{"id":411,"date":"2021-04-30T06:48:03","date_gmt":"2021-04-29T22:48:03","guid":{"rendered":"http:\/\/www.wayln.com\/?p=411"},"modified":"2021-05-09T13:23:27","modified_gmt":"2021-05-09T05:23:27","slug":"1-5%e7%ba%bf%e6%80%a7%e6%96%b9%e7%a8%8b%e7%bb%84%e7%9a%84%e8%a7%a3%e9%9b%86","status":"publish","type":"post","link":"https:\/\/www.wayln.com\/?p=411","title":{"rendered":"1.5\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u89e3\u96c6"},"content":{"rendered":"<div id=\"toc_container\" class=\"toc_transparent no_bullets\"><p class=\"toc_title\">Contents<\/p><ul class=\"toc_list\"><li><a href=\"#i\"><span class=\"toc_number toc_depth_1\">1<\/span> \u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4<\/a><\/li><li><a href=\"#i-2\"><span class=\"toc_number toc_depth_1\">2<\/span> \u53c2\u6570\u5411\u91cf\u5f62\u5f0f<\/a><\/li><li><a href=\"#i-3\"><span class=\"toc_number toc_depth_1\">3<\/span> \u975e\u9f50\u6b21\u65b9\u7a0b\u7ec4\u7684\u89e3<\/a><\/li><\/ul><\/div>\n<h2><span id=\"i\">\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4<\/span><\/h2>\n<p>\u7ebf\u6027\u65b9\u7a0b\u7ec4\u79f0\u4e3a<strong>\u9f50\u6b21<\/strong>\u7684\uff0c\u82e5\u5b83\u53ef\u5199\u6210<span class=\"katex math inline\">Ax=0<\/span>\u7684\u5f62\u5f0f\uff0c\u5176\u4e2dA\u662fmxn\u77e9\u9635\u800c0\u662f<span class=\"katex math inline\">R^m<\/span>\u4e2d\u7684\u96f6\u5411\u91cf\uff0c\u8fd9\u6837\u7684\u65b9\u7a0b\u81f3\u5c11\u6709\u4e00\u4e2a\u89e3\uff0c\u5373<span class=\"katex math inline\">x=0(R^n\u4e2d\u7684\u96f6\u5411\u91cf)<\/span>,\u8fd9\u4e2a\u89e3\u79f0\u4e3a\u5b83\u7684\u5e73\u51e1\u89e3\uff0c\u5bf9\u7ed9\u5b9a\u65b9\u7a0b<span class=\"katex math inline\">Ax=0<\/span>\u91cd\u8981\u7684\u662f\u4ed6\u662f\u5426\u6709\u975e\u5e73\u51e1\u89e3\uff0c\u5373\u6ee1\u8db3<span class=\"katex math inline\">Ax=0<\/span>\u7684\u975e\u96f6\u5411\u91cfx\u3002\u7531\u89e3\u7684\u5b58\u5728\u6027\u4e0e\u552f\u4e00\u6027\u5b9a\u7406\uff08\u5b9a\u74062\uff09\uff0c\u5f97\u51fa\u4ee5\u4e0b\u4e8b\u5b9e\uff1a<\/p>\n<blockquote><p>\n  \u9f50\u6b21\u65b9\u7a0b<span class=\"katex math inline\">Ax=0<\/span>\u6709\u975e\u5e73\u51e1\u89e3\uff0c\u5f53\u4e14\u4ec5\u5f53\u65b9\u7a0b\u81f3\u5c11\u6709\u4e00\u4e2a\u81ea\u7531\u53d8\u91cf\u3002\n<\/p><\/blockquote>\n<p>\u4f8b2\uff0c\u5f53\u4ee5\u65b9\u7a0b\u4e5f\u53ef\u770b\u505a\u65b9\u7a0b\u7ec4\uff0c\u63cf\u8ff0\u4e0b\u5217\u9f50\u6b21\u65b9\u7a0b\u7ec4\u7684\u89e3\u96c6<\/p>\n<div class=\"katex math multi-line no-emojify\">10x_1-3x_2-2x_3=0 \\tag{1}\n<\/div>\n<p><strong>\u89e3<\/strong> \u8fd9\u91cc\u65e0\u9700\u77e9\u9635\u8bb0\u53f7\uff0c\u7528\u81ea\u7531\u53d8\u91cf\u8868\u793a\u57fa\u672c\u53d8\u91cf<span class=\"katex math inline\">x_1<\/span>.\u901a\u89e3\u4e3a\uff1a<br \/>\n<span class=\"katex math inline\">x_1=0.3x_2+0.2x_3<\/span><br \/>\n<span class=\"katex math inline\">x_2\u548cx_3<\/span>\u4e3a\u81ea\u7531\u53d8\u91cf\uff0c\u5199\u6210\u5411\u91cf\u5f62\u5f0f\uff0c\u901a\u89e3\u4e3a\uff1a<\/p>\n<div class=\"katex math multi-line no-emojify\">x=\\begin{bmatrix}<br \/>\nx_1\\\\x_2\\\\x_3\\\\<br \/>\n\\end{bmatrix}=<br \/>\n\\begin{bmatrix}<br \/>\n0.3x_2+0.2x_3\\\\<br \/>\nx_2\\\\<br \/>\nx_3\\\\<br \/>\n\\end{bmatrix}=<br \/>\n\\begin{bmatrix}<br \/>\n0.3x_2\\\\<br \/>\nx_2\\\\<br \/>\n0<br \/>\n\\end{bmatrix}+<br \/>\n\\begin{bmatrix}<br \/>\n0.2x_3\\\\<br \/>\n0\\\\<br \/>\nx_3\\\\<br \/>\n\\end{bmatrix}\\\\<br \/>\n=x2\\underbrace{\\begin{bmatrix}<br \/>\n0.3\\\\<br \/>\n1\\\\<br \/>\n0\\\\<br \/>\n\\end{bmatrix}}_u+<br \/>\nx3\\underbrace{\\begin{bmatrix}<br \/>\n0.2\\\\<br \/>\n0\\\\<br \/>\n1\\\\<br \/>\n\\end{bmatrix}}_v(x_2,x_3\u4e3a\u81ea\u7531\u53d8\u91cf) \\tag{2}\n<\/div>\n<p>\u8ba1\u7b97\u8868\u660e\uff0c\u65b9\u7a0b\uff081\uff09\u7684\u6bcf\u4e2a\u89e3\u90fd\u662f\u5411\u91cfu\u548cv\u7684\u7ebf\u6027\u7ec4\u5408\uff0c\u5982(2)\u5f0f\u6240\u793a\u3002\u5373\u89e3\u96c6\u4e3a<span class=\"katex math inline\">Span&#92;{u,v&#92;}<\/span>\uff0c\u56e0\u4e3a\uff0cu\u4e0d\u662fv\u7684\u500d\u6570\uff0c\u89e3\u96c6\u662f\u901a\u8fc7\u539f\u70b9\u7684\u4e00\u4e2a\u5e73\u9762\uff0c\u89c1\u4e0b\u56fe.<br \/>\n<a class=\"wp-editor-md-post-content-link\" href=\"http:\/\/www.wayln.com\/wp-content\/uploads\/2021\/04\/wp_editor_md_744b54b2abbf690552d0ee89d08030e8.jpg\"><img decoding=\"async\" src=\"http:\/\/www.wayln.com\/wp-content\/uploads\/2021\/04\/wp_editor_md_744b54b2abbf690552d0ee89d08030e8.jpg\" alt=\"\" \/><\/a><\/p>\n<h2><span id=\"i-2\">\u53c2\u6570\u5411\u91cf\u5f62\u5f0f<\/span><\/h2>\n<p>\u6700\u521d\u7684\u65b9\u7a0b(1)\u662f\u4f8b2\u4e2d\u7684\u5e73\u9762\u7684\u9690\u5f0f\u63cf\u8ff0\uff0c\u89e3\u6b64\u65b9\u7a0b\u5c31\u662f\u8981\u627e\u8fd9\u4e2a\u5e73\u9762\u7684\u663e\u5f0f\u63cf\u8ff0\uff0c\u5c31\u662f\u8bf4\uff0c\u5c06\u5b83\u4f5c\u4e3au\u548cv\u6240\u751f\u6210\u7684\u5b50\u96c6\uff0c\u65b9\u7a0b(2)\u79f0\u4e3a\u5e73\u9762\u7684\u53c2\u6570\u5411\u91cf\u65b9\u7a0b\uff0c\u6709\u65f6\u4e5f\u53ef\u5199\u4e3a\uff1a<br \/>\n<span class=\"katex math inline\">x=su+tv (s,t\u4e3a\u5b9e\u6570)<\/span><\/p>\n<h2><span id=\"i-3\">\u975e\u9f50\u6b21\u65b9\u7a0b\u7ec4\u7684\u89e3<\/span><\/h2>\n<p>\u5f53\u975e\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4\u6709\u8bb8\u591a\u89e3\u65f6\uff0c\u4e00\u822c\u53ef\u8868\u793a\u4e3a\u53c2\u6570\u7684\u5411\u91cf\u5f62\u5f0f\uff0c\u5373\u7531\u4e00\u4e2a\u5411\u91cf\u52a0\u4e0a\u6ee1\u8db3\u5bf9\u5e94\u7684\u9f50\u6b21\u65b9\u7a0b\u7684\u4e00\u4e9b\u5411\u91cf\u7684\u4efb\u610f\u7ebf\u6027\u7ec4\u5408\u7684\u5f62\u5f0f\u3002<br \/>\n\u4f8b3  \u63cf\u8ff0<span class=\"katex math inline\">Ax=b<\/span>\u7684\u89e3\uff0c\u5176\u4e2d<\/p>\n<div class=\"katex math multi-line no-emojify\">A=<br \/>\n\\begin{bmatrix}<br \/>\n3&amp;5&amp;-4\\\\<br \/>\n-3&amp;-2&amp;4\\\\<br \/>\n6&amp;1&amp;-8\\\\<br \/>\n\\end{bmatrix},b=<br \/>\n\\begin{bmatrix}<br \/>\n7\\\\-1\\\\-4\\\\<br \/>\n\\end{bmatrix}\n<\/div>\n<p><strong>\u89e3<\/strong> \u8fd9\u91ccA\u5c31\u662f\u4f8b1\u7684\u7cfb\u6570\u77e9\u9635\u3002\u5bf9<span class=\"katex math inline\">\\begin{bmatrix}A&amp;b\\end{bmatrix}<\/span>\u4f5c\u884c\u53d8\u6362\u5f97\uff1a<\/p>\n<div class=\"katex math multi-line no-emojify\">\\left[\\begin{array}{rrrr}<br \/>\n3&amp;5&amp;-4&amp;7\\\\<br \/>\n-3&amp;-2&amp;4&amp;-1\\\\<br \/>\n6&amp;1&amp;-8&amp;-4\\\\<br \/>\n\\end{array}\\right]<br \/>\n\\sim<br \/>\n\\left[\\begin{array}{lcr r}<br \/>\n1&amp;0&amp;-\\frac{4}{3}&amp;-1\\\\<br \/>\n0&amp;1&amp;0&amp;2\\\\<br \/>\n0&amp;0&amp;0&amp;0\\\\<br \/>\n\\end{array}\\right]<br \/>\n\\begin{aligned}<br \/>\nx_1&amp; &amp;-\\frac{4}{3}x_3=-1\\\\<br \/>\n&amp;x_2&amp;=2\\\\<br \/>\n&amp; &amp;0=0<br \/>\n\\end{aligned}\n<\/div>\n<p>\u6240\u4ee5<span class=\"katex math inline\">x_1=-1+\\frac{4}{3}x_3,x_2=2<\/span>\uff0c<span class=\"katex math inline\">x_3<\/span>\u4e3a\u81ea\u7531\u53d8\u91cf,<span class=\"katex math inline\">Ax=b<\/span>\u7684\u901a\u89e3\u53ef\u5199\u6210\u5411\u91cf\u5f62\u5f0f<\/p>\n<div class=\"katex math multi-line no-emojify\">x=<br \/>\n\\left[\\begin{array}{r}<br \/>\nx_1\\\\<br \/>\nx_2\\\\<br \/>\nx_3\\\\<br \/>\n\\end{array}\\right]=<br \/>\n\\left[\\begin{array}{c}<br \/>\n-1+\\frac{4}{3}x_3\\\\<br \/>\n2\\\\<br \/>\nx_3\\\\<br \/>\n\\end{array}\\right]<br \/>\n=<br \/>\n\\left[\\begin{array}{r}<br \/>\n-1\\\\<br \/>\n2\\\\<br \/>\n0\\\\<br \/>\n\\end{array}\\right]+<\/p>\n<p>\\left[\\begin{array}{c}<br \/>\n\\frac{4}{3}x_3\\\\<br \/>\n0\\\\<br \/>\nx_3\\\\<br \/>\n\\end{array}\\right]\\\\=<br \/>\n\\underbrace{<br \/>\n\\left[\\begin{array}{r}<br \/>\n-1\\\\<br \/>\n2\\\\<br \/>\n0\\\\<br \/>\n\\end{array}\\right]}_p<br \/>\n+x_3<br \/>\n\\underbrace{<br \/>\n\\left[\\begin{array}{c}<br \/>\n\\frac{4}{3}\\\\0\\\\1<br \/>\n\\end{array}\\right]<br \/>\n}_v\n<\/p><\/div>\n<p>\u65b9\u7a0b<span class=\"katex math inline\">x=p+x_3v<\/span>\uff0c\u6216\u7528t\u8868\u793a\u4e00\u822c\u53c2\u6570\uff0c<\/p>\n<div class=\"katex math multi-line no-emojify\">x=p+tv (t\u4e3a\u5b9e\u6570) \\tag{3}\n<\/div>\n<p>\u5c31\u662f\u7528\u53c2\u6570\u5411\u91cf\u5f62\u5f0f\u8868\u793a<span class=\"katex math inline\">Ax=b<\/span>\u7684\u89e3\u96c6\u3002\u56de\u5fc6\u4f8b1\u4e2d<span class=\"katex math inline\">Ax=0<\/span>\u7684\u89e3\u96c6\u6709\u53c2\u6570\u5411\u91cf\u5f62\u5f0f<\/p>\n<div class=\"katex math multi-line no-emojify\">x=tv \\tag{4}\n<\/div>\n<p>(v\u4e0e\uff083\uff09\u4e2d\u7684v\u76f8\u540c)\uff0c\u6545<span class=\"katex math inline\">Ax=b<\/span>\u7684\u89e3\u53ef\u7531\u5411\u91cf<span class=\"katex math inline\">p<\/span>\u52a0\u4e0a<span class=\"katex math inline\">Ax=0<\/span>\u7684\u89e3\u5f97\u5230\uff0c\u5411\u91cf<span class=\"katex math inline\">p<\/span>\u672c\u8eab\u4e5f\u662f<span class=\"katex math inline\">Ax=b<\/span>\u7684\u4e00\u4e2a\u7279\u89e3(\u5728(3)\u4e2d\u5bf9\u5e94<span class=\"katex math inline\">t=0<\/span>)\u3002<br \/>\n\u79f0\uff083\uff09\u4e3a\u901a\u8fc7p\u5e73\u884c\u4e8ev\u7684\u76f4\u7ebf\u65b9\u7a0b\u3002\u4e8e\u662f<span class=\"katex math inline\">Ax=b<\/span>\u7684\u89e3\u96c6\u662f\u4e00\u6761\u901a\u8fc7<span class=\"katex math inline\">p<\/span>\u800c\u5e73\u884c\u4e8e<span class=\"katex math inline\">Ax=0<\/span>\u7684\u76f4\u7ebf<br \/>\n<a class=\"wp-editor-md-post-content-link\" href=\"http:\/\/www.wayln.com\/wp-content\/uploads\/2021\/04\/wp_editor_md_ad8f775ea18955c90861fe122d536aba.jpg\"><img decoding=\"async\" src=\"http:\/\/www.wayln.com\/wp-content\/uploads\/2021\/04\/wp_editor_md_ad8f775ea18955c90861fe122d536aba.jpg\" alt=\"\" \/><\/a><\/p>\n<blockquote><p>\n  \u5b9a\u74066 \u8bbe\u65b9\u7a0b<span class=\"katex math inline\">Ax=b<\/span>\u5bf9\u67d0\u4e2ab\u662f\u76f8\u5bb9\u7684\uff0cp\u4e3a\u4e00\u4e2a\u7279\u89e3\uff0c\u5219<span class=\"katex math inline\">Ax=b<\/span>\u7684\u89e3\u96c6\u662f\u6240\u6709\u5f62\u5982<span class=\"katex math inline\">w=p+v_h<\/span>\u7684\u5411\u91cf\u7684\u96c6\uff0c\u5176\u4e2d\uff0c<span class=\"katex math inline\">v_h<\/span>\u662f\u9f50\u6b21\u65b9\u7a0b<span class=\"katex math inline\">Ax=0<\/span>\u7684\u4efb\u610f\u4e00\u4e2a\u89e3.<br \/>\n  \u5b9a\u74066\u8bf4\u660e\u82e5Ax=b\u6709\u89e3\uff0c\u5219\u89e3\u96c6\u53ef\u7531<span class=\"katex math inline\">Ax=0<\/span>\u7684\u89e3\u5e73\u79fb\u5411\u91cfp\u5f97\u5230\uff0cp\u662f<span class=\"katex math inline\">Ax=b<\/span>\u7684\u4efb\u610f\u4e00\u4e2a\u7279\u89e3\u3002\n<\/p><\/blockquote>\n<p><strong>\u628a\u76f8\u5bb9\u65b9\u7a0b\u7ec4\u7684\u89e3\u96c6\u8868\u793a\u6210\u53c2\u6570\u5411\u91cf\u5f62\u5f0f<\/strong><\/p>\n<div class=\"mermaid mermaid-diagram no-emojify\"><script type=\"text\/javascript\">document.write(\"graph TD\\na(\u628a\u589e\u5e7f\u77e9\u9635\u884c\u5316\u7b80\u4e3a\u7b80\u5316\u9636\u68af\u5f62) --&gt; b(\u628a\u6bcf\u4e2a\u57fa\u672c\u53d8\u91cf\u7528\u81ea\u7531\u53d8\u91cf\u8868\u793a)\\nb--&gt;c(\u628a\u4e00\u822c\u89e3x\u8868\u793a\u6210\u5411\u91cf,\u5982\u679c\u6709\u81ea\u7531\u53d8\u91cf,\u5176\u5143\u7d20\u4f9d\u8d56\u4e8e\u81ea\u7531\u53d8\u91cf)\\nc--&gt;d(\u628ax\u5206\u89e3\u4e3a\u5411\u91cf\\\\\\\\\u5143\u7d20\u4e3a\u5e38\u6570\\\\\\\\\u7684\u7ebf\u6027\u7ec4\u5408,\u7528\u81ea\u7531\u53d8\u91cf\u4f5c\u4e3a\u53c2\u6570)\\n\");<\/script><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Contents1 \u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec42 \u53c2\u6570\u5411\u91cf\u5f62\u5f0f3 \u975e\u9f50\u6b21\u65b9\u7a0b\u7ec4\u7684\u89e3 \u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4 \u7ebf\u6027\u65b9\u7a0b\u7ec4\u79f0\u4e3a\u9f50\u6b21\u7684\uff0c [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,22,26],"tags":[],"class_list":["post-411","post","type-post","status-publish","format-standard","hentry","category-21","category-22","category-26"],"_links":{"self":[{"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/411","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=411"}],"version-history":[{"count":3,"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/411\/revisions"}],"predecessor-version":[{"id":448,"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/411\/revisions\/448"}],"wp:attachment":[{"href":"https:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=411"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=411"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=411"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}