \u0646\u062d\u0635\u0644 \u0639\u0644\u0649 <strong>\u0641\u0636\u0627\u0621 \u0625\u0642\u0644\u064a\u062f\u064a.<\/strong><\/p>\n<p>\u0628\u0646\u0627\u0621\u064b \u0639\u0644\u0649 \u0630\u0644\u0643\u060c \u0646\u0642\u0648\u0644 \u0625\u0646 <strong>\u0627\u0644\u0645\u062a\u0631\u064a\u0629 \u0641\u064a \u0627\u0644\u0641\u0636\u0627\u0621 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a \u0647\u064a \u0627\u0644\u0645\u062a\u0631\u064a\u0629 \u0627\u0644\u0645\u0633\u062a\u062d\u062b\u0629 \u0645\u0646 \u0627\u0644\u0645\u0639\u064a\u0627\u0631 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a.<\/strong><\/p>\n<h3>\u062e\u0635\u0627\u0626\u0635 \u0627\u0644\u0645\u0639\u064a\u0627\u0631 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a<\/h3>\n<p><strong>\u0628\u0645\u0627 \u0623\u0646 \u062f\u0631\u0627\u0633\u062a\u0646\u0627 \u062a\u0631\u0643\u0632 \u062a\u062d\u062f\u064a\u062f\u064b\u0627 \u0639\u0644\u0649 \u0627\u0644\u0641\u0636\u0627\u0621 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a\u060c \u0633\u064a\u0643\u0648\u0646 \u0645\u0646 \u0627\u0644\u0645\u0641\u064a\u062f \u0645\u0631\u0627\u062c\u0639\u0629 \u062e\u0635\u0627\u0626\u0635 \u0627\u0644\u0645\u0639\u064a\u0627\u0631 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a.<\/strong><\/p>\n<h4>\u0645\u062a\u0628\u0627\u064a\u0646\u0629 \u0643\u0648\u0634\u064a-\u0634\u0648\u0627\u0631\u062a\u0632<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vTFqDBEyU4Y&#038;t=1624s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u0625\u0630\u0627 \u0643\u0627\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x},\\vec{y}\\in\\mathbb{R}^n,<\/span><\/span><\/span><\/strong><\/a> \u0641\u0625\u0646 \u0627\u0644\u062e\u0627\u0635\u064a\u0629 \u0627\u0644\u062a\u0627\u0644\u064a\u0629 \u062a\u062a\u062d\u0642\u0642:<\/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>\u0627\u0644\u0625\u062b\u0628\u0627\u062a:<\/strong><\/p>\n<p>\u0644\u062a\u0643\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\lambda = (\\vec{x}\\cdot\\vec{y})\/\\|\\vec{y}\\|^2,<\/span><\/span> \u0639\u0646\u062f\u0647\u0627 \u0644\u062f\u064a\u0646\u0627:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n0\\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}\n\n\\end{array}<\/span>\n<p>\u0648\u0628\u0627\u0644\u062a\u0627\u0644\u064a \u064a\u0645\u0643\u0646\u0646\u0627 \u0627\u0644\u0642\u0648\u0644:<\/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>\u0648\u0628\u0627\u0644\u062a\u0627\u0644\u064a:<\/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>\u0648\u0623\u062e\u064a\u0631\u064b\u0627\u060c \u0628\u0623\u062e\u0630 \u0627\u0644\u062c\u0630\u0631 \u0627\u0644\u062a\u0631\u0628\u064a\u0639\u064a\u060c \u0646\u062d\u0635\u0644 \u0639\u0644\u0649 \u0627\u0644\u0646\u062a\u064a\u062c\u0629 \u0627\u0644\u0645\u0637\u0644\u0648\u0628\u0629:<\/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>\u0645\u062a\u0628\u0627\u064a\u0646\u0629 \u0627\u0644\u0645\u062b\u0644\u062b<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vTFqDBEyU4Y&#038;t=2065s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u0644\u062a\u0643\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x},\\vec{y}\\in\\mathbb{R}^n,<\/span><\/span><\/span><\/strong><\/a> \u0641\u0625\u0646 \u0647\u0630\u064a\u0646 \u0627\u0644\u0645\u062a\u062c\u0647\u064a\u0646 \u064a\u062d\u0642\u0642\u0627\u0646 \u0627\u0644\u0639\u0644\u0627\u0642\u0629 \u0627\u0644\u062a\u0627\u0644\u064a\u0629:<\/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>\u0627\u0644\u0625\u062b\u0628\u0627\u062a:<\/strong><\/p>\n<p>\u0623\u0648\u0644\u0627\u064b\u060c \u0646\u0644\u0627\u062d\u0638 \u0623\u0646:<\/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>\u0648\u0646\u0638\u0631\u064b\u0627 \u0644\u0644\u0639\u0644\u0627\u0642\u0627\u062a \u0627\u0644\u062a\u0627\u0644\u064a\u0629:<\/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>\u064a\u0645\u0643\u0646\u0646\u0627 \u0643\u062a\u0627\u0628\u0629 \u0645\u0627 \u064a\u0644\u064a:<\/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 \\\\ \\\\  &amp;\\leq  \\left(\\|\\vec{x}\\|  + \\|\\vec{y}\\| \\right)^2\n\n\\end{array}<\/span>\n<p>\u0648\u0623\u062e\u064a\u0631\u064b\u0627\u060c \u0628\u0623\u062e\u0630 \u0627\u0644\u062c\u0630\u0631 \u0627\u0644\u062a\u0631\u0628\u064a\u0639\u064a \u0646\u062d\u0635\u0644 \u0639\u0644\u0649 \u0627\u0644\u0646\u062a\u064a\u062c\u0629 \u0627\u0644\u0645\u0637\u0644\u0648\u0628\u0629:<\/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>\u0627\u0644\u062e\u0627\u062a\u0645\u0629<\/h2>\n<p>\u0639\u0644\u0649 \u0645\u062f\u0627\u0631 \u0647\u0630\u0627 \u0627\u0644\u062f\u0631\u0633\u060c \u0627\u0633\u062a\u0643\u0634\u0641\u0646\u0627 \u0627\u0644\u062e\u0635\u0627\u0626\u0635 \u0627\u0644\u0623\u0633\u0627\u0633\u064a\u0629 \u0644\u0644\u0641\u0636\u0627\u0621 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span>\u060c \u0645\u062a\u0646\u0627\u0648\u0644\u064a\u0646 \u0628\u0646\u064a\u062a\u0647 \u0627\u0644\u062c\u0628\u0631\u064a\u0629 \u0648\u0627\u0644\u0645\u062a\u0631\u064a\u0629. \u0628\u062f\u0623\u0646\u0627 \u0628\u062a\u0639\u0631\u064a\u0641 \u0627\u0644\u0639\u0645\u0644\u064a\u0627\u062a \u0627\u0644\u0623\u0633\u0627\u0633\u064a\u0629\u060c \u0645\u062b\u0644 \u062c\u0645\u0639 \u0627\u0644\u0645\u062a\u062c\u0647\u0627\u062a \u0648\u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0639\u062f\u062f\u064a\u060c \u0645\u0645\u0627 \u064a\u062b\u0628\u062a \u0623\u0646\u0647 \u0641\u0636\u0627\u0621 \u0645\u062a\u062c\u0647\u064a. \u0628\u0639\u062f \u0630\u0644\u0643\u060c \u062a\u0639\u0645\u0642\u0646\u0627 \u0641\u064a \u0645\u0641\u0647\u0648\u0645 \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0642\u064a\u0627\u0633\u064a \u0648\u0623\u0647\u0645\u064a\u062a\u0647 \u0641\u064a \u0627\u0644\u0647\u0646\u062f\u0633\u0629 \u062f\u0627\u062e\u0644 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span>\u060c \u0645\u0639 \u0627\u0644\u062a\u0631\u0643\u064a\u0632 \u0639\u0644\u0649 \u062a\u0641\u0633\u064a\u0631\u0647 \u0627\u0644\u0645\u0635\u0641\u0648\u0641\u064a \u0648\u0639\u0644\u0627\u0642\u062a\u0647 \u0628\u0627\u0644\u0648\u0638\u0627\u0626\u0641 \u0627\u0644\u062e\u0637\u064a\u0629.<\/p>\n<p>\u062b\u0645 \u0627\u0646\u062a\u0642\u0644\u0646\u0627 \u0625\u0644\u0649 \u062f\u0631\u0627\u0633\u0629 \u0627\u0644\u0645\u0639\u064a\u0627\u0631 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a \u0648\u0627\u0644\u0645\u0633\u0627\u0641\u0629 \u0627\u0644\u062a\u064a \u064a\u064f\u062d\u062f\u062b\u0647\u0627\u060c \u0645\u0633\u0644\u0637\u064a\u0646 \u0627\u0644\u0636\u0648\u0621 \u0639\u0644\u0649 \u0643\u064a\u0641\u064a\u0629 \u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0647\u0630\u0647 \u0627\u0644\u0623\u062f\u0648\u0627\u062a \u0644\u0642\u064a\u0627\u0633 \u0627\u0644\u0623\u0637\u0648\u0627\u0644 \u0648\u0627\u0644\u0645\u0633\u0627\u0641\u0627\u062a \u0641\u064a \u0647\u0630\u0627 \u0627\u0644\u0641\u0636\u0627\u0621. \u0643\u0645\u0627 \u0627\u0633\u062a\u0639\u0631\u0636\u0646\u0627 \u062e\u0635\u0627\u0626\u0635 \u0623\u0633\u0627\u0633\u064a\u0629 \u0645\u062b\u0644 \u0645\u062a\u0628\u0627\u064a\u0646\u0629 \u0643\u0648\u0634\u064a-\u0634\u0648\u0627\u0631\u062a\u0632:<\/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>\u0648\u0645\u062a\u0628\u0627\u064a\u0646\u0629 \u0627\u0644\u0645\u062b\u0644\u062b:<\/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>\u0648\u0627\u0644\u062a\u064a \u062a\u0639\u062a\u0628\u0631 \u0623\u0633\u0627\u0633\u064b\u0627 \u0645\u0647\u0645\u064b\u0627 \u0644\u062a\u0637\u0648\u064a\u0631 \u0646\u0638\u0631\u064a\u0627\u062a \u0623\u0643\u062b\u0631 \u062a\u0642\u062f\u0645\u064b\u0627 \u0641\u064a \u0627\u0644\u062a\u062d\u0644\u064a\u0644 \u0648\u0627\u0644\u0647\u0646\u062f\u0633\u0629.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u0627\u0644\u0641\u0636\u0627\u0621 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a \u0641\u064a \u0647\u0630\u0647 \u0627\u0644\u062f\u0631\u0633\u060c \u0646\u0633\u062a\u0643\u0634\u0641 \u0627\u0644\u0641\u0636\u0627\u0621 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a \u060c \u0647\u064a\u0643\u0644\u0647 \u0627\u0644\u062c\u0628\u0631\u064a \u0648\u062e\u0635\u0627\u0626\u0635\u0647 \u0627\u0644\u0645\u062a\u0631\u064a\u0629. \u0633\u062a\u062a\u0639\u0644\u0645 \u0639\u0646 \u0627\u0644\u0639\u0645\u0644\u064a\u0627\u062a \u0627\u0644\u0645\u062a\u062c\u0647\u064a\u0629\u060c \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0642\u064a\u0627\u0633\u064a\u060c \u0627\u0644\u0645\u0639\u064a\u0627\u0631 \u0648\u0627\u0644\u0645\u0633\u0627\u0641\u0629 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a\u0629\u060c \u0648\u0647\u064a \u0645\u0641\u0627\u0647\u064a\u0645 \u0623\u0633\u0627\u0633\u064a\u0629 \u0641\u064a \u0627\u0644\u0647\u0646\u062f\u0633\u0629 \u0648\u0627\u0644\u062a\u062d\u0644\u064a\u0644. \u0645\u0639 \u0634\u0631\u0648\u062d\u0627\u062a \u0648\u0627\u0636\u062d\u0629 \u0648\u0623\u0645\u062b\u0644\u0629 \u062a\u0648\u0636\u064a\u062d\u064a\u0629\u060c \u0633\u064a\u0645\u0643\u0646\u0643 \u0647\u0630\u0627 \u0627\u0644\u0645\u062d\u062a\u0648\u0649 \u0645\u0646 \u0641\u0647\u0645 \u0643\u064a\u0641\u064a\u0629 \u062a\u0645\u062b\u064a\u0644 \u0627\u0644\u0641\u0636\u0627\u0621 \u0631\u064a\u0627\u0636\u064a\u064b\u0627 \u0641\u064a \u0623\u0628\u0639\u0627\u062f \u0645\u062a\u0639\u062f\u062f\u0629. \u0623\u0647\u062f\u0627\u0641 \u0627\u0644\u062a\u0639\u0644\u0645: \u0628\u0639\u062f \u0627\u0644\u0627\u0646\u062a\u0647\u0627\u0621 \u0645\u0646 \u0647\u0630\u0627 \u0627\u0644\u062f\u0631\u0633\u060c \u0633\u064a\u0643\u0648\u0646 \u0627\u0644\u0637\u0627\u0644\u0628 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":32541,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":34,"footnotes":""},"categories":[797,1122,565],"tags":[],"class_list":["post-32556","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-797","category-1122","category-565"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>\u0627\u0644\u0641\u0636\u0627\u0621 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a \u211d\u207f - toposuranos.com\/material<\/title>\n<meta name=\"description\" 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Mi objetivo es hacer que estos campos sean f\u00e1cilmente comprensibles para todos, proporcionando las herramientas para explorar no solo el mundo que nos rodea, sino tambi\u00e9n las profundidades de nuestra propia existencia y el orden natural que nos conecta con el cosmos.","sameAs":["http:\/\/toposuranos.com\/material"],"url":"https:\/\/toposuranos.com\/material\/author\/giorgio-reveco\/"}]}},"_links":{"self":[{"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts\/32556","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/comments?post=32556"}],"version-history":[{"count":0,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts\/32556\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/media\/32541"}],"wp:attachment":[{"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/media?parent=32556"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/categories?post=32556"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/tags?post=32556"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}