{"id":397,"date":"2021-04-29T21:01:40","date_gmt":"2021-04-29T13:01:40","guid":{"rendered":"http:\/\/www.wayln.com\/?p=397"},"modified":"2021-04-29T21:38:26","modified_gmt":"2021-04-29T13:38:26","slug":"4-3-%e7%ba%bf%e6%80%a7%e6%97%a0%e5%85%b3%e9%9b%86%e5%92%8c%e5%9f%ba","status":"publish","type":"post","link":"https:\/\/www.wayln.com\/?p=397","title":{"rendered":"4.3 \u7ebf\u6027\u65e0\u5173\u96c6\u548c\u57fa"},"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> \u751f\u6210\u96c6\u5b9a\u7406<\/a><\/li><li><a href=\"#NulAColA\"><span class=\"toc_number toc_depth_1\">2<\/span> NulA\u548cColA\u7684\u57fa<\/a><\/li><li><a href=\"#i-2\"><span class=\"toc_number toc_depth_1\">3<\/span> \u5173\u4e8e\u57fa\u7684\u4e24\u70b9\u89c2\u5bdf<\/a><\/li><\/ul><\/div>\n<p>V\u4e2d\u5411\u91cf\u7684\u4e00\u4e2a\u6307\u6807\u96c6<span class=\"katex math inline\">&#92;{v_1,\\dots,v_p&#92;}<\/span>\u79f0\u4e3a\u662f<strong>\u7ebf\u6027\u65e0\u5173<\/strong>\u7684\uff0c\u5982\u679c\u5411\u91cf\u65b9\u7a0b<\/p>\n<div class=\"katex math multi-line no-emojify\">c_1v_1+c_2v_2+\\dots+c_pv_p=0 \\tag{1}\n<\/div>\n<p>\u53ea\u6709\u5e73\u51e1\u89e3\uff0c\u5373<span class=\"katex math inline\">c_1=0,\\dots,c_p=0<\/span><br \/>\n\u3000\u96c6\u5408<span class=\"katex math inline\">&#92;{v_1,\\dots,v_p&#92;}<\/span>\u79f0\u4e3a<strong>\u7ebf\u6027\u76f8\u5173<\/strong>,\u5982\u679c(1)\u6709\u4e00\u4e2a\u975e\u5e73\u51e1\u89e3\uff0c\u5373\u5b58\u5728\u67d0\u4e9b\u6743<span class=\"katex math inline\">c_1,\\dots,c_p<\/span>\u4e0d\u5168\u4e3a0\uff0c\u4f7f\u5f97\uff081\uff09\u5f0f\u6210\u7acb\uff0c\u6b64\u65f6(1)\u5f0f\u79f0\u4e3a<span class=\"katex math inline\">v_1,\\dots,v_p<\/span>\u4e4b\u95f4\u7684\u4e00\u4e2a\u7ebf\u6027\u76f8\u5173\u5173\u7cfb\u3002<br \/>\n\u3000\u4e0e<span class=\"katex math inline\">R^n<\/span>\u4e2d\u4e00\u6837\uff0c\u4e00\u4e2a\u4ec5\u542b\u4e00\u4e2a\u5411\u91cfv\u7684\u96c6\u662f\u7ebf\u6027\u65e0\u5173\u7684\u5f53\u4e14\u4ec5\u5f53<span class=\"katex math inline\">v \\neq 0<\/span>\uff1b\u4e00\u4e2a\u4ec5\u542b\u4e24\u4e2a\u5411\u91cf\u7684\u96c6\u5408\u662f\u7ebf\u6027\u76f8\u5173\u7684\u5f53\u4e14\u4ec5\u5f53\u5176\u4e2d\u4e00\u4e2a\u5411\u91cf\u662f\u53e6\u4e00\u4e2a\u7684\u500d\u6570\uff1b\u4efb\u4f55\u542b\u6709\u96f6\u5411\u91cf\u7684\u96c6\u5408\u662f\u7ebf\u6027\u76f8\u5173\u7684\u3002<\/p>\n<blockquote><p>\n  <strong>\u5b9a\u74064<\/strong>\u3000\u4e0d\u5c11\u4e8e\u4e24\u4e2a\u6709\u7f16\u53f7\u7684\u5411\u91cf\u7684\u96c6\u5408<span class=\"katex math inline\">&#92;{v_1,\\dots,v_p&#92;}<\/span>,\u5982\u679c\u6709<span class=\"katex math inline\">v_1 \\neq 0<\/span>\uff0c\u5219<span class=\"katex math inline\">&#92;{v_1,\\dots,v_p&#92;}<\/span>\u662f\u7ebf\u6027\u76f8\u5173\u7684\uff0c\u5f53\u4e14\u4ec5\u5f53\u67d0<span class=\"katex math inline\">v_j(j>1)<\/span>\u662f\u5176\u524d\u9762\u5411\u91cf<span class=\"katex math inline\">v_1,\\dots,v_{j-1}<\/span>\u7684\u7ebf\u6027\u7ec4\u5408\n<\/p><\/blockquote>\n<p>\u3000\u4e00\u822c\u5411\u91cf\u7a7a\u95f4\u4e2d\u7684\u7ebf\u6027\u76f8\u5173\u4e0e<span class=\"katex math inline\">R^n<\/span>\u4e2d\u7684\u7ebf\u6027\u76f8\u5173\u7684\u4e3b\u8981\u4e0d\u540c\u70b9\u5728\u4e8e\u5411\u91cf\u4e0d\u662fn\u5143\u6570\u7ec4\u65f6\uff0c\u9f50\u6b21\u65b9\u7a0b(1)\u901a\u5e38\u4e0d\u80fd\u88ab\u5199\u6210\u4e00\u4e2an\u5143\u7ebf\u6027\u65b9\u7a0b\u7ec4\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u4e3a\u4e86\u7814\u7a76\u65b9\u7a0b<span class=\"katex math inline\">Ax=0<\/span>\uff0c\u5411\u91cf\u4e0d\u80fd\u4ece\u4e00\u4e2a\u77e9\u9635A\u7684\u5217\u4e2d\u5f97\u5230\uff0c\u53d6\u800c\u4ee3\u4e4b\u7684\u662f\u6211\u4eec\u5fc5\u987b\u8981\u4f9d\u9760\u7ebf\u6027\u76f8\u5173\u7684\u5b9a\u4e49\u548c\u5b9a\u74064\u3002<\/p>\n<blockquote><p>\n  \u3000<strong>\u5b9a\u4e49<\/strong>\u3000\u4ee4<span class=\"katex math inline\">H<\/span>\u662f\u5411\u91cf\u7a7a\u95f4<span class=\"katex math inline\">V<\/span>\u7684\u4e00\u4e2a\u5b50\u7a7a\u95f4\uff0c<span class=\"katex math inline\">V<\/span>\u4e2d\u5411\u91cf\u7684\u6307\u6807\u96c6<span class=\"katex math inline\">\\beta=&#92;{b_1,\\dots,b_p&#92;}<\/span>\u79f0\u4e3aH\u7684\u4e00\u4e2a\u57fa\uff0c\u5982\u679c<br \/>\n   (1)\u3000<span class=\"katex math inline\">\\beta<\/span>\u65f6\u4e00\u7ebf\u6027\u65e0\u5173\u96c6<br \/>\n   (2)\u3000\u7531<span class=\"katex math inline\">\\beta<\/span>\u751f\u6210\u7684\u5b50\u7a7a\u95f4\u4e0eH\u76f8\u540c\uff0c\u5373<span class=\"katex math inline\">H=Span&#92;{b_1,\\dots,b_p&#92;}<\/span>\n<\/p><\/blockquote>\n<h1><span id=\"i\">\u751f\u6210\u96c6\u5b9a\u7406<\/span><\/h1>\n<p>\u4e00\u4e2a\u57fa\u662f\u4e00\u4e2a\u4e0d\u5305\u542b\u4e0d\u5fc5\u8981\u7684\u5411\u91cf\u7684\u201c\u9ad8\u6548\u7387\u201d\u7684\u751f\u6210\u96c6\u3002\u4e8b\u5b9e\u4e0a\uff0c\u4e00\u4e2a\u57fa\u53ef\u4ee5\u901a\u8fc7\u7531\u4e00\u4e2a\u751f\u6210\u96c6\u4e2d\u53bb\u6389\u4e0d\u9700\u8981\u7684\u5411\u91cf\u6784\u9020\u51fa\u6765\u3002<\/p>\n<blockquote><p>\n  <strong>\u5b9a\u74065<\/strong>\u3000\uff08\u751f\u6210\u96c6\u5b9a\u7406\uff09<br \/>\n  \u4ee4<span class=\"katex math inline\">S=&#92;{v_1,\\dots,v_p&#92;}<\/span>\u662fV\u4e2d\u7684\u5411\u91cf\u96c6\uff0c<span class=\"katex math inline\">H=Span&#92;{v_1,\\dots,v_p&#92;}<\/span><br \/>\n  a.\u3000\u82e5S\u4e2d\u67d0\u4e00\u4e2a\u5411\u91cf\uff0c\u6bd4\u5982\u8bf4<span class=\"katex math inline\">v_k<\/span>\uff0c\u662fS\u4e2d\u5176\u4f59\u5411\u91cf\u7684\u7ebf\u6027\u7ec4\u5408\uff0c\u5219S\u4e2d\u53bb\u6389<span class=\"katex math inline\">V_k<\/span>\u540e\u5f62\u6210\u7684\u96c6\u5408\u4ecd\u7136\u53ef\u4ee5\u751f\u6210H.<br \/>\n  b.\u3000\u82e5<span class=\"katex math inline\">H \\neq &#92;{0&#92;}<\/span>,\u5219S\u7684\u67d0\u4e00\u5b50\u96c6\u662fH\u7684\u4e00\u4e2a\u57fa\u3002\n<\/p><\/blockquote>\n<h1><span id=\"NulAColA\">NulA\u548cColA\u7684\u57fa<\/span><\/h1>\n<blockquote><p>\n  <strong>\u5b9a\u74066<\/strong>\u3000\u77e9\u9635A\u7684\u4e3b\u5143\u5217\u6784\u6210ColA\u7684\u4e00\u4e2a\u57fa\n<\/p><\/blockquote>\n<div class=\"katex math multi-line no-emojify\">{\\color{red}\u5bf9ColA\u7684\u57fa\uff0c\u8981\u614e\u91cd\u4f7f\u7528A\u672c\u8eab\u7684\u4e3b\u5143\u5217\uff0c\u9636\u68af\u5f62B\u7684\u4e3b\u5143\u5217\u901a\u5e38\u4e0d\u5728A\u7684\u5217\u7a7a\u95f4}\n<\/div>\n<h1><span id=\"i-2\">\u5173\u4e8e\u57fa\u7684\u4e24\u70b9\u89c2\u5bdf<\/span><\/h1>\n<p>\u3000\u4f7f\u7528\u751f\u6210\u96c6\u5b9a\u7406\u65f6\uff0c\u4ece\u751f\u6210\u96c6\u4e2d\u5220\u9664\u5411\u91cf\uff0c\u5f53\u96c6\u5408\u7f16\u7a0b\u7ebf\u6027\u65e0\u5173\u65f6\u5fc5\u987b\u505c\u6b62\uff0c\u56e0\u4e3a\u5982\u679c\u518d\u591a\u5220\u4e00\u4e2a\u5411\u91cf\uff0c\u8be5\u5411\u91cf\u5c06\u4e0d\u662f\u5269\u4e0b\u5411\u91cf\u7684\u7ebf\u6027\u7ec4\u5408\uff0c\u4ece\u800c\u8fd9\u4e2a\u8f83\u5c0f\u7684\u96c6\u5408\u5c06\u4e0d\u518d\u751f\u6210V\uff0c\u6240\u4ee5\u4e00\u4e2a\u57fa\u662f\u4e00\u4e2a\u5c3d\u53ef\u80fd\u5c0f\u7684\u751f\u6210\u96c6\u3002<br \/>\n \u3000\u57fa\u8fd8\u662f\u5c3d\u53ef\u80fd\u5927\u7684\u7ebf\u6027\u65e0\u5173\u96c6\uff0c\u82e5S\u662fV\u7684\u4e00\u4e2a\u57fa\uff0c\u5728S\u4e2d\u518d\u6dfb\u52a0\u8fdb\u4e00\u4e2a\u65b0\u7684\u5411\u91cf\u3002\u6bd4\u5982\u4eceV\u4e2d\u53d6\u4e00\u4e2aw,\u5219\u65b0\u7684\u96c6\u5408\u4e0d\u518d\u662f\u7ebf\u6027\u65e0\u5173\u4e86\uff0c\u8fd9\u662f\u56e0\u4e3aS\u751f\u6210V\uff0c\u56e0\u6b64w\u662fS\u4e2d\u5143\u7d20\u7684\u7ebf\u6027\u7ec4\u5408\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Contents1 \u751f\u6210\u96c6\u5b9a\u74062 NulA\u548cColA\u7684\u57fa3 \u5173\u4e8e\u57fa\u7684\u4e24\u70b9\u89c2\u5bdf V\u4e2d\u5411\u91cf\u7684\u4e00\u4e2a\u6307\u6807\u96c6&#92;{ [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":399,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,22,26],"tags":[],"class_list":["post-397","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-21","category-22","category-26"],"_links":{"self":[{"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/397","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=397"}],"version-history":[{"count":6,"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/397\/revisions"}],"predecessor-version":[{"id":404,"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/397\/revisions\/404"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/media\/399"}],"wp:attachment":[{"href":"https:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=397"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=397"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=397"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}