{"id":432,"date":"2021-05-08T17:11:33","date_gmt":"2021-05-08T09:11:33","guid":{"rendered":"http:\/\/www.wayln.com\/?p=432"},"modified":"2021-05-08T20:54:04","modified_gmt":"2021-05-08T12:54:04","slug":"4-4-%e5%9d%90%e6%a0%87%e7%b3%bb","status":"publish","type":"post","link":"http:\/\/www.wayln.com\/?p=432","title":{"rendered":"4.4 \u5750\u6807\u7cfb"},"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> \u5750\u6807\u7684\u51e0\u4f55\u610f\u4e49<\/a><\/li><li><a href=\"#Rn\"><span class=\"toc_number toc_depth_1\">2<\/span> R^n\u4e2d\u7684\u5750\u6807<\/a><\/li><li><a href=\"#i-2\"><span class=\"toc_number toc_depth_1\">3<\/span> \u5750\u6807\u6620\u5c04<\/a><\/li><\/ul><\/div>\n<p>\u5bf9\u4e00\u4e2a\u5411\u91cf\u7a7a\u95f4V\uff0c\u660e\u786e\u6307\u5b9a\u4e00\u4e2a\u57fa<span class=\"katex math inline\">\\beta<\/span>\u7684\u4e00\u4e2a\u91cd\u8981\u539f\u56e0\u662f\u5728V\u4e0a\u5f3a\u52a0\u4e00\u4e2a\u201c\u5750\u6807\u7cfb\u201d\u3002\u672c\u8282\u6211\u4eec\u5c06\u8bc1\u660e\u5982\u679c<span class=\"katex math inline\">\\beta<\/span>\u5305\u542bn\u4e2a\u5411\u91cf\uff0c\u5219\u5750\u6807\u7cfb\u5c06\u4f7fV\u50cf<span class=\"katex math inline\">R^n<\/span>\u4e00\u6837\u4fbf\u4e8e\u64cd\u4f5c\u3002\u82e5V\u5c31\u662f<span class=\"katex math inline\">R^n<\/span>\u672c\u8eab\uff0c\u5219<span class=\"katex math inline\">\\beta<\/span>\u5c06\u786e\u5b9a\u4e00\u4e2a\u5750\u6807\u7cfb\uff0c\u5b83\u7ed9V\u4ee5\u4e00\u4e2a\u65b0\u7684\u201c\u89c6\u91ce\u201d<\/p>\n<blockquote><p>\n  <strong>\u5b9a\u74067<\/strong>\u3000\uff08\u552f\u4e00\u6807\u793a\u5b9a\u7406\uff09<br \/>\n  \u4ee4<span class=\"katex math inline\">\\beta=&#92;{b_1,\\dots,b_n&#92;}<\/span>\u662f\u5411\u91cf\u7a7a\u95f4<span class=\"katex math inline\">V<\/span>\u7684\u4e00\u4e2a\u57fa\uff0c\u5219\u5bf9V\u4e2d\u6bcf\u4e2a\u5411\u91cfx\uff0c\u5b58\u5728\u552f\u4e00\u7684\u4e00\u7ec4\u6570<span class=\"katex math inline\">c_1,\\dots,c_n<\/span>\u4f7f\u5f97<\/p>\n<div class=\"katex math multi-line no-emojify\">x=c_1b_1+\\dots+c_nb_n \\tag{1}\n  <\/div>\n<p>  <strong>\u5b9a\u4e49<\/strong>\u3000\u5047\u8bbe\u96c6\u5408<span class=\"katex math inline\">\\beta=&#92;{b_1,\\dots,b_n&#92;}<\/span>\u662fV\u7684\u4e00\u4e2a\u57fa\uff0cx\u5728V\u4e2d\uff0cx\u76f8\u5bf9\u4e8e\u57fa<span class=\"katex math inline\">\\beta<\/span>\u7684\u5750\u6807(\u6216x\u7684<span class=\"katex math inline\">\\beta<\/span>-\u5750\u6807)\u662f\u4f7f\u5f97<span class=\"katex math inline\">x=c_1b_1+\\dots+c_nb_n<\/span>\u7684\u6743<span class=\"katex math inline\">c_1,\\dots,c_n<\/span>\u3002<br \/>\n  \u82e5<span class=\"katex math inline\">c_1,\\dots,c_n<\/span>\u662fx\u7684<span class=\"katex math inline\">\\beta-<\/span>\u5750\u6807\uff0c\u5219<span class=\"katex math inline\">R^n<\/span>\u4e2d\u7684\u5411\u91cf<\/p>\n<div class=\"katex math multi-line no-emojify\">\\begin{bmatrix}<br \/>\n  x<br \/>\n  \\end{bmatrix}_{\\beta} =<br \/>\n  \\left[\\begin{array}{}<br \/>\n  c_1\\\\<br \/>\n  \\vdots\\\\<br \/>\n  c_n<br \/>\n  \\end{array}\\right]\n  <\/div>\n<p>  \u662f<strong>x(\u76f8\u5bf9\u4e8e<span class=\"katex math inline\">\\beta<\/span>)\u7684\u5750\u6807\u5411\u91cf<\/strong>\uff0c\u6216x\u7684<span class=\"katex math inline\">\\beta-<\/span>\u5750\u6807\u5411\u91cf\uff0c\u6620\u5c04<span class=\"katex math inline\">x \\mapsto \\begin{bmatrix}x\\end{bmatrix}_{\\beta}<\/span>\u79f0\u4e3a\uff08\u7531<span class=\"katex math inline\">\\beta<\/span>\u786e\u5b9a\u7684\uff09\u5750\u6807\u6620\u5c04\u3002\n<\/p><\/blockquote>\n<h1><span id=\"i\">\u5750\u6807\u7684\u51e0\u4f55\u610f\u4e49<\/span><\/h1>\n<p>\u4e00\u4e2a\u96c6\u5408\u4e0a\u7684\u5750\u6807\u7cfb\u7531\u6b64\u96c6\u5408\u4e2d\u70b9\u5230<span class=\"katex math inline\">R^n<\/span>\u4e2d\u7684\u4e00\u4e00\u6620\u5c04\u7ec4\u6210\u3002\u4f8b\u5982\uff0c\u5f53\u6211\u4eec\u9009\u53d6\u5782\u76f4\u7684\u8f74\u540c\u65f6\u5728\u6bcf\u4e2a\u8f74\u4e0a\u53d6\u4e00\u4e2a\u76f8\u540c\u7684\u5ea6\u91cf\u5355\u4f4d\u65f6\uff0c\u901a\u5e38\u7684\u56fe\u7eb8\u7ed9\u51fa\u4e86\u5e73\u9762\u4e0a\u7684\u4e00\u4e2a\u5750\u6807\u7cfb\u3002<br \/>\n<a class=\"wp-editor-md-post-content-link\" href=\"http:\/\/www.wayln.com\/wp-content\/uploads\/2021\/05\/wp_editor_md_651291ae4b4e8f37cd21ed51a6277421.jpg\"><img decoding=\"async\" src=\"http:\/\/www.wayln.com\/wp-content\/uploads\/2021\/05\/wp_editor_md_651291ae4b4e8f37cd21ed51a6277421.jpg\" alt=\"\" \/><\/a><\/p>\n<h1><span id=\"Rn\">R^n\u4e2d\u7684\u5750\u6807<\/span><\/h1>\n<p>\u5f53<span class=\"katex math inline\">R^n<\/span>\u4e2d\u7684\u4e00\u7ec4\u57fa<span class=\"katex math inline\">\\beta<\/span>\u56fa\u5b9a\u65f6\uff0c\u5bb9\u6613\u6c42\u51fa\u4efb\u4e00\u6307\u5b9a\u7684\u5411\u91cfx\u7684<span class=\"katex math inline\">\\beta-<\/span>\u5750\u6807\u3002<\/p>\n<div class=\"katex math multi-line no-emojify\">\\left[\\begin{array}{rr}<br \/>\n2&amp;-1\\\\<br \/>\n1&amp;1\\\\<br \/>\n\\end{array}\\right]<br \/>\n\\left[\\begin{array}{rr}<br \/>\nc_1\\\\<br \/>\nc_2\\\\<br \/>\n\\end{array}\\right]=<br \/>\n\\left[\\begin{array}{rr}<br \/>\n4\\\\<br \/>\n5\\\\<br \/>\n\\end{array}\\right] \\tag{3}\n<\/div>\n<p>(3)\u5f0f\u4e2d\u7684\u77e9\u9635\u5c06\u5411\u91cfx\u7684<span class=\"katex math inline\">\\beta-<\/span>\u5750\u6807\u53d8\u4e3ax\u7684\u6807\u51c6\u5750\u6807\u3002\u5bf9<span class=\"katex math inline\">R^n<\/span>\u4e2d\u7684\u4e00\u4e2a\u57fa<span class=\"katex math inline\">\\beta=&#92;{b_1,\\dots,b_n&#92;}<\/span>,\u53ef\u4ee5\u5b9e\u65bd\u7c7b\u4f3c\u7684\u5750\u6807\u53d8\u6362\u3002\u4ee4<\/p>\n<div class=\"katex math multi-line no-emojify\">P_\\beta=\\begin{bmatrix}b_1&amp;b_2&amp;\\dots b_n\\end{bmatrix}\n<\/div>\n<p>\u5219\u5411\u91cf\u65b9\u7a0b<\/p>\n<div class=\"katex math multi-line no-emojify\">x=c_1b_1+c_2b_2+\\dots+c_nb_n\n<\/div>\n<p>\u7b49\u4ef7\u4e8e<\/p>\n<div class=\"katex math multi-line no-emojify\">x=P_{\\beta}\\begin{bmatrix}x\\end{bmatrix}_{\\beta}  \\tag{4}\n<\/div>\n<p>\u6211\u4eec\u79f0<span class=\"katex math inline\">P_\\beta<\/span>\u4e3a\u4ece<span class=\"katex math inline\">\\beta<\/span>\u5230<span class=\"katex math inline\">R^n<\/span>\u4e2d\u6807\u51c6\u57fa\u7684<strong>\u5750\u6807\u53d8\u6362\u77e9\u9635<\/strong>\u3002\u901a\u8fc7\u5de6\u4e58<span class=\"katex math inline\">P_\\beta<\/span>\u5c06\u5750\u6807\u5411\u91cf$\\begin{bmatrix}x\\end{bmatrix}<em>\\beta<span class=\"katex math inline\">\u53d8\u6362\u5230<\/span>x<span class=\"katex math inline\">.\u5750\u6807\u53d8\u6362\u65b9\u7a0b(4)\u662f\u91cd\u8981\u7684\u3002<br \/>\n\u7531\u4e8e<\/span>P<\/em>\\beta<span class=\"katex math inline\">\u7684\u5217\u6784\u6210<\/span>R^n<span class=\"katex math inline\">\u7684\u4e00\u4e2a\u57fa\uff0c<\/span>P_\\beta<span class=\"katex math inline\">\u662f\u53ef\u9006\u7684\uff08\u7531\u53ef\u9006\u77e9\u9635\u5b9a\u7406\uff09\uff0c\u901a\u8fc7\u5de6\u4e58<\/span>P^{-1}_{\\beta}<span class=\"katex math inline\">\u53c8\u5c06<\/span>x<span class=\"katex math inline\">\u53d8\u56de<\/span>\\beta-$\u5750\u6807\u5411\u91cf\uff1a<\/p>\n<div class=\"katex math multi-line no-emojify\">P^{-1}_{\\beta}=\\begin{bmatrix}x\\end{bmatrix}_\\beta\n<\/div>\n<p>\u8fd9\u91cc\u53c8$P^{-1}<em>{\\beta}<span class=\"katex math inline\">\u4ea7\u751f\u7684\u6620\u5c04<\/span>x \\mapsto \\begin{bmatrix}x\\end{bmatrix}<\/em>\\beta<span class=\"katex math inline\">\u662f\u524d\u9762\u63d0\u5230\u7684\u5750\u6807\u6620\u5c04\u3002\u56e0\u4e3a<\/span>P^{-1}_\\beta<span class=\"katex math inline\">\u662f\u4e00\u53ef\u9006\u77e9\u9635\uff0c\u7531\u53ef\u9006\u77e9\u9635\u5b9a\u7406\uff08\u4e5f\u53ef\u4ee5\u53c2\u89c11.9\u8282\u5b9a\u740612\uff09\uff0c\u6b64\u5750\u6807\u6620\u5c04\u662f\u4e00\u4e2a\u7531<\/span>R^n\u5230R^n$\u4e0a\u7684\u4e00\u4e00\u7ebf\u6027\u53d8\u6362\u3002<\/p>\n<h1><span id=\"i-2\">\u5750\u6807\u6620\u5c04<\/span><\/h1>\n<p>\u5bf9\u5411\u91cf\u7a7a\u95f4V\u9009\u5b9a\u4e00\u4e2a\u57fa<span class=\"katex math inline\">\\beta=&#92;{b_1,\\dots,b_n&#92;}<\/span>,\u5b83\u5f15\u51faV\u4e2d\u4e00\u4e2a\u5750\u6807\u7cfb\u3002\u5750\u6807\u6620\u5c04<span class=\"katex math inline\">x \\mapsto \\begin{bmatrix}x\\end{bmatrix}_\\beta<\/span>\u5c06\u53ef\u80fd\u4e0d\u719f\u6089\u7684\u7a7a\u95f4V\u4e0e\u719f\u6089\u7684\u7a7a\u95f4<span class=\"katex math inline\">R^n<\/span>\u8054\u7cfb\u4e86\u8d77\u6765\u3002\u89c1\u4e0b\u56fe\uff0cV\u4e2d\u7684\u70b9\u73b0\u5728\u53ef\u4ee5\u7531\u4ed6\u4eec\u7684\u65b0\u201c\u540d\u5b57\u201d\u6765\u786e\u5b9a\u3002<br \/>\n<a class=\"wp-editor-md-post-content-link\" href=\"http:\/\/www.wayln.com\/wp-content\/uploads\/2021\/05\/wp_editor_md_7ee406916e3b29f9995e0fe946a9f4a3.jpg\"><img decoding=\"async\" src=\"http:\/\/www.wayln.com\/wp-content\/uploads\/2021\/05\/wp_editor_md_7ee406916e3b29f9995e0fe946a9f4a3.jpg\" alt=\"\" \/><\/a><\/p>\n<blockquote><p>\n  <strong>\u5b9a\u74068<\/strong>\u3000\u4ee4<span class=\"katex math inline\">\\beta=&#92;{b_1,\\dots,b_n&#92;}<\/span>\u662f\u5411\u91cf\u7a7a\u95f4V\u7684\u4e00\u4e2a\u57fa\uff0c\u5219\u5750\u6807\u6620\u5c04<span class=\"katex math inline\">x \\mapsto \\begin{bmatrix}x\\end{bmatrix}_\\beta<\/span>\u662f\u4e00\u4e2a\u7531<span class=\"katex math inline\">V<\/span>\u6620\u4e0a\u5230<span class=\"katex math inline\">R^n<\/span>\u7684\u4e00\u5bf9\u4e00\u7684\u7ebf\u6027\u53d8\u6362\u3002\n<\/p><\/blockquote>\n<p>\u5b9a\u74068\u4e2d\u7684\u5750\u6807\u6620\u5c04\u662f\u4e00\u4e2a\u7531<span class=\"katex math inline\">V<\/span>\u5230<span class=\"katex math inline\">R^n<\/span>\u4e0a\u540c\u6784\u7684\u91cd\u8981\u4f8b\u5b50\u3002\u4e00\u822c\u800c\u8a00\uff0c\u4ece\u4e00\u4e2a\u5411\u91cf\u7a7a\u95f4V\u6620\u4e0a\u5230\u53e6\u4e00\u4e2a\u5411\u91cf\u7a7a\u95f4W\u7684\u4e00\u4e00\u7ebf\u6027\u53d8\u6362\u79f0\u4e3a\u4eceV\u5230W\u4e0a\u7684\u4e00\u4e2a\u540c\u6784\u3002\u6bcf\u4e00\u4e2a\u5728V\u4e2d\u7684\u5411\u91cf\u7a7a\u95f4\u7684\u8ba1\u7b97\u53ef\u4ee5\u5b8c\u5168\u76f8\u540c\u5730\u51fa\u73b0\u5728W\u4e2d\uff0c\u53cd\u4e4b\u4ea6\u7136\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Contents1 \u5750\u6807\u7684\u51e0\u4f55\u610f\u4e492 R^n\u4e2d\u7684\u5750\u68073 \u5750\u6807\u6620\u5c04 \u5bf9\u4e00\u4e2a\u5411\u91cf\u7a7a\u95f4V\uff0c\u660e\u786e\u6307\u5b9a\u4e00\u4e2a\u57fa\\beta\u7684 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":388,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,22,26],"tags":[],"class_list":["post-432","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\/432","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=432"}],"version-history":[{"count":6,"href":"http:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/432\/revisions"}],"predecessor-version":[{"id":438,"href":"http:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/432\/revisions\/438"}],"wp:featuredmedia":[{"embeddable":true,"href":"http:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/media\/388"}],"wp:attachment":[{"href":"http:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=432"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=432"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=432"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}