{"id":306,"date":"2021-04-13T16:12:20","date_gmt":"2021-04-13T08:12:20","guid":{"rendered":"http:\/\/www.wayln.com\/?p=306"},"modified":"2021-04-15T09:51:20","modified_gmt":"2021-04-15T01:51:20","slug":"2-8-rn%e7%9a%84%e5%ad%90%e7%a9%ba%e9%97%b4","status":"publish","type":"post","link":"http:\/\/www.wayln.com\/?p=306","title":{"rendered":"2.8 R^n\u7684\u5b50\u7a7a\u95f4"},"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> \u77e9\u9635\u7684\u5217\u7a7a\u95f4\u4e0e\u96f6\u7a7a\u95f4<\/a><\/li><li><a href=\"#i-2\"><span class=\"toc_number toc_depth_1\">2<\/span> \u5b50\u7a7a\u95f4\u7684\u57fa<\/a><\/li><\/ul><\/div>\n<p>\u672c\u8282\u8ba8\u8bba<span class=\"katex math inline\">R^n<\/span>\u4e2d\u91cd\u8981\u7684\u5411\u91cf\u5b50\u96c6\uff0c\u79f0\u4e3a\u5b50\u7a7a\u95f4\u3002\u901a\u5e38\u5b50\u7a7a\u95f4\u4e0e\u67d0\u4e2a\u77e9\u9635A\u6709\u5173\uff0c\u4ed6\u4eec\u63d0\u4f9b\u4e86\u5173\u4e8e\u65b9\u7a0b<span class=\"katex math inline\">Ax=b<\/span>\u7684\u6709\u7528\u4fe1\u606f\u3002<\/p>\n<blockquote><p>\n  <strong>\u5b9a\u4e49<\/strong>\u3000<span class=\"katex math inline\">R^n<\/span>\u4e2d\u7684\u4e00\u4e2a\u5b50\u7a7a\u95f4\u662f<span class=\"katex math inline\">R^n<\/span>\u4e2d\u7684\u96c6\u5408H\uff0c\u5177\u6709\u4ee5\u4e0b\u4e09\u4e2a\u6027\u8d28\uff1a<br \/>\n  a.\u3000\u96f6\u5411\u91cf\u5c5e\u4e8eH<br \/>\n  b.\u3000\u5bf9H\u4e2d\u4efb\u610f\u7684\u5411\u91cf<span class=\"katex math inline\">u\u548cv,u+v\u5c5e\u4e8eH<\/span><br \/>\n  c.\u3000\u5bf9H\u4e2d\u4efb\u610f\u5411\u91cf<span class=\"katex math inline\">u\u548c\u6570c,cu\u5c5e\u4e8eH<\/span>\n<\/p><\/blockquote>\n<p>\u6362\u53e5\u8bdd\u8bf4\uff0c\u5b50\u7a7a\u95f4\u5bf9\u52a0\u6cd5\u548c\u6807\u91cf\u4e58\u6cd5\u8fd0\u7b97\u662f\u5c01\u95ed\u7684<br \/>\n<a class=\"wp-editor-md-post-content-link\" href=\"http:\/\/www.wayln.com\/wp-content\/uploads\/2021\/04\/wp_editor_md_dc0fb02101862cac2b9133eaeb8b7a8c.jpg\"><img decoding=\"async\" src=\"http:\/\/www.wayln.com\/wp-content\/uploads\/2021\/04\/wp_editor_md_dc0fb02101862cac2b9133eaeb8b7a8c.jpg\" alt=\"\" \/><\/a><\/p>\n<h1><span id=\"i\">\u77e9\u9635\u7684\u5217\u7a7a\u95f4\u4e0e\u96f6\u7a7a\u95f4<\/span><\/h1>\n<blockquote><p>\n  <strong>\u5b9a\u4e49<\/strong>\u3000\u77e9\u9635A\u7684\u5217\u7a7a\u95f4\u662fA\u7684\u5404\u5217\u7684\u7ebf\u6027\u7ec4\u5408\u7684\u96c6\u5408\uff0c\u8bb0\u4f5c<span class=\"katex math inline\">ColA<\/span>\u3002\n<\/p><\/blockquote>\n<p>\u82e5<span class=\"katex math inline\">A=\\begin{bmatrix}a_1 \\dots a_n\\end{bmatrix}<\/span>,\u4ed6\u4eec\u5404\u5217\u5c5e\u4e8e<span class=\"katex math inline\">R^m<\/span>\uff0c\u5219<span class=\"katex math inline\">ColA\u548cSpan&#92;{a_1,\\dots,a_n&#92;}<\/span>\u76f8\u540c,\u4f8b4\u8bf4\u660emxn\u77e9\u9635\u7684\u5217\u7a7a\u95f4\u662f<span class=\"katex math inline\">R^m<\/span>\u7684\u5b50\u7a7a\u95f4\u3002\u6ce8\u610f\uff0c\u4ec5\u5f53A\u7684\u5217\u751f\u6210<span class=\"katex math inline\">R^m<\/span>\u65f6\uff0c<span class=\"katex math inline\">ColA\u7b49\u4e8eR^m<\/span>\u3002\u5426\u5219\uff0c<span class=\"katex math inline\">ColA\u4ec5\u662fR^m<\/span>\u7684\u4e00\u90e8\u5206\u3002<br \/>\n<strong>\u4f8b 4<\/strong>\u3000\u8bbe<span class=\"katex math inline\">A=<br \/>\n\\left[\\begin{array}{rrr}1&amp;-3&amp;4&#92;&#92;-4&amp;6&amp;-2&#92;&#92;-3&amp;7&amp;6\\end{array}\\right]<\/span>,<span class=\"katex math inline\">b=\\left[\\begin{array}{r}3&#92;&#92;3&#92;&#92;-4\\end{array}\\right]<\/span>,\u786e\u5b9ab\u662f\u5426\u5c5e\u4e8eA\u7684\u5217\u7a7a\u95f4\u3002<br \/>\n<strong>\u89e3<\/strong>\u3000\u5411\u91cfb\u662fA\u7684\u5404\u5217\u7684\u7ebf\u6027\u7ec4\u5408\uff0c\u5f53\u4e14\u4ec5\u5f53b\u53ef\u5199\u6210Ax\u7684\u5f62\u52bf\uff0cx\u5c5e\u4e8e<span class=\"katex math inline\">R^3<\/span>,\u4e5f\u5c31\u662f\u8bf4\uff0c\u5f53\u4e14\u4ec5\u5f53\u65b9\u7a0b<span class=\"katex math inline\">Ax=b<\/span>\u6709\u89e3\uff0c\u628a\u589e\u5e7f\u77e9\u9635<span class=\"katex math inline\">\\begin{bmatrix}A&amp;b\\end{bmatrix}<\/span>\u8fdb\u884c\u53d8\u6362\uff1a<\/p>\n<blockquote><p>\n  <strong>\u5b9a\u4e49<\/strong>\u3000\u77e9\u9635A\u7684\u96f6\u7a7a\u95f4\u662f\u9f50\u6b21\u65b9\u7a0b<span class=\"katex math inline\">Ax=0<\/span>\u7684\u6240\u6709\u89e3\u7684\u96c6\u5408\uff0c\u8bb0\u4e3a<span class=\"katex math inline\">NulA<\/span>\u3002\n<\/p><\/blockquote>\n<p>\u5f53A\u6709n\u5217\u65f6\uff0c<span class=\"katex math inline\">Ax=0<\/span>\u7684\u89e3\u5c5e\u4e8e<span class=\"katex math inline\">R^n<\/span>,A\u7684\u96f6\u7a7a\u95f4\u662f<span class=\"katex math inline\">R^n<\/span>\u7684\u5b50\u96c6\uff0c\u4e8b\u5b9e\u4e0a\uff0c<span class=\"katex math inline\">NulA\u5177\u6709R^n<\/span>\u7684\u5b50\u7a7a\u95f4\u7684\u6027\u8d28\u3002<\/p>\n<blockquote><p>\n  <strong>\u5b9a\u7406 12<\/strong>\u3000mxn\u77e9\u9635A\u7684\u96f6\u7a7a\u95f4\u662f<span class=\"katex math inline\">R^n<\/span>\u7684\u5b50\u7a7a\u95f4\u3002\u7b49\u4ef7\u5730\uff0cn\u4e2a\u672a\u77e5\u6570\u7684m\u4e2a\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7684\u89e3\u7684\u5168\u4f53\u662f<span class=\"katex math inline\">R^n<\/span>\u7684\u5b50\u7a7a\u95f4\n<\/p><\/blockquote>\n<h1><span id=\"i-2\">\u5b50\u7a7a\u95f4\u7684\u57fa<\/span><\/h1>\n<p>\u56e0\u4e3a\u5b50\u7a7a\u95f4\u4e00\u822c\u542b\u6709\u65e0\u7a77\u591a\u4e2a\u5411\u91cf\uff0c\u5b50\u7a7a\u95f4\u4e2d\u7684\u95ee\u9898\u6700\u597d\u80fd\u591f\u901a\u8fc7\u7814\u7a76\u751f\u6210\u8fd9\u4e2a\u5b50\u7a7a\u95f4\u7684\u4e00\u4e2a\u5c0f\u7684\u6709\u9650\u96c6\u5408\u6765\u89e3\u51b3\uff0c\u8fd9\u4e2a\u96c6\u5408\u8d8a\u5c0f\u8d8a\u597d\u3002\u53ef\u4ee5\u8bc1\u660e\uff0c\u6700\u5c0f\u53ef\u80fd\u7684\u751f\u6210\u96c6\u5408\u5fc5\u662f\u7ebf\u6027\u65e0\u5173\u7684\u3002<\/p>\n<blockquote><p>\n  <strong>\u5b9a\u4e49<\/strong>\u3000<span class=\"katex math inline\">R^n<\/span>\u4e2d\u5b50\u7a7a\u95f4H\u7684\u4e00\u7ec4\u57fa\u662fH\u4e2d\u4e00\u4e2a\u7ebf\u6027\u65e0\u5173\u96c6\uff0c\u5b83\u751f\u6210H\u3002\n<\/p><\/blockquote>\n<p><strong>\u4f8b 5<\/strong> \u53ef\u9006nxn\u77e9\u9635\u7684\u5404\u5217\u6784\u6210<span class=\"katex math inline\">R^n<\/span>\u7684\u4e00\u7ec4\u57fa\u3002\u56e0\u4e3a\u4ed6\u4eec\u7ebf\u6027\u65e0\u5173\uff0c\u800c\u4e14\u751f\u6210<span class=\"katex math inline\">R^n<\/span>,\u8fd9\u7531\u9006\u77e9\u9635\u5b9a\u7406\u53ef\u77e5\uff0c\u4e00\u4e2a\u8fd9\u6837\u7684\u77e9\u9635\u662fnxn\u5355\u4f4d\u77e9\u9635\uff0c\u4ed6\u7684\u5404\u5217\u7528<span class=\"katex math inline\">e_1,\\dots,e_n<\/span>\u8868\u793a\uff1a<\/p>\n<div class=\"katex math multi-line no-emojify\">e_1=\\begin{bmatrix}<br \/>\n1\\\\0\\\\\\vdots\\\\0<br \/>\n\\end{bmatrix},<br \/>\ne_2=\\begin{bmatrix}<br \/>\n0\\\\1\\\\\\vdots\\\\0<br \/>\n\\end{bmatrix},\u3000\\dots\u3000,<br \/>\ne_n=\\begin{bmatrix}<br \/>\n0\\\\0\\\\\\vdots\\\\1<br \/>\n\\end{bmatrix}\n<\/div>\n<p><span class=\"katex math inline\">&#92;{e_1,\\dots,e_n&#92;}<\/span>\u79f0\u4e3a<span class=\"katex math inline\">R^n<\/span>\u7684\u6807\u51c6\u57fa\u3002<\/p>\n<p>\u4e0b\u4f8b\u8bf4\u660e\uff0c\u6c42\u51fa\u65b9\u7a0b<span class=\"katex math inline\">Ax=0<\/span>\u7684\u89e3\u7684\u53c2\u6570\u5411\u91cf\u5f62\u5f0f\u5b9e\u9645\u4e0a\u662f\u786e\u5b9a<span class=\"katex math inline\">NulA<\/span>\u7684\u57fa<br \/>\n<strong>\u4f8b6<\/strong>\u3000\u6c42\u51fa\u4e0b\u5217\u77e9\u9635\u7684\u96f6\u7a7a\u95f4\u7684\u57fa<\/p>\n<div class=\"katex math multi-line no-emojify\">A=\\left[\\begin{array}{rrrrr}<br \/>\n-3&amp;6&amp;-1&amp;1&amp;-7\\\\<br \/>\n1&amp;-2&amp;2&amp;3&amp;-1\\\\<br \/>\n2&amp;-4&amp;5&amp;8&amp;-4<br \/>\n\\end{array}\\right]\n<\/div>\n<p><strong>\u89e3<\/strong> \u9996\u5148\u628a\u65b9\u7a0b<span class=\"katex math inline\">Ax=0<\/span>\u7684\u89e3\u5199\u6210\u53c2\u6570\u5411\u91cf\u5f62\u5f0f<\/p>\n<div class=\"katex math multi-line no-emojify\">A \\sim<br \/>\n\\left[\\begin{array}{rrrrr}<br \/>\n1&amp;-2&amp;0&amp;-1&amp;3&amp;0\\\\<br \/>\n0&amp;0&amp;1&amp;2&amp;-2&amp;0\\\\<br \/>\n0&amp;0&amp;0&amp;0&amp;0&amp;0<br \/>\n\\end{array}\\right]\n<\/div>\n<div class=\"katex math multi-line no-emojify\">\\begin{cases}<br \/>\nx_1-2x_2-x_4+3x_5=0\\\\<br \/>\nx_3+2x_4-2x_5=0\\\\<br \/>\n0=0<br \/>\n\\end{cases}\n<\/div>\n<p>\u901a\u89e3\u4e3a<span class=\"katex math inline\">x_1=2x_2+x_4-3x+5,x_3=-2x_4+2x_5,x_2,x_4,x_5<\/span>\u4e3a\u81ea\u7531\u53d8\u91cf\u3002<\/p>\n<div class=\"katex math multi-line no-emojify\">\\begin{bmatrix}<br \/>\nx_1\\\\x_2\\\\x_3\\\\x_4\\\\x_5<br \/>\n\\end{bmatrix}=<br \/>\n\\begin{bmatrix}<br \/>\n2x_1+x_4-3x_5\\\\<br \/>\nx_2\\\\<br \/>\n-2x_4+2x_5\\\\<br \/>\nx_4\\\\<br \/>\nx_5\\\\<br \/>\n\\end{bmatrix}\n<\/div>\n<div class=\"katex math multi-line no-emojify\">=<br \/>\nx_2\\underbrace{\\left[\\begin{array}{r}<br \/>\n2\\\\1\\\\0\\\\0\\\\0<br \/>\n\\end{array}\\right]}_u<br \/>\n+x_4\\underbrace{\\left[\\begin{array}{r}<br \/>\n1\\\\0\\\\-2\\\\1\\\\0<br \/>\n\\end{array}\\right]}_v<br \/>\n+<br \/>\nx_5\\underbrace{\\left[\\begin{array}{r}<br \/>\n-3\\\\0\\\\2\\\\0\\\\1<br \/>\n\\end{array}\\right]}_w \\tag{1}\n<\/div>\n<div class=\"katex math multi-line no-emojify\">=x_2u+x_4v+x_5w\n<\/div>\n<p>\u65b9\u7a0b(1)\u8bf4\u660e<span class=\"katex math inline\">NulA\u4e0eu,v,w<\/span>\u7684\u6240\u6709\u7ebf\u6027\u7ec4\u5408\u7684\u96c6\u5408\u662f\u4e00\u81f4\u7684,\u5373<span class=\"katex math inline\">&#92;{u,v,w&#92;}<\/span>\u751f\u6210<span class=\"katex math inline\">Nul A<\/span>\uff0c\u4e8b\u5b9e\u4e0a\uff0cu,v,w\u7684\u6784\u9020\u4fdd\u8bc1\u4e86\u4ed6\u4eec\u7ebf\u6027\u65e0\u5173\uff0c\u56e0\u4e3a(1)\u5f0f\u8bf4\u660e\uff0c<\/p>\n<div class=\"katex math multi-line no-emojify\">0=x_2u+x_4v+x_5w\n<\/div>\n<p>\u4ec5\u5f53\u6743<span class=\"katex math inline\">x_2,x_4,x_5<\/span>\u7b49\u4e8e\u96f6\u5219\u6210\u7acb\uff0c\u56e0\u6b64<span class=\"katex math inline\">&#92;{u,v,w&#92;}<\/span>\u662f<span class=\"katex math inline\">Nul A<\/span>\u7684\u4e00\u7ec4\u57fa\u3002<\/p>\n<blockquote><p>\n  <strong>\u5b9a\u740613<\/strong>\u77e9\u9635A\u7684\u4e3b\u5143\u5217\u6784\u6210\u5217\u7a7a\u95f4\u7684\u57fa\u3002\n<\/p><\/blockquote>\n<div class=\"katex math multi-line no-emojify\">{\\color{red}\u8b66\u544a\u3000\u5c0f\u5fc3\uff0c\u8981\u7528A\u7684\u4e3b\u5143\u5217\u672c\u8eab\u4f5c\u4e3aColA\u7684\u57fa\uff0c\u9636\u68af\u5f62B\u7684\u5217\u672c\u8eab\u901a\u5e38\u5e76\u4e0d\u5728A\u7684\u5217\u7a7a\u95f4\u4e2d}\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Contents1 \u77e9\u9635\u7684\u5217\u7a7a\u95f4\u4e0e\u96f6\u7a7a\u95f42 \u5b50\u7a7a\u95f4\u7684\u57fa \u672c\u8282\u8ba8\u8bbaR^n\u4e2d\u91cd\u8981\u7684\u5411\u91cf\u5b50\u96c6\uff0c\u79f0\u4e3a\u5b50\u7a7a\u95f4\u3002\u901a\u5e38\u5b50\u7a7a\u95f4 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":334,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,22,26],"tags":[],"class_list":["post-306","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-21","category-22","category-26"],"_links":{"self":[{"href":"http:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/306","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=306"}],"version-history":[{"count":9,"href":"http:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/306\/revisions"}],"predecessor-version":[{"id":317,"href":"http:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/306\/revisions\/317"}],"wp:featuredmedia":[{"embeddable":true,"href":"http:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/media\/334"}],"wp:attachment":[{"href":"http:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=306"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=306"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=306"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}