{"id":32560,"date":"2022-03-08T13:00:45","date_gmt":"2022-03-08T13:00:45","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=32560"},"modified":"2025-03-10T02:24:52","modified_gmt":"2025-03-10T02:24:52","slug":"%e0%a4%af%e0%a5%82%e0%a4%95%e0%a5%8d%e0%a4%b2%e0%a4%bf%e0%a4%a1%e0%a4%bf%e0%a4%af%e0%a4%a8-%e0%a4%b8%e0%a5%8d%e0%a4%a5%e0%a4%be%e0%a4%a8-r%e2%81%bf","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/hi\/%e0%a4%af%e0%a5%82%e0%a4%95%e0%a5%8d%e0%a4%b2%e0%a4%bf%e0%a4%a1%e0%a4%bf%e0%a4%af%e0%a4%a8-%e0%a4%b8%e0%a5%8d%e0%a4%a5%e0%a4%be%e0%a4%a8-r%e2%81%bf\/","title":{"rendered":"\u092f\u0942\u0915\u094d\u0932\u093f\u0921\u093f\u092f\u0928 \u0938\u094d\u0925\u093e\u0928 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>\u092f\u0942\u0915\u094d\u0932\u093f\u0921\u093f\u092f\u0928 \u0938\u094d\u0925\u093e\u0928 <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>\u0907\u0938 \u0915\u0915\u094d\u0937\u093e \u092e\u0947\u0902 \u0939\u092e <strong>\u092f\u0942\u0915\u094d\u0932\u093f\u0921\u093f\u092f\u0928 \u0938\u094d\u0925\u093e\u0928 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span><\/strong>, \u0907\u0938\u0915\u0940 \u092c\u0940\u091c\u0940\u092f \u0938\u0902\u0930\u091a\u0928\u093e \u0914\u0930 \u092e\u0940\u091f\u094d\u0930\u093f\u0915 \u0917\u0941\u0923\u094b\u0902 \u0915\u093e \u0905\u0928\u094d\u0935\u0947\u0937\u0923 \u0915\u0930\u0947\u0902\u0917\u0947\u0964 \u0906\u092a \u0935\u0947\u0915\u094d\u091f\u0930 \u0938\u0902\u0915\u094d\u0930\u093f\u092f\u093e\u0913\u0902, <strong>\u0921\u0949\u091f \u0909\u0924\u094d\u092a\u093e\u0926<\/strong>, <strong>\u0928\u0949\u0930\u094d\u092e<\/strong> \u0914\u0930 <strong>\u092f\u0942\u0915\u094d\u0932\u093f\u0921\u093f\u092f\u0928 \u0926\u0942\u0930\u0940<\/strong> \u0915\u0947 \u092c\u093e\u0930\u0947 \u092e\u0947\u0902 \u091c\u093e\u0928\u0947\u0902\u0917\u0947, \u091c\u094b \u091c\u094d\u092f\u093e\u092e\u093f\u0924\u093f \u0914\u0930 \u0935\u093f\u0936\u094d\u0932\u0947\u0937\u0923 \u092e\u0947\u0902 \u0906\u0935\u0936\u094d\u092f\u0915 \u0905\u0935\u0927\u093e\u0930\u0923\u093e\u090f\u0901 \u0939\u0948\u0902\u0964 \u0938\u094d\u092a\u0937\u094d\u091f \u0935\u094d\u092f\u093e\u0916\u094d\u092f\u093e\u0913\u0902 \u0914\u0930 \u0938\u0939\u091c \u0909\u0926\u093e\u0939\u0930\u0923\u094b\u0902 \u0915\u0947 \u0938\u093e\u0925, \u092f\u0939 \u0938\u093e\u092e\u0917\u094d\u0930\u0940 \u0906\u092a\u0915\u094b \u092f\u0939 \u0938\u092e\u091d\u0928\u0947 \u092e\u0947\u0902 \u0938\u0915\u094d\u0937\u092e \u092c\u0928\u093e\u090f\u0917\u0940 \u0915\u093f \u092c\u0939\u0941-\u0906\u092f\u093e\u092e\u0940 \u0938\u094d\u0925\u093e\u0928 \u0915\u094b \u0917\u0923\u093f\u0924\u0940\u092f \u0930\u0942\u092a \u0938\u0947 \u0915\u0948\u0938\u0947 \u092e\u0949\u0921\u0932 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/em><\/p>\n<p style=\"text-align:center;\" dir=\"ltr\">\n<strong>\u0905\u0927\u094d\u092f\u092f\u0928 \u0932\u0915\u094d\u0937\u094d\u092f:<\/strong><br \/>\n\u0907\u0938 \u0915\u0915\u094d\u0937\u093e \u0915\u094b \u092a\u0942\u0930\u093e \u0915\u0930\u0928\u0947 \u0915\u0947 \u092c\u093e\u0926, \u091b\u093e\u0924\u094d\u0930 \u0938\u0915\u094d\u0937\u092e \u0939\u094b\u0917\u093e:\n<\/p>\n<ol>\n<li><strong>\u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u0930\u0947\u0902<\/strong> \u092f\u0942\u0915\u094d\u0932\u093f\u0921\u093f\u092f\u0928 \u0938\u094d\u0925\u093e\u0928 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u0914\u0930 \u0907\u0938\u0915\u0940 \u092e\u094c\u0932\u093f\u0915 \u0935\u093f\u0936\u0947\u0937\u0924\u093e\u090f\u0901\u0964<\/li>\n<li><strong>\u0938\u092e\u091d\u093e\u090f\u0901<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u0915\u0940 \u0935\u0947\u0915\u094d\u091f\u0930 \u0938\u0902\u0930\u091a\u0928\u093e \u0914\u0930 \u0907\u0938\u0915\u0940 \u092e\u0942\u0932\u092d\u0942\u0924 \u0938\u0902\u0915\u094d\u0930\u093f\u092f\u093e\u090f\u0901\u0964<\/li>\n<li><strong>\u0932\u093e\u0917\u0942 \u0915\u0930\u0947\u0902<\/strong> \u0921\u0949\u091f \u0909\u0924\u094d\u092a\u093e\u0926 \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u094b\u0923\u094b\u0902 \u0914\u0930 \u0935\u0947\u0915\u094d\u091f\u0930\u094b\u0902 \u0915\u0947 \u092a\u094d\u0930\u0915\u094d\u0937\u0947\u092a\u0923 \u0915\u0940 \u0917\u0923\u0928\u093e \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f\u0964<\/li>\n<li><strong>\u0938\u093e\u092c\u093f\u0924 \u0915\u0930\u0947\u0902<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u092e\u0947\u0902 \u0921\u0949\u091f \u0909\u0924\u094d\u092a\u093e\u0926 \u0915\u0947 \u092c\u0940\u091c\u0940\u092f \u0914\u0930 \u092e\u0940\u091f\u094d\u0930\u093f\u0915 \u0917\u0941\u0923\u0964<\/li>\n<li><strong>\u092a\u094d\u0930\u092f\u094b\u0917 \u0915\u0930\u0947\u0902<\/strong> \u092f\u0942\u0915\u094d\u0932\u093f\u0921\u093f\u092f\u0928 \u0928\u0949\u0930\u094d\u092e \u0935\u0947\u0915\u094d\u091f\u0930 \u0915\u0940 \u092a\u0930\u093f\u092e\u093e\u0923 \u0928\u093f\u0930\u094d\u0927\u093e\u0930\u093f\u0924 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f\u0964<\/li>\n<li><strong>\u0917\u0923\u0928\u093e \u0915\u0930\u0947\u0902<\/strong> \u0926\u094b \u092c\u093f\u0902\u0926\u0941\u0913\u0902 \u0915\u0947 \u092c\u0940\u091a \u0915\u0940 \u092f\u0942\u0915\u094d\u0932\u093f\u0921\u093f\u092f\u0928 \u0926\u0942\u0930\u0940 \u0914\u0930 \u0909\u0938\u0915\u0947 \u091c\u094d\u092f\u093e\u092e\u093f\u0924\u0940\u092f \u0905\u0930\u094d\u0925 \u0915\u093e \u0935\u093f\u0936\u094d\u0932\u0947\u0937\u0923 \u0915\u0930\u0947\u0902\u0964<\/li>\n<li><strong>\u0938\u0924\u094d\u092f\u093e\u092a\u093f\u0924 \u0915\u0930\u0947\u0902<\/strong> \u0915\u094b\u0936\u0940-\u0936\u094d\u0935\u093e\u0930\u094d\u091c \u0905\u0938\u092e\u093e\u0928\u0924\u093e \u0914\u0930 \u0924\u094d\u0930\u093f\u092d\u0941\u091c \u0905\u0938\u092e\u093e\u0928\u0924\u093e \u091c\u0948\u0938\u0940 \u092e\u094c\u0932\u093f\u0915 \u0905\u0938\u092e\u093e\u0928\u0924\u093e\u0913\u0902 \u0915\u0940 \u0935\u0948\u0927\u0924\u093e\u0964<\/li>\n<\/ol>\n<p style=\"text-align:center;\" dir=\"ltr\"><strong>\u0905\u0928\u0941\u0915\u094d\u0930\u092e\u0923\u093f\u0915\u093e<\/strong><br \/>\n<a href=\"#1\">\u092f\u0942\u0915\u094d\u0932\u093f\u0921\u093f\u092f\u0928 \u0938\u094d\u0925\u093e\u0928 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span><\/a><br \/>\n<a href=\"#2\">\u0921\u0949\u091f \u0909\u0924\u094d\u092a\u093e\u0926<\/a><br \/>\n<a href=\"#3\">\u0928\u0949\u0930\u094d\u092e \u0914\u0930 \u092f\u0942\u0915\u094d\u0932\u093f\u0921\u093f\u092f\u0928 \u0926\u0942\u0930\u0940<\/a><br \/>\n<a href=\"#4\">\u0928\u093f\u0937\u094d\u0915\u0930\u094d\u0937<\/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>\u0935\u0947\u0915\u094d\u091f\u0930 \u0938\u094d\u0925\u093e\u0928 <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;\">\u0928\u093f\u0936\u094d\u091a\u093f\u0924 \u0930\u0942\u092a \u0938\u0947, \u0907\u0938 \u092c\u093f\u0902\u0926\u0941 \u092a\u0930 \u092a\u0939\u0941\u0901\u091a\u0928\u0947 \u0938\u0947 \u092a\u0939\u0932\u0947, \u0906\u092a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R},<\/span><\/span> \u092f\u093e \u0938\u092e\u0924\u0932 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^2,<\/span><\/span> \u092f\u093e \u0938\u094d\u0925\u093e\u0928 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3.<\/span><\/span> \u0915\u0940 \u0935\u093f\u0936\u0947\u0937\u0924\u093e\u0913\u0902 \u0938\u0947 \u092a\u0930\u093f\u091a\u093f\u0924 \u0939\u094b\u0902\u0917\u0947\u0964<\/span><\/strong><\/a> \u092f\u0947 \u0938\u092d\u0940 \u0935\u093f\u091a\u093e\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n.<\/span><\/span> \u0915\u094b \u0938\u092e\u091d\u0928\u0947 \u092e\u0947\u0902 \u0938\u0939\u093e\u092f\u0915 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902\u0964 \u092e\u0941\u0916\u094d\u092f \u0930\u0942\u092a \u0938\u0947, \u0938\u0947\u091f <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> \u091c\u093f\u0938\u092e\u0947\u0902 \u0935\u0947\u0915\u094d\u091f\u0930 \u092f\u094b\u0917 \u0914\u0930 \u0938\u094d\u0915\u0947\u0932\u0930 \u0917\u0941\u0923\u093e \u0915\u0940 \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0938\u0902\u0915\u094d\u0930\u093f\u092f\u093e\u090f\u0901 \u0939\u094b\u0924\u0940 \u0939\u0948\u0902, \u090f\u0915 \u0935\u0947\u0915\u094d\u091f\u0930 \u0938\u094d\u0925\u093e\u0928 \u0939\u0948\u0964 \u0907\u0938\u0947 \u0938\u092e\u091d\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u0915\u0940 \u092e\u0942\u0932\u092d\u0942\u0924 \u0938\u0902\u0915\u094d\u0930\u093f\u092f\u093e\u0913\u0902 \u0915\u0940 \u0938\u092e\u0940\u0915\u094d\u0937\u093e \u0915\u0930\u0924\u0947 \u0939\u0948\u0902\u0964<\/p>\n<h3><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u0915\u0940 \u092e\u0942\u0932\u092d\u0942\u0924 \u0938\u0902\u0915\u094d\u0930\u093f\u092f\u093e\u090f\u0901<\/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;\">\u092f\u0926\u093f <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> \u0935\u0947\u0915\u094d\u091f\u0930 \u0939\u0948\u0902 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u092e\u0947\u0902 \u0914\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> \u0915\u094b\u0908 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u0938\u094d\u0915\u0947\u0932\u0930 \u0939\u0948,<\/span><\/strong> <\/a>\u0924\u094b <strong>\u0935\u0947\u0915\u094d\u091f\u0930 \u092f\u094b\u0917<\/strong> \u0914\u0930 <strong>\u0938\u094d\u0915\u0947\u0932\u0930 \u0917\u0941\u0923\u093e<\/strong> \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0930\u0942\u092a \u092e\u0947\u0902 \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902:<\/p>\n<p><strong>\u0935\u0947\u0915\u094d\u091f\u0930 \u092f\u094b\u0917:<\/strong> \u0935\u0947\u0915\u094d\u091f\u0930 \u092f\u094b\u0917 \u0915\u094b \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u092b\u0932\u0928 \u0926\u094d\u0935\u093e\u0930\u093e \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948:<\/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>\u0938\u094d\u0915\u0947\u0932\u0930 \u0917\u0941\u0923\u093e:<\/strong> \u0938\u094d\u0915\u0947\u0932\u0930 \u0917\u0941\u0923\u093e \u0915\u094b \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u092b\u0932\u0928 \u0926\u094d\u0935\u093e\u0930\u093e \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948:<\/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> \u0915\u093e \u0935\u0947\u0915\u094d\u091f\u0930 \u0938\u094d\u0925\u093e\u0928\u0940\u092f \u0917\u0941\u0923<\/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;\">\u0938\u094d\u0925\u093e\u0928 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u0909\u092a\u0930\u094b\u0915\u094d\u0924 \u0938\u0902\u0915\u094d\u0930\u093f\u092f\u093e\u0913\u0902 \u0915\u0947 \u0938\u093e\u0925<\/span><\/strong><\/a> \u090f\u0915 <strong>\u0935\u0947\u0915\u094d\u091f\u0930 \u0938\u094d\u0925\u093e\u0928<\/strong> \u0939\u0948 \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u0907\u0938\u0915\u0940 \u0938\u0902\u0915\u094d\u0930\u093f\u092f\u093e\u090f\u0901 \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0917\u0941\u0923\u094b\u0902 \u0915\u094b \u0938\u0902\u0924\u0941\u0937\u094d\u091f \u0915\u0930\u0924\u0940 \u0939\u0948\u0902:<\/p>\n<p>\u092a\u0939\u0932\u0947 \u0939\u092e <strong>\u092a\u0930\u093f\u0935\u0930\u094d\u0924\u0928\u0940\u092f\u0924\u093e<\/strong> \u0914\u0930 <strong>\u0938\u0939\u0932\u0917\u094d\u0928\u0924\u093e<\/strong> \u0917\u0941\u0923 \u0926\u0947\u0916\u0924\u0947 \u0939\u0948\u0902\u0964<\/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>\u0938\u094d\u0915\u0947\u0932\u0930\u094b\u0902 \u0915\u093e \u092f\u094b\u0917 \u0938\u094d\u0915\u0947\u0932\u0930 \u0917\u0941\u0923\u093e \u0915\u0947 \u0938\u0902\u092c\u0902\u0927 \u092e\u0947\u0902 \u0935\u093f\u0924\u0930\u093f\u0924 \u0939\u094b\u0924\u093e \u0939\u0948 \u0914\u0930 \u0935\u0947\u0915\u094d\u091f\u0930 \u092f\u094b\u0917 \u0938\u094d\u0915\u0947\u0932\u0930 \u0917\u0941\u0923\u093e \u0915\u0947 \u0938\u0902\u092c\u0902\u0927 \u092e\u0947\u0902 \u0935\u093f\u0924\u0930\u093f\u0924 \u0939\u094b\u0924\u093e \u0939\u0948;<\/strong> \u0905\u0930\u094d\u0925\u093e\u0924, \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0938\u092e\u093e\u0928\u0924\u093e\u090f\u0901 \u0938\u0902\u0924\u0941\u0937\u094d\u091f \u0939\u094b\u0924\u0940 \u0939\u0948\u0902:<\/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>\u090f\u0915 <strong>\u0938\u0902\u0935\u0930\u094d\u0926\u094d\u0927\u0915 \u0924\u091f\u0938\u094d\u0925 \u0924\u0924\u094d\u0935<\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{0}=(0,\\cdots, 0)<\/span> \u092e\u094c\u091c\u0942\u0926 \u0939\u094b\u0924\u093e \u0939\u0948, \u091c\u094b \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0917\u0941\u0923 \u0915\u094b \u0938\u0902\u0924\u0941\u0937\u094d\u091f \u0915\u0930\u0924\u093e \u0939\u0948:<\/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>\u0938\u094d\u0915\u0947\u0932\u0930 \u0917\u0941\u0923\u093e \u0915\u0947 \u0932\u093f\u090f \u090f\u0915 <strong>\u0917\u0941\u0923\u093e\u0924\u094d\u092e\u0915 \u0924\u091f\u0938\u094d\u0925 \u0924\u0924\u094d\u0935<\/strong> \u092e\u094c\u091c\u0942\u0926 \u0939\u094b\u0924\u093e \u0939\u0948:<\/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>\u0914\u0930 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0935\u0947\u0915\u094d\u091f\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\in\\mathbb{R}^n<\/span><\/span> \u0915\u0947 \u092a\u093e\u0938 \u090f\u0915 <strong>\u0935\u093f\u0932\u094b\u092a\u0928\u093e\u0924\u094d\u092e\u0915 \u0935\u094d\u092f\u0941\u0924\u094d\u0915\u094d\u0930\u092e<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-\\vec{x},<\/span><\/span> \u0939\u094b\u0924\u093e \u0939\u0948, \u091c\u094b \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0917\u0941\u0923 \u0915\u094b \u0938\u0902\u0924\u0941\u0937\u094d\u091f \u0915\u0930\u0924\u093e \u0939\u0948:<\/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>\u0921\u0949\u091f \u0909\u0924\u094d\u092a\u093e\u0926<\/h2>\n<p>\u092f\u0926\u093f \u0939\u092e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u0915\u094b \u090f\u0915 \u0935\u0947\u0915\u094d\u091f\u0930 \u0938\u094d\u0925\u093e\u0928 \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0926\u0947\u0916\u0924\u0947 \u0939\u0948\u0902, \u0924\u094b \u0939\u092e \u0926\u0947\u0916\u0947\u0902\u0917\u0947 \u0915\u093f \u0907\u0938\u092e\u0947\u0902 \u0935\u0947\u0915\u094d\u091f\u0930\u094b\u0902 \u0915\u0947 \u092c\u0940\u091a \u0915\u094b\u0908 \u0917\u0941\u0923\u093e \u0938\u0902\u0915\u094d\u0930\u093f\u092f\u093e \u0928\u0939\u0940\u0902 \u0939\u094b\u0924\u0940; \u092e\u0942\u0932 \u0930\u0942\u092a \u0938\u0947, \u0939\u092e \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0930\u0942\u092a \u0938\u0947 \u0926\u094b \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0940 \u0924\u0930\u0939 \u0935\u0947\u0915\u094d\u091f\u0930\u094b\u0902 \u0915\u094b \u00ab\u0917\u0941\u0923\u093e\u00bb \u0928\u0939\u0940\u0902 \u0915\u0930 \u0938\u0915\u0924\u0947\u0964 \u0939\u093e\u0932\u093e\u0901\u0915\u093f, \u0935\u0947\u0915\u094d\u091f\u0930\u094b\u0902 \u0915\u0947 \u092c\u0940\u091a \u0907\u0938 \u0938\u0902\u0915\u094d\u0930\u093f\u092f\u093e \u0915\u094b \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u0930\u0928\u093e \u0938\u0902\u092d\u0935 \u0939\u0948, \u0914\u0930 \u0907\u0938\u0947 <strong>\u0921\u0949\u091f \u0909\u0924\u094d\u092a\u093e\u0926<\/strong> \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u091c\u093e\u0928\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/p>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=HL85aSpHdsI&#038;t=349s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u0921\u0949\u091f \u0909\u0924\u094d\u092a\u093e\u0926 \u0915\u094b \u0938\u094d\u0915\u0947\u0932\u0930 \u0917\u0941\u0923\u093e \u0938\u0947 \u092d\u094d\u0930\u092e\u093f\u0924 \u0928\u0939\u0940\u0902 \u0915\u093f\u092f\u093e \u091c\u093e\u0928\u093e \u091a\u093e\u0939\u093f\u090f,<\/span><\/strong><\/a> \u092a\u0939\u0932\u093e \u0926\u094b \u0935\u0947\u0915\u094d\u091f\u0930\u094b\u0902 \u0915\u0947 \u092c\u0940\u091a \u0915\u093e \u0917\u0941\u0923\u093e \u0939\u0948 \u091c\u094b \u090f\u0915 \u0938\u094d\u0915\u0947\u0932\u0930 \u0909\u0924\u094d\u092a\u0928\u094d\u0928 \u0915\u0930\u0924\u093e \u0939\u0948, \u091c\u092c\u0915\u093f \u0926\u0942\u0938\u0930\u093e \u090f\u0915 \u0938\u094d\u0915\u0947\u0932\u0930 \u0914\u0930 \u0935\u0947\u0915\u094d\u091f\u0930 \u0915\u0947 \u092c\u0940\u091a \u0915\u093e \u0917\u0941\u0923\u093e \u0939\u0948 \u091c\u094b \u090f\u0915 \u0928\u092f\u093e \u0935\u0947\u0915\u094d\u091f\u0930 \u0909\u0924\u094d\u092a\u0928\u094d\u0928 \u0915\u0930\u0924\u093e \u0939\u0948\u0964 \u0906\u0907\u090f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n:<\/span><\/span> \u0915\u0947 \u0926\u094b \u0935\u0947\u0915\u094d\u091f\u0930\u094b\u0902 \u092a\u0930 \u0935\u093f\u091a\u093e\u0930 \u0915\u0930\u0947\u0902: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}=(x_1, \\cdots, x_n)<\/span><\/span> \u0914\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}=(y_1, \\cdots, y_n).<\/span><\/span> \u0907\u0928\u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0915\u0947, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span><\/span> \u0914\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span><\/span> \u0915\u0947 \u092c\u0940\u091a \u0915\u093e \u0921\u0949\u091f \u0909\u0924\u094d\u092a\u093e\u0926 \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0938\u0942\u0924\u094d\u0930 \u0926\u094d\u0935\u093e\u0930\u093e \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948:<\/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><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u0915\u0947 \u0935\u0947\u0915\u094d\u091f\u0930\u094b\u0902 \u0915\u0947 \u092c\u0940\u091a \u0921\u0949\u091f \u0909\u0924\u094d\u092a\u093e\u0926 \u0915\u094b \u0935\u094d\u092f\u0915\u094d\u0924 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0915\u0908 \u0924\u0930\u0940\u0915\u0947 \u0939\u0948\u0902, \u090f\u0915 \u0924\u0930\u0940\u0915\u093e \u0939\u092e\u0928\u0947 \u0905\u092d\u0940 \u0926\u0947\u0916\u093e \u0939\u0948, \u091c\u092c\u0915\u093f \u0926\u0942\u0938\u0930\u093e \u090f\u0915 \u0906\u0927\u093e\u0930 \u0914\u0930 <strong>\u0906\u0907\u0902\u0938\u094d\u091f\u0940\u0928 \u0938\u0902\u0915\u0947\u0924\u0928 \u0938\u092e\u094d\u092e\u0924\u093f<\/strong> \u0915\u094b \u0927\u094d\u092f\u093e\u0928 \u092e\u0947\u0902 \u0930\u0916\u0924\u0947 \u0939\u0941\u090f \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948: \u092f\u0926\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\hat{e}_i\\}_{i=\\overline{1,n}}<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u0915\u093e \u090f\u0915 \u0906\u0927\u093e\u0930 \u0939\u0948 (\u0906\u092e\u0924\u094c\u0930 \u092a\u0930 \u0915\u0948\u0928\u094b\u0928\u093f\u0915\u0932 \u0906\u0927\u093e\u0930), \u0924\u094b \u0935\u0947\u0915\u094d\u091f\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span><\/span> \u0914\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span><\/span> \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0930\u0942\u092a \u092e\u0947\u0902 \u0932\u093f\u0916\u0947 \u091c\u093e \u0938\u0915\u0924\u0947 \u0939\u0948\u0902:<\/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>\u0907\u0938\u092e\u0947\u0902 \u0938\u094d\u092a\u0937\u094d\u091f \u0930\u0942\u092a \u0938\u0947 \u092f\u0939 \u0926\u0930\u094d\u0936\u093e\u092f\u093e \u0917\u092f\u093e \u0939\u0948 \u0915\u093f \u0935\u0947\u0915\u094d\u091f\u0930\u094b\u0902 \u0915\u0947 \u0917\u0941\u0923\u093e\u0902\u0915 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_i<\/span><\/span> \u0914\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y_i<\/span><\/span> \u0938\u094d\u0925\u093e\u0928 \u0915\u0947 \u0906\u0927\u093e\u0930 \u0915\u0947 \u0938\u093e\u092a\u0947\u0915\u094d\u0937 \u0939\u0948\u0902\u0964<\/p>\n<h3>\u0906\u0907\u0902\u0938\u094d\u091f\u0940\u0928 \u0938\u0902\u0915\u0947\u0924\u0928 \u0938\u092e\u094d\u092e\u0924\u093f<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=HL85aSpHdsI&#038;t=518s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u0906\u0907\u0902\u0938\u094d\u091f\u0940\u0928 \u0938\u0902\u0915\u0947\u0924\u0928 \u0938\u092e\u094d\u092e\u0924\u093f<\/span><\/strong><\/a> \u0939\u092e\u0947\u0902 \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0930\u0942\u092a \u0938\u0947 \u0935\u0947\u0915\u094d\u091f\u0930\u094b\u0902 \u0914\u0930 \u0935\u093f\u0936\u0947\u0937 \u0930\u0942\u092a \u0938\u0947 \u0921\u0949\u091f \u0909\u0924\u094d\u092a\u093e\u0926 \u0915\u0940 \u0905\u092d\u093f\u0935\u094d\u092f\u0915\u094d\u0924\u093f \u0915\u094b \u0938\u0930\u0932 \u092c\u0928\u093e\u0928\u0947 \u0915\u0940 \u0905\u0928\u0941\u092e\u0924\u093f \u0926\u0947\u0924\u093e \u0939\u0948\u0964 \u092f\u0926\u093f \u0939\u092e \u092a\u093f\u091b\u0932\u0947 \u0926\u094b \u0938\u092e\u0940\u0915\u0930\u0923\u094b\u0902 \u0915\u094b \u0926\u0947\u0916\u0947\u0902, \u0924\u094b \u0939\u092e \u0926\u0947\u0916\u0947\u0902\u0917\u0947 \u0915\u093f \u0909\u092a-\u0938\u0942\u091a\u0915\u093e\u0902\u0915 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">i<\/span><\/span> \u0935\u0947\u0915\u094d\u091f\u0930 \u0917\u0941\u0923\u093e\u0902\u0915 \u0914\u0930 \u0906\u0927\u093e\u0930 \u0924\u0924\u094d\u0935 \u0926\u094b\u0928\u094b\u0902 \u092e\u0947\u0902 \u0926\u094b\u0939\u0930\u093e\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u0906\u0907\u0902\u0938\u094d\u091f\u0940\u0928 \u0915\u0947 \u0905\u0928\u0941\u0938\u093e\u0930, \u0926\u094b\u0939\u0930\u093e\u090f \u0917\u090f \u0907\u0902\u0921\u0947\u0915\u094d\u0938 \u0915\u0940 \u0909\u092a\u0938\u094d\u0925\u093f\u0924\u093f \u0907\u0938 \u092f\u094b\u0917 \u0915\u094b \u092e\u093e\u0928\u094d\u092f \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u092a\u0930\u094d\u092f\u093e\u092a\u094d\u0924 \u0939\u094b\u0924\u0940 \u0939\u0948, \u091c\u093f\u0938\u0938\u0947 \u0907\u0938\u0947 \u0938\u0902\u0915\u094d\u0937\u0947\u092a \u092e\u0947\u0902 \u0932\u093f\u0916\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948:<\/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>\u0907\u0938 \u0938\u0902\u0915\u0947\u0924\u0928 \u0938\u092e\u094d\u092e\u0924\u093f \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0924\u0947 \u0939\u0941\u090f, \u0921\u0949\u091f \u0909\u0924\u094d\u092a\u093e\u0926 \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0930\u0942\u092a \u0932\u0947\u0924\u093e \u0939\u0948:<\/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>\u0907\u0938 \u0905\u0902\u0924\u093f\u092e \u0938\u092e\u093e\u0928\u0924\u093e \u092e\u0947\u0902 \u092f\u0939 \u092e\u093e\u0928\u093e \u0917\u092f\u093e \u0939\u0948 \u0915\u093f \u0939\u092e \u0915\u0948\u0928\u094b\u0928\u093f\u0915\u0932 \u0906\u0927\u093e\u0930 \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930 \u0930\u0939\u0947 \u0939\u0948\u0902\u0964<\/p>\n<h3>\u0921\u0949\u091f \u0909\u0924\u094d\u092a\u093e\u0926 \u0915\u0947 \u0905\u0928\u094d\u092f \u0938\u0902\u0915\u0947\u0924\u0928<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=HL85aSpHdsI&#038;t=825s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u0935\u0947\u0915\u094d\u091f\u0930\u094b\u0902 \u0914\u0930 \u0909\u0928\u0915\u0940 \u0938\u0902\u0915\u094d\u0930\u093f\u092f\u093e\u0913\u0902 \u0915\u0947 \u0932\u093f\u090f \u0909\u092a\u092f\u094b\u0917 \u0915\u093f\u092f\u093e \u091c\u093e\u0928\u0947 \u0935\u093e\u0932\u093e \u0938\u0902\u0915\u0947\u0924\u0928 \u0938\u092d\u0940 \u0938\u0902\u0926\u0930\u094d\u092d\u094b\u0902 \u092e\u0947\u0902 \u0938\u092e\u093e\u0928 \u0928\u0939\u0940\u0902 \u0939\u094b\u0924\u093e,<\/span><\/strong><\/a> \u092e\u0948\u0902\u0928\u0947 \u0907\u0938 \u092a\u094d\u0930\u0935\u093f\u0937\u094d\u091f\u093f \u0915\u0947 \u092a\u0939\u0932\u0947 \u092a\u0948\u0930\u093e\u0917\u094d\u0930\u093e\u092b\u093c\u094b\u0902 \u092e\u0947\u0902 \u091c\u094b \u0938\u0902\u0915\u0947\u0924\u0928 \u0909\u092a\u092f\u094b\u0917 \u0915\u093f\u092f\u093e \u0939\u0948, \u0935\u0939 \u0924\u092c \u0938\u092c\u0938\u0947 \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0939\u0948 \u091c\u092c \u0939\u092e \u0915\u0948\u0932\u0915\u0941\u0932\u0938 \u092e\u0947\u0902 \u0915\u093e\u092e \u0915\u0930 \u0930\u0939\u0947 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902\u0964 \u091c\u092c \u0930\u0947\u0916\u0940\u092f \u092c\u0940\u091c\u0917\u0923\u093f\u0924 \u092e\u0947\u0902 \u0915\u093e\u092e \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948, \u0924\u094b \u0915\u092d\u0940-\u0915\u092d\u0940 \u0935\u0947\u0915\u094d\u091f\u0930 \u0914\u0930 \u0915\u094b\u0935\u0947\u0915\u094d\u091f\u0930 \u092e\u0947\u0902 \u092d\u0947\u0926 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948:<\/p>\n<p>\u091c\u092c \u0939\u092e \u0935\u0947\u0915\u094d\u091f\u0930\u094b\u0902 \u0915\u0940 \u092c\u093e\u0924 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902, \u0924\u094b \u0939\u092e\u093e\u0930\u093e \u0924\u093e\u0924\u094d\u092a\u0930\u094d\u092f \u00ab\u0938\u094d\u0924\u0902\u092d \u0935\u0947\u0915\u094d\u091f\u0930\u00bb \u0938\u0947 \u0939\u094b\u0924\u093e \u0939\u0948, \u091c\u093f\u0938\u0947 \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0930\u0942\u092a \u092e\u0947\u0902 \u0917\u0923\u093f\u0924\u0940\u092f \u0930\u0942\u092a \u0938\u0947 \u0926\u0930\u094d\u0936\u093e\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948:<\/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>\u091c\u092c\u0915\u093f \u091c\u092c \u0939\u092e \u0915\u094b\u0935\u0947\u0915\u094d\u091f\u0930\u094b\u0902 \u0915\u0940 \u092c\u093e\u0924 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902, \u0924\u094b \u0939\u092e\u093e\u0930\u093e \u0924\u093e\u0924\u094d\u092a\u0930\u094d\u092f \u00ab\u092a\u0902\u0915\u094d\u0924\u093f \u0935\u0947\u0915\u094d\u091f\u0930\u00bb \u0938\u0947 \u0939\u094b\u0924\u093e \u0939\u0948, \u091c\u093f\u0938\u0947 \u0917\u0923\u093f\u0924\u0940\u092f \u0930\u0942\u092a \u0938\u0947 \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0930\u0942\u092a \u092e\u0947\u0902 \u0926\u0930\u094d\u0936\u093e\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948:<\/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>\u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930, \u0926\u094b \u0935\u0947\u0915\u094d\u091f\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}=(x_1,\\cdots,x_n)<\/span><\/span> \u0914\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}=(y_1,\\cdots,y_n)<\/span><\/span> \u0915\u0947 \u092c\u0940\u091a \u0915\u093e \u0921\u0949\u091f \u0909\u0924\u094d\u092a\u093e\u0926 \u00ab\u0915\u094b\u0935\u0947\u0915\u094d\u091f\u0930\u00bb <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_i<\/span><\/span> \u0914\u0930 \u0935\u0947\u0915\u094d\u091f\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y^i,<\/span><\/span> \u0915\u0947 \u092c\u0940\u091a \u0915\u093e \u0917\u0923\u093f\u0924\u0940\u092f \u0917\u0941\u0923\u0928 \u0939\u0948, \u091c\u093f\u0938\u0938\u0947 \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0924\u0940 \u0939\u0948:<\/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>\u0927\u094d\u092f\u093e\u0928 \u0926\u0947\u0902 \u0915\u093f \u0907\u0938 \u0905\u0902\u0924\u093f\u092e \u0938\u092e\u093e\u0928\u0924\u093e \u092e\u0947\u0902 \u0906\u0907\u0902\u0938\u094d\u091f\u0940\u0928 \u092f\u094b\u0917 \u0938\u0902\u0915\u0947\u0924\u0928 \u092b\u093f\u0930 \u0938\u0947 \u092a\u094d\u0930\u0915\u091f \u0939\u094b\u0924\u093e \u0939\u0948, \u091c\u0939\u093e\u0902 \u0926\u094b\u0939\u0930\u093e\u090f \u0917\u090f \u0938\u0942\u091a\u0915\u093e\u0902\u0915 \u0907\u0902\u0917\u093f\u0924 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f \u0905\u0902\u0924\u093f\u092e \u092a\u0930\u093f\u0923\u093e\u092e \u090f\u0915 \u092f\u094b\u0917 \u0939\u094b\u0917\u093e\u0964<\/p>\n<p>\u0938\u0902\u0915\u0947\u0924\u0928, \u091c\u094b \u0935\u0947\u0915\u094d\u091f\u0930 \u0914\u0930 \u0915\u094b\u0935\u0947\u0915\u094d\u091f\u0930 \u0915\u094b \u0909\u092a-\u0938\u0942\u091a\u0915\u093e\u0902\u0915 \u0914\u0930 \u0938\u0941\u092a\u0930-\u0938\u0942\u091a\u0915\u093e\u0902\u0915 \u0915\u0947 \u092e\u093e\u0927\u094d\u092f\u092e \u0938\u0947 \u0905\u0932\u0917 \u0915\u0930\u0928\u0947 \u0915\u0940 \u0905\u0928\u0941\u092e\u0924\u093f \u0926\u0947\u0924\u093e \u0939\u0948, \u0915\u094b \u00ab\u0915\u094b\u0935\u0947\u0930\u093f\u090f\u0902\u091f \u0938\u0902\u0915\u0947\u0924\u0928\u00bb \u092f\u093e \u00ab\u091f\u0947\u0902\u0938\u0930 \u0938\u0902\u0915\u0947\u0924\u0928\u00bb \u0915\u0939\u093e \u091c\u093e\u0924\u093e \u0939\u0948 \u0914\u0930 \u092f\u0939 \u0935\u093f\u0936\u0947\u0937 \u0914\u0930 \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0938\u093e\u092a\u0947\u0915\u094d\u0937\u0924\u093e \u0938\u093f\u0926\u094d\u0927\u093e\u0902\u0924\u094b\u0902 \u0915\u0947 \u0905\u0927\u094d\u092f\u092f\u0928 \u092e\u0947\u0902 \u0935\u094d\u092f\u093e\u092a\u0915 \u0930\u0942\u092a \u0938\u0947 \u0909\u092a\u092f\u094b\u0917 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u0907\u0938\u0915\u0947 \u0905\u0932\u093e\u0935\u093e, \u092f\u0939 \u091f\u0947\u0902\u0938\u0930\u094b\u0902 \u0915\u0947 \u0938\u093e\u0925 \u0915\u093e\u0930\u094d\u092f \u0915\u0930\u0928\u0947 \u0915\u094b \u0938\u0930\u0932 \u092c\u0928\u093e\u0924\u093e \u0939\u0948, \u091c\u094b \u0935\u0947 \u091a\u0940\u091c\u0947\u0902 \u0939\u0948\u0902 \u091c\u093f\u0928\u0915\u0940 \u0939\u092e\u0928\u0947 \u0905\u092d\u0940 \u0938\u092e\u0940\u0915\u094d\u0937\u093e \u0915\u0940 \u0939\u0948 \u0914\u0930 \u091c\u093f\u0928\u094d\u0939\u0947\u0902 \u0939\u092e \u092c\u093e\u0926 \u092e\u0947\u0902 \u0935\u093f\u0938\u094d\u0924\u093e\u0930 \u0938\u0947 \u0926\u0947\u0916\u0947\u0902\u0917\u0947\u0964 \u0905\u0928\u094d\u092f \u0935\u093f\u0937\u092f\u094b\u0902 \u091c\u0948\u0938\u0947 \u0915\u094d\u0935\u093e\u0902\u091f\u092e \u092f\u093e\u0902\u0924\u094d\u0930\u093f\u0915\u0940 \u092e\u0947\u0902, \u092c\u094d\u0930\u093e-\u0915\u0948\u091f \u0938\u0902\u0915\u0947\u0924\u0928 \u0915\u094b \u092a\u094d\u0930\u093e\u0925\u092e\u093f\u0915\u0924\u093e \u0926\u0940 \u091c\u093e\u0924\u0940 \u0939\u0948, \u091c\u0939\u093e\u0902:<\/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>\u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930, \u0921\u0949\u091f \u0909\u0924\u094d\u092a\u093e\u0926 \u0915\u094b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left&lt;x|y\\right&gt;<\/span><\/span> \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0926\u0930\u094d\u0936\u093e\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/p>\n<h3>\u0921\u0949\u091f \u0909\u0924\u094d\u092a\u093e\u0926 \u0915\u0947 \u0917\u0941\u0923<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=HL85aSpHdsI&#038;t=1083s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u0921\u0949\u091f \u0909\u0924\u094d\u092a\u093e\u0926 \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e \u0938\u0947 \u0939\u092e \u0915\u0908 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0917\u0941\u0923 \u0928\u093f\u0915\u093e\u0932 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902,<\/span><\/strong><\/a> \u091c\u094b \u0906\u0917\u0947 \u0939\u092e\u093e\u0930\u0947 \u0932\u093f\u090f \u0905\u0924\u094d\u092f\u0902\u0924 \u0909\u092a\u092f\u094b\u0917\u0940 \u0939\u094b\u0902\u0917\u0947\u0964<\/p>\n<p>\u092f\u0926\u093f \u0939\u092e \u0921\u0949\u091f \u0909\u0924\u094d\u092a\u093e\u0926 \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0915\u0947 \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u092b\u0932\u0928 \u0915\u094b \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u0930\u0947\u0902: <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> \u0924\u094b \u0939\u092e \u0926\u0947\u0916\u0947\u0902\u0917\u0947 \u0915\u093f \u092f\u0939 \u092b\u0932\u0928 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tilde{\\omega}<\/span><\/span> \u0938\u092d\u0940 \u0930\u0947\u0916\u0940\u092f \u092b\u0932\u0928\u094b\u0902 \u0915\u0947 \u0917\u0941\u0923\u094b\u0902 \u0915\u094b \u0938\u0902\u0924\u0941\u0937\u094d\u091f \u0915\u0930\u0924\u093e \u0939\u0948\u0964 \u0907\u0938\u0947 \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0928\u093e \u0938\u0930\u0932 \u0939\u094b\u0917\u093e \u0915\u093f:<\/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>\u0907\u0938\u0940 \u0915\u093e\u0930\u0923 \u0938\u0947, \u0909\u0924\u094d\u092a\u093e\u0926 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tilde{\\omega}<\/span><\/span> \u0938\u0947 \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0935\u0938\u094d\u0924\u0941\u0913\u0902 \u0915\u094b <strong>\u0930\u0947\u0916\u0940\u092f \u092b\u0932\u0928 (\u0932\u093f\u0928\u093f\u092f\u0930 \u092b\u093c\u0902\u0915\u094d\u0936\u0928\u0932)<\/strong> \u0915\u0939\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u091c\u0948\u0938\u093e \u0915\u093f \u0939\u092e \u091c\u093e\u0928\u0924\u0947 \u0939\u0948\u0902, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span><\/span> \u090f\u0915 \u0935\u0947\u0915\u094d\u091f\u0930 \u0939\u0948 \u091c\u094b <strong>\u0935\u0947\u0915\u094d\u091f\u0930 \u0938\u094d\u0925\u093e\u0928<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u0915\u093e \u0938\u0926\u0938\u094d\u092f \u0939\u0948, \u0914\u0930 \u091c\u0948\u0938\u093e \u0915\u093f \u0905\u0928\u094d\u092f \u0938\u0902\u0926\u0930\u094d\u092d\u094b\u0902 \u092e\u0947\u0902 \u0926\u0947\u0916\u093e \u091c\u093e\u090f\u0917\u093e, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tilde{\\omega}<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u0915\u0947 <strong>\u0926\u094d\u0935\u0902\u0926\u094d\u0935 \u0938\u094d\u0925\u093e\u0928 (\u0921\u0941\u0905\u0932 \u0938\u094d\u092a\u0947\u0938)<\/strong> \u0915\u093e \u090f\u0915 \u0924\u0924\u094d\u0935 \u0939\u0948\u0964<\/p>\n<p>\u0907\u0938\u0938\u0947 \u092f\u0939 \u0938\u094d\u092a\u0937\u094d\u091f \u0939\u094b\u0924\u093e \u0939\u0948 \u0915\u093f \u0921\u0949\u091f \u0909\u0924\u094d\u092a\u093e\u0926 \u0914\u0930 \u0930\u0947\u0916\u0940\u092f \u092b\u0932\u0928\u094b\u0902 \u0915\u0947 \u092c\u0940\u091a \u0917\u0939\u0930\u093e \u0938\u0902\u092c\u0902\u0927 \u0939\u0948; \u0935\u093e\u0938\u094d\u0924\u0935 \u092e\u0947\u0902, \u090f\u0915 \u0938\u092e\u0940\u0915\u0930\u0923 \u091c\u094b \u0921\u0949\u091f \u0909\u0924\u094d\u092a\u093e\u0926 \u0915\u0947 \u0938\u092d\u0940 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0917\u0941\u0923\u094b\u0902 \u0915\u094b \u0938\u0902\u0915\u094d\u0937\u0947\u092a \u092e\u0947\u0902 \u092a\u094d\u0930\u0938\u094d\u0924\u0941\u0924 \u0915\u0930\u0924\u093e \u0939\u0948, \u0935\u0939 \u092f\u0939 \u0939\u0948: <em><strong>\u00ab\u0921\u0949\u091f \u0909\u0924\u094d\u092a\u093e\u0926 \u090f\u0915 \u0926\u094d\u0935\u093f\u0930\u0947\u0916\u0940\u092f, \u0938\u092e\u092e\u093f\u0924, \u0938\u0915\u093e\u0930\u093e\u0924\u094d\u092e\u0915 \u0914\u0930 \u0905\u092a\u0938\u0943\u091c\u0928\u0930\u0939\u093f\u0924 \u0930\u0942\u092a \u0939\u0948\u0964\u00bb<\/strong><\/em> \u0906\u0907\u090f \u0926\u0947\u0916\u0947\u0902 \u0915\u093f \u0907\u0938 \u0915\u0925\u0928 \u0915\u093e \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u092d\u093e\u0917 \u0915\u094d\u092f\u093e \u0926\u0930\u094d\u0936\u093e\u0924\u093e \u0939\u0948:<\/p>\n<p>\u091c\u092c \u0939\u092e \u0915\u0939\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f <strong>\u0921\u0949\u091f \u0909\u0924\u094d\u092a\u093e\u0926 \u090f\u0915 \u0926\u094d\u0935\u093f\u0930\u0947\u0916\u0940\u092f \u0930\u0942\u092a \u0939\u0948,<\/strong> \u0924\u094b \u0907\u0938\u0915\u093e \u0905\u0930\u094d\u0925 \u092f\u0939 \u0939\u0948 \u0915\u093f \u092f\u0926\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x},\\vec{y}<\/span><\/span> \u0914\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}<\/span><\/span> \u0935\u0947\u0915\u094d\u091f\u0930 \u0939\u0948\u0902 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u092e\u0947\u0902, \u0914\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha,\\beta \\in \\mathbb{R},<\/span><\/span> \u0924\u094b \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0926\u094b \u0938\u092e\u093e\u0928\u0924\u093e\u090f\u0901 \u0938\u0902\u0924\u0941\u0937\u094d\u091f \u0939\u094b\u0924\u0940 \u0939\u0948\u0902:<\/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>\u0921\u0949\u091f \u0909\u0924\u094d\u092a\u093e\u0926 <strong>\u0938\u092e\u092e\u093f\u0924<\/strong> \u0939\u094b\u0924\u093e \u0939\u0948 \u0915\u094d\u092f\u094b\u0902\u0915\u093f:<\/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>\u092f\u0939 <strong>\u0938\u0915\u093e\u0930\u093e\u0924\u094d\u092e\u0915 \u0930\u0942\u092a \u0938\u0947 \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924<\/strong> \u0939\u0948 \u0915\u094d\u092f\u094b\u0902\u0915\u093f:<\/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>\u0914\u0930 \u0905\u0902\u0924\u0924\u0903, \u092f\u0939 <strong>\u0905\u092a\u0938\u0943\u091c\u0928\u0930\u0939\u093f\u0924<\/strong> \u0939\u0948 \u0915\u094d\u092f\u094b\u0902\u0915\u093f:<\/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>\u0928\u0949\u0930\u094d\u092e \u0914\u0930 \u092f\u0942\u0915\u094d\u0932\u093f\u0921\u093f\u092f\u0928 \u0926\u0942\u0930\u0940<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vTFqDBEyU4Y&#038;t=174s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u0928\u0949\u0930\u094d\u092e \u090f\u0915 \u0935\u0947\u0915\u094d\u091f\u0930 \u0915\u0940 \u092a\u0930\u093f\u092e\u093e\u0923 \u0915\u094b \u092e\u093e\u092a\u0928\u0947 \u0915\u093e \u090f\u0915 \u0924\u0930\u0940\u0915\u093e \u0939\u0948,<\/span><\/strong><\/a> \u091c\u092c \u0915\u093f\u0938\u0940 \u0935\u0947\u0915\u094d\u091f\u0930 \u0938\u094d\u0925\u093e\u0928 \u092e\u0947\u0902 \u090f\u0915 \u0928\u0949\u0930\u094d\u092e \u0939\u094b\u0924\u0940 \u0939\u0948, \u0924\u094b \u0907\u0938\u0947 \u090f\u0915 <strong>\u0928\u0949\u0930\u094d\u092e\u0921 \u0935\u0947\u0915\u094d\u091f\u0930 \u0938\u094d\u092a\u0947\u0938<\/strong> \u0915\u0939\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u092f\u0926\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x},\\vec{y}\\in\\mathbb{R}^n<\/span><\/span> \u0914\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\lambda\\in\\mathbb{R},<\/span><\/span> \u0924\u094b \u092b\u093c\u0902\u0915\u094d\u0936\u0928 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Norm( . )<\/span><\/span> \u090f\u0915 \u0928\u0949\u0930\u094d\u092e \u0939\u0948 \u092f\u0926\u093f \u092f\u0939 \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0917\u0941\u0923\u094b\u0902 \u0915\u094b \u0938\u0902\u0924\u0941\u0937\u094d\u091f \u0915\u0930\u0924\u093e \u0939\u0948:<\/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;\">\u0921\u0949\u091f \u0909\u0924\u094d\u092a\u093e\u0926 \u0915\u093e \u090f\u0915 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u092a\u0939\u0932\u0942<\/span><\/strong><\/a> \u092f\u0939 \u0939\u0948 \u0915\u093f \u092f\u0939 \u0935\u093f\u0936\u0947\u0937 \u0930\u0942\u092a \u0938\u0947 \u090f\u0915 \u0917\u0923\u093f\u0924\u0940\u092f \u0926\u0942\u0930\u0940 \u0915\u094b \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0909\u092a\u092f\u094b\u0917\u0940 \u0939\u094b\u0924\u093e \u0939\u0948, \u091c\u094b \u0938\u0939\u091c \u0930\u0942\u092a \u0938\u0947 \u0939\u092e\u093e\u0930\u0940 \u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u092e\u091d \u0915\u0947 \u0905\u0928\u0941\u0930\u0942\u092a \u0939\u094b\u0924\u0940 \u0939\u0948 \u0915\u093f \u0926\u094b \u092c\u093f\u0902\u0926\u0941\u0913\u0902 \u0915\u0947 \u092c\u0940\u091a \u0915\u0940 \u0926\u0942\u0930\u0940 \u0915\u094b \u0915\u0948\u0938\u0947 \u092e\u093e\u092a\u093e \u091c\u093e\u090f\u0964 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\in\\mathbb{R}^n<\/span><\/span> \u0915\u0947 \u0932\u093f\u090f, \u0907\u0938\u0915\u0940 <strong>\u092f\u0942\u0915\u094d\u0932\u093f\u0921\u093f\u092f\u0928 \u0928\u0949\u0930\u094d\u092e<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x}\\|<\/span><\/span> \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0938\u092e\u0940\u0915\u0930\u0923 \u0926\u094d\u0935\u093e\u0930\u093e \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0939\u094b\u0924\u0940 \u0939\u0948:<\/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>\u0907\u0938\u0938\u0947 \u0939\u092e \u0915\u0939\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f <strong>\u092f\u0942\u0915\u094d\u0932\u093f\u0921\u093f\u092f\u0928 \u0928\u0949\u0930\u094d\u092e \u0921\u0949\u091f \u0909\u0924\u094d\u092a\u093e\u0926 \u0926\u094d\u0935\u093e\u0930\u093e \u092a\u094d\u0930\u0947\u0930\u093f\u0924 \u0928\u0949\u0930\u094d\u092e \u0939\u0948\u0964<\/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;\">\u090f\u0915 \u0926\u0942\u0930\u0940, \u092f\u093e \u092e\u0940\u091f\u094d\u0930\u093f\u0915,<\/span><\/strong><\/a> \u090f\u0915 \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0939\u094b\u0924\u093e \u0939\u0948 \u091c\u094b \u0939\u092e\u0947\u0902 \u00ab\u0915\u093f\u0938\u0940 \u0938\u0947\u091f \u0915\u0947 \u0926\u094b \u0924\u0924\u094d\u0935\u094b\u0902 \u0915\u0947 \u092c\u0940\u091a \u0915\u0940 \u092a\u0943\u0925\u0915\u094d\u0915\u0930\u0923\u00bb \u0915\u0947 \u092c\u093e\u0930\u0947 \u092e\u0947\u0902 \u092c\u0924\u093e\u0924\u093e \u0939\u0948\u0964 \u092f\u0926\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}, \\vec{y}, \\vec{z}\\in\\mathbb{R}^n<\/span><\/span> \u0914\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\lambda\\in\\mathbb{R},<\/span><\/span> \u0924\u094b \u092b\u093c\u0902\u0915\u094d\u0936\u0928 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Dist( . )<\/span><\/span> \u090f\u0915 \u0926\u0942\u0930\u0940 \u0939\u094b\u0924\u0940 \u0939\u0948 \u092f\u0926\u093f \u092f\u0939 \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0917\u0941\u0923\u094b\u0902 \u0915\u094b \u0938\u0902\u0924\u0941\u0937\u094d\u091f \u0915\u0930\u0924\u0940 \u0939\u0948:<\/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>\u0905\u0902\u0924\u093f\u092e \u0905\u092d\u093f\u0935\u094d\u092f\u0915\u094d\u0924\u093f \u0915\u094b <strong>\u0924\u094d\u0930\u093f\u092d\u0941\u091c \u0905\u0938\u092e\u093e\u0928\u0924\u093e<\/strong> \u0915\u0939\u093e \u091c\u093e\u0924\u093e \u0939\u0948, \u0914\u0930 \u092f\u0926\u093f \u092f\u0939 \u0938\u0902\u0924\u0941\u0937\u094d\u091f \u0928\u0939\u0940\u0902 \u0939\u094b\u0924\u0940, \u0924\u094b \u092b\u093c\u0902\u0915\u094d\u0936\u0928 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Dist(.)<\/span><\/span> \u0915\u094b \u00ab\u091b\u0926\u094d\u092e \u0926\u0942\u0930\u0940\u00bb \u092f\u093e \u00ab\u091b\u0926\u094d\u092e \u092e\u0940\u091f\u094d\u0930\u093f\u0915\u00bb \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u091c\u093e\u0928\u093e \u091c\u093e\u090f\u0917\u093e\u0964 \u090f\u0915 \u0935\u0947\u0915\u094d\u091f\u0930 \u0938\u094d\u0925\u093e\u0928 \u091c\u093f\u0938\u092e\u0947\u0902 \u090f\u0915 \u0926\u0942\u0930\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0939\u094b\u0924\u0940 \u0939\u0948, \u0909\u0938\u0947 <strong>\u092e\u0940\u091f\u094d\u0930\u093f\u0915 \u0938\u094d\u092a\u0947\u0938<\/strong> \u0915\u0939\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/p>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vTFqDBEyU4Y&#038;t=1013s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u092f\u0942\u0915\u094d\u0932\u093f\u0921\u093f\u092f\u0928 \u0928\u0949\u0930\u094d\u092e \u0915\u0947 \u0906\u0927\u093e\u0930 \u092a\u0930<\/span><\/strong><\/a> \u0926\u094b \u0935\u0947\u0915\u094d\u091f\u0930\u094b\u0902 \u0915\u0947 \u092c\u0940\u091a <strong>\u092f\u0942\u0915\u094d\u0932\u093f\u0921\u093f\u092f\u0928 \u0926\u0942\u0930\u0940<\/strong> \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u0940 \u091c\u093e\u0924\u0940 \u0939\u0948\u0964 \u092f\u0926\u093f \u0939\u092e\u093e\u0930\u0947 \u092a\u093e\u0938 \u0926\u094b \u0935\u0947\u0915\u094d\u091f\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x},\\vec{y}\\in\\mathbb{R}^n,<\/span><\/span> \u0939\u0948\u0902, \u0924\u094b \u0907\u0928 \u0926\u094b\u0928\u094b\u0902 \u0935\u0947\u0915\u094d\u091f\u0930\u094b\u0902 \u0915\u0947 \u092c\u0940\u091a \u0915\u0940 \u092f\u0942\u0915\u094d\u0932\u093f\u0921\u093f\u092f\u0928 \u0926\u0942\u0930\u0940 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">dist_e(\\vec{x},\\vec{y})<\/span><\/span> \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0930\u0942\u092a \u092e\u0947\u0902 \u0926\u0940 \u091c\u093e\u0924\u0940 \u0939\u0948:<\/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>\u092f\u0926\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}=(x_1,\\cdots,x_n)<\/span><\/span> \u0914\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}=(y_1,\\cdots, y_n),<\/span><\/span> \u0924\u094b \u092f\u0939 \u0921\u0949\u091f \u0909\u0924\u094d\u092a\u093e\u0926 \u0914\u0930 \u0928\u0949\u0930\u094d\u092e \u0915\u0947 \u0917\u0941\u0923\u094b\u0902 \u0938\u0947 \u0906\u0938\u093e\u0928\u0940 \u0938\u0947 \u0938\u093f\u0926\u094d\u0927 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948 \u0915\u093f:<\/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>\u092f\u0926\u093f \u0939\u092e \u0935\u0947\u0915\u094d\u091f\u0930 \u0938\u094d\u0925\u093e\u0928 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u0915\u094b \u092f\u0942\u0915\u094d\u0932\u093f\u0921\u093f\u092f\u0928 \u0926\u0942\u0930\u0940 \u0938\u0947 \u0938\u0941\u0938\u091c\u094d\u091c\u093f\u0924 \u0915\u0930\u0947\u0902, \u0924\u094b \u091c\u094b \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0924\u093e \u0939\u0948 \u0935\u0939 \u090f\u0915 <strong>\u092f\u0942\u0915\u094d\u0932\u093f\u0921\u093f\u092f\u0928 \u0938\u094d\u0925\u093e\u0928<\/strong> \u0915\u0939\u0932\u093e\u0924\u093e \u0939\u0948\u0964<\/p>\n<p>\u0907\u0938\u0938\u0947 \u0939\u092e \u0915\u0939\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f <strong>\u092f\u0942\u0915\u094d\u0932\u093f\u0921\u093f\u092f\u0928 \u0938\u094d\u0925\u093e\u0928 \u0915\u0940 \u092e\u0940\u091f\u094d\u0930\u093f\u0915 \u092f\u0942\u0915\u094d\u0932\u093f\u0921\u093f\u092f\u0928 \u0928\u0949\u0930\u094d\u092e \u0926\u094d\u0935\u093e\u0930\u093e \u092a\u094d\u0930\u0947\u0930\u093f\u0924 \u092e\u0940\u091f\u094d\u0930\u093f\u0915 \u0939\u0948\u0964<\/strong><\/p>\n<h3>\u092f\u0942\u0915\u094d\u0932\u093f\u0921\u093f\u092f\u0928 \u0928\u0949\u0930\u094d\u092e \u0915\u0947 \u0917\u0941\u0923<\/h3>\n<p><strong>\u091a\u0942\u0902\u0915\u093f \u0939\u092e\u093e\u0930\u093e \u0905\u0927\u094d\u092f\u092f\u0928 \u0935\u093f\u0936\u0947\u0937 \u0930\u0942\u092a \u0938\u0947 \u092f\u0942\u0915\u094d\u0932\u093f\u0921\u093f\u092f\u0928 \u0938\u094d\u0925\u093e\u0928 \u092a\u0930 \u0915\u0947\u0902\u0926\u094d\u0930\u093f\u0924 \u0939\u0948, \u0907\u0938\u0932\u093f\u090f \u092f\u0942\u0915\u094d\u0932\u093f\u0921\u093f\u092f\u0928 \u0928\u0949\u0930\u094d\u092e \u0915\u0947 \u0917\u0941\u0923\u094b\u0902 \u0915\u0940 \u0938\u092e\u0940\u0915\u094d\u0937\u093e \u0915\u0930\u0928\u093e \u0909\u092a\u092f\u094b\u0917\u0940 \u0930\u0939\u0947\u0917\u093e\u0964<\/strong><\/p>\n<h4>\u0915\u094b\u0936\u0940-\u0936\u094d\u0935\u093e\u0930\u094d\u091c \u0905\u0938\u092e\u093e\u0928\u0924\u093e<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vTFqDBEyU4Y&#038;t=1624s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u092f\u0926\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x},\\vec{y}\\in\\mathbb{R}^n,<\/span><\/span><\/span><\/strong><\/a> \u0924\u094b \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0917\u0941\u0923 \u0938\u0902\u0924\u0941\u0937\u094d\u091f \u0939\u094b\u0924\u093e \u0939\u0948:<\/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>\u0938\u093f\u0926\u094d\u0927\u093e\u0902\u0924:<\/strong><\/p>\n<p>\u092e\u093e\u0928 \u0932\u0947\u0902 \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\lambda = (\\vec{x}\\cdot\\vec{y})\/\\|\\vec{y}\\|^2,<\/span><\/span> \u0924\u092c \u0939\u092e\u093e\u0930\u0947 \u092a\u093e\u0938:<\/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})\\\\ \\\\ &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\\displaystyle &amp;= \\|\\vec{x}\\|^2 - \\frac{\\left(\\vec{x}\\cdot\\vec{y}\\right)^2}{\\|\\vec{y}\\|^2} \\end{array}<\/span>\n<p>\u0907\u0938\u0938\u0947 \u0939\u092e \u0928\u093f\u0937\u094d\u0915\u0930\u094d\u0937 \u0928\u093f\u0915\u093e\u0932 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f:<\/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>\u0914\u0930 \u0907\u0938\u0932\u093f\u090f:<\/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>\u0914\u0930 \u0905\u0902\u0924\u0924\u0903, \u0926\u094b\u0928\u094b\u0902 \u092a\u0915\u094d\u0937\u094b\u0902 \u0915\u0940 \u092e\u0942\u0932 \u0928\u093f\u0915\u093e\u0932\u0928\u0947 \u092a\u0930 \u0939\u092e\u0947\u0902 \u0935\u0939 \u092e\u093f\u0932\u0924\u093e \u0939\u0948 \u091c\u093f\u0938\u0947 \u0939\u092e \u0938\u093e\u092c\u093f\u0924 \u0915\u0930\u0928\u093e \u091a\u093e\u0939\u0924\u0947 \u0925\u0947:<\/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>\u0924\u094d\u0930\u093f\u092d\u0941\u091c \u0905\u0938\u092e\u093e\u0928\u0924\u093e<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vTFqDBEyU4Y&#038;t=2065s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u092e\u093e\u0928 \u0932\u0947\u0902 \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x},\\vec{y}\\in\\mathbb{R}^n,<\/span><\/span><\/span><\/strong><\/a> \u0924\u094b \u092f\u0947 \u0935\u0947\u0915\u094d\u091f\u0930 \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0938\u0902\u092c\u0902\u0927 \u0915\u094b \u0938\u0902\u0924\u0941\u0937\u094d\u091f \u0915\u0930\u0924\u0947 \u0939\u0948\u0902:<\/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>\u0938\u093f\u0926\u094d\u0927\u093e\u0902\u0924:<\/strong><\/p>\n<p>\u0938\u092c\u0938\u0947 \u092a\u0939\u0932\u0947, \u0939\u092e \u0927\u094d\u092f\u093e\u0928 \u0926\u0947\u0902 \u0915\u093f:<\/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>\u0915\u094d\u092f\u094b\u0902\u0915\u093f \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0938\u0902\u092c\u0902\u0927 \u092e\u093e\u0928\u094d\u092f \u0939\u0948\u0902:<\/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>\u0939\u092e \u0907\u0938\u0947 \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0930\u0942\u092a \u092e\u0947\u0902 \u092a\u0941\u0928\u0930\u094d\u0932\u0947\u0916\u093f\u0924 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902:<\/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  \\|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>\u0905\u0902\u0924\u0924\u0903, \u0926\u094b\u0928\u094b\u0902 \u092a\u0915\u094d\u0937\u094b\u0902 \u0915\u0940 \u092e\u0942\u0932 \u0928\u093f\u0915\u093e\u0932\u0928\u0947 \u092a\u0930, \u0939\u092e\u0947\u0902 \u0935\u0939 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0924\u093e \u0939\u0948 \u091c\u093f\u0938\u0947 \u0939\u092e \u0938\u093e\u092c\u093f\u0924 \u0915\u0930\u0928\u093e \u091a\u093e\u0939\u0924\u0947 \u0925\u0947:<\/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>\u0928\u093f\u0937\u094d\u0915\u0930\u094d\u0937<\/h2>\n<p>\u0907\u0938 \u0915\u0915\u094d\u0937\u093e \u092e\u0947\u0902, \u0939\u092e\u0928\u0947 \u092f\u0942\u0915\u094d\u0932\u093f\u0921\u093f\u092f\u0928 \u0938\u094d\u0925\u093e\u0928 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u0915\u0947 \u092e\u094c\u0932\u093f\u0915 \u0917\u0941\u0923\u094b\u0902 \u0915\u093e \u0905\u0927\u094d\u092f\u092f\u0928 \u0915\u093f\u092f\u093e, \u0907\u0938\u0915\u0940 \u092c\u0940\u091c\u0940\u092f \u0914\u0930 \u092e\u0940\u091f\u094d\u0930\u093f\u0915 \u0938\u0902\u0930\u091a\u0928\u093e\u0913\u0902 \u0915\u0940 \u0938\u092e\u0940\u0915\u094d\u0937\u093e \u0915\u0940\u0964 \u0939\u092e\u0928\u0947 \u0907\u0938\u0915\u0940 \u092e\u0942\u0932\u092d\u0942\u0924 \u0938\u0902\u0915\u094d\u0930\u093f\u092f\u093e\u090f\u0901 \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u0930\u0915\u0947 \u0936\u0941\u0930\u0941\u0906\u0924 \u0915\u0940, \u091c\u0948\u0938\u0947 \u0915\u093f \u0935\u0947\u0915\u094d\u091f\u0930 \u092f\u094b\u0917 \u0914\u0930 \u0938\u094d\u0915\u0947\u0932\u0930 \u0917\u0941\u0923\u093e, \u091c\u093f\u0938\u0938\u0947 \u0907\u0938\u0915\u093e \u0935\u0947\u0915\u094d\u091f\u0930 \u0938\u094d\u0925\u093e\u0928 \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0938\u094d\u0935\u0930\u0942\u092a \u0938\u094d\u0925\u093e\u092a\u093f\u0924 \u0939\u0941\u0906\u0964 \u0907\u0938\u0915\u0947 \u092c\u093e\u0926, \u0939\u092e\u0928\u0947 \u0921\u0949\u091f \u0909\u0924\u094d\u092a\u093e\u0926 \u0915\u0940 \u0905\u0935\u0927\u093e\u0930\u0923\u093e \u0915\u093e \u0917\u0939\u0930\u093e\u0908 \u0938\u0947 \u0905\u0927\u094d\u092f\u092f\u0928 \u0915\u093f\u092f\u093e \u0914\u0930 \u0907\u0938\u0915\u0940 \u0917\u0923\u093f\u0924\u0940\u092f \u0935\u094d\u092f\u093e\u0916\u094d\u092f\u093e \u0924\u0925\u093e \u0930\u0947\u0916\u0940\u092f \u092b\u0932\u0928\u094b\u0902 \u0915\u0947 \u0938\u093e\u0925 \u0907\u0938\u0915\u0947 \u0938\u0902\u092c\u0902\u0927 \u0915\u094b \u0909\u091c\u093e\u0917\u0930 \u0915\u093f\u092f\u093e\u0964<\/p>\n<p>\u0907\u0938\u0915\u0947 \u092c\u093e\u0926, \u0939\u092e\u0928\u0947 \u092f\u0942\u0915\u094d\u0932\u093f\u0921\u093f\u092f\u0928 \u0928\u0949\u0930\u094d\u092e \u0914\u0930 \u0907\u0938\u0938\u0947 \u092a\u094d\u0930\u0947\u0930\u093f\u0924 \u0926\u0942\u0930\u0940 \u0915\u093e \u0935\u093f\u0936\u094d\u0932\u0947\u0937\u0923 \u0915\u093f\u092f\u093e, \u0907\u0938 \u092a\u0930 \u092c\u0932 \u0926\u0947\u0924\u0947 \u0939\u0941\u090f \u0915\u093f \u092f\u0947 \u0909\u092a\u0915\u0930\u0923 \u0939\u092e\u0947\u0902 \u0907\u0938 \u0938\u094d\u0925\u093e\u0928 \u092e\u0947\u0902 \u0932\u0902\u092c\u093e\u0907\u092f\u094b\u0902 \u0914\u0930 \u0926\u0942\u0930\u093f\u092f\u094b\u0902 \u0915\u094b \u092e\u093e\u0924\u094d\u0930\u093e\u0924\u094d\u092e\u0915 \u0930\u0942\u092a \u0938\u0947 \u092e\u093e\u092a\u0928\u0947 \u0915\u0940 \u0905\u0928\u0941\u092e\u0924\u093f \u0926\u0947\u0924\u0947 \u0939\u0948\u0902\u0964 \u0938\u093e\u0925 \u0939\u0940, \u0939\u092e\u0928\u0947 \u0915\u0941\u091b \u092e\u0942\u0932\u092d\u0942\u0924 \u0905\u0938\u092e\u093e\u0928\u0924\u093e\u0913\u0902 \u0915\u0940 \u0938\u092e\u0940\u0915\u094d\u0937\u093e \u0915\u0940, \u091c\u0948\u0938\u0947 \u0915\u093f \u0915\u094b\u0936\u0940-\u0936\u094d\u0935\u093e\u0930\u094d\u091c \u0905\u0938\u092e\u093e\u0928\u0924\u093e:<\/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>\u0914\u0930 \u0924\u094d\u0930\u093f\u092d\u0941\u091c \u0905\u0938\u092e\u093e\u0928\u0924\u093e:<\/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>\u091c\u094b \u0935\u093f\u0936\u094d\u0932\u0947\u0937\u0923 \u0914\u0930 \u091c\u094d\u092f\u093e\u092e\u093f\u0924\u093f \u092e\u0947\u0902 \u0909\u0928\u094d\u0928\u0924 \u0938\u093f\u0926\u094d\u0927\u093e\u0902\u0924\u094b\u0902 \u0915\u0947 \u0935\u093f\u0915\u093e\u0938 \u0915\u0947 \u0932\u093f\u090f \u0906\u0935\u0936\u094d\u092f\u0915 \u0939\u0948\u0902\u0964<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u092f\u0942\u0915\u094d\u0932\u093f\u0921\u093f\u092f\u0928 \u0938\u094d\u0925\u093e\u0928 \u0907\u0938 \u0915\u0915\u094d\u0937\u093e \u092e\u0947\u0902 \u0939\u092e \u092f\u0942\u0915\u094d\u0932\u093f\u0921\u093f\u092f\u0928 \u0938\u094d\u0925\u093e\u0928 , \u0907\u0938\u0915\u0940 \u092c\u0940\u091c\u0940\u092f \u0938\u0902\u0930\u091a\u0928\u093e \u0914\u0930 \u092e\u0940\u091f\u094d\u0930\u093f\u0915 \u0917\u0941\u0923\u094b\u0902 \u0915\u093e \u0905\u0928\u094d\u0935\u0947\u0937\u0923 \u0915\u0930\u0947\u0902\u0917\u0947\u0964 \u0906\u092a \u0935\u0947\u0915\u094d\u091f\u0930 \u0938\u0902\u0915\u094d\u0930\u093f\u092f\u093e\u0913\u0902, \u0921\u0949\u091f \u0909\u0924\u094d\u092a\u093e\u0926, \u0928\u0949\u0930\u094d\u092e \u0914\u0930 \u092f\u0942\u0915\u094d\u0932\u093f\u0921\u093f\u092f\u0928 \u0926\u0942\u0930\u0940 \u0915\u0947 \u092c\u093e\u0930\u0947 \u092e\u0947\u0902 \u091c\u093e\u0928\u0947\u0902\u0917\u0947, \u091c\u094b \u091c\u094d\u092f\u093e\u092e\u093f\u0924\u093f \u0914\u0930 \u0935\u093f\u0936\u094d\u0932\u0947\u0937\u0923 \u092e\u0947\u0902 \u0906\u0935\u0936\u094d\u092f\u0915 \u0905\u0935\u0927\u093e\u0930\u0923\u093e\u090f\u0901 \u0939\u0948\u0902\u0964 \u0938\u094d\u092a\u0937\u094d\u091f \u0935\u094d\u092f\u093e\u0916\u094d\u092f\u093e\u0913\u0902 \u0914\u0930 \u0938\u0939\u091c \u0909\u0926\u093e\u0939\u0930\u0923\u094b\u0902 \u0915\u0947 \u0938\u093e\u0925, \u092f\u0939 \u0938\u093e\u092e\u0917\u094d\u0930\u0940 \u0906\u092a\u0915\u094b \u092f\u0939 \u0938\u092e\u091d\u0928\u0947 \u092e\u0947\u0902 \u0938\u0915\u094d\u0937\u092e \u092c\u0928\u093e\u090f\u0917\u0940 \u0915\u093f 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