{"id":235,"date":"2021-04-07T22:09:34","date_gmt":"2021-04-07T14:09:34","guid":{"rendered":"http:\/\/www.wayln.com\/?p=235"},"modified":"2021-04-08T14:23:33","modified_gmt":"2021-04-08T06:23:33","slug":"235","status":"publish","type":"post","link":"http:\/\/www.wayln.com\/?p=235","title":{"rendered":"1.8 \u7ebf\u6027\u53d8\u6362\u4ecb\u7ecd"},"content":{"rendered":"<div id=\"toc_container\" class=\"toc_transparent no_bullets\"><p class=\"toc_title\">Contents<\/p><ul class=\"toc_list\"><li><a href=\"#18\"><span class=\"toc_number toc_depth_1\">1<\/span> 1.8 \u7ebf\u6027\u53d8\u6362\u4ecb\u7ecd<\/a><ul><li><a href=\"#i\"><span class=\"toc_number toc_depth_2\">1.1<\/span> \u77e9\u9635\u53d8\u6362<\/a><\/li><li><a href=\"#i-2\"><span class=\"toc_number toc_depth_2\">1.2<\/span> \u7ebf\u6027\u53d8\u6362<\/a><\/li><\/ul><\/li><\/ul><\/div>\n\n<h1><span id=\"18\">1.8 \u7ebf\u6027\u53d8\u6362\u4ecb\u7ecd<\/span><\/h1>\n<p>\u901a\u5e38\u662f\u628a\u77e9\u9635A\u5f53\u505a\u4e00\u79cd\u201c\u5bf9\u8c61\u201d,\u5b83\u901a\u8fc7\u4e58\u6cd5\u201c\u4f5c\u7528\u201d\u4e8e\u5411\u91cfx,\u4ea7\u751f\u7684\u65b0\u5411\u91cf\u79f0\u4e3aAx\u3002<br \/>\n\u4f8b\u5982\uff0c\u65b9\u7a0b:<\/p>\n<div class=\"katex math multi-line no-emojify\">\\underbrace{\\left[\\begin{array}{rrrr}<br \/>\n4&amp;-3&amp;1&amp;3\\\\<br \/>\n2&amp;0&amp;5&amp;1\\\\<br \/>\n\\end{array}\\right]}_A<br \/>\n\\underbrace{\\begin{bmatrix}<br \/>\n1\\\\1\\\\1\\\\1\\\\<br \/>\n\\end{bmatrix}}_x=<br \/>\n\\underbrace{\\begin{bmatrix}<br \/>\n5\\\\8<br \/>\n\\end{bmatrix}}_b\u548c<br \/>\n\\underbrace{\\left[\\begin{array}{rrrr}<br \/>\n4&amp;-3&amp;1&amp;3\\\\<br \/>\n2&amp;0&amp;5&amp;1\\\\<br \/>\n\\end{array}\\right]}_A<br \/>\n\\underbrace{<br \/>\n\\left[\\begin{array}{r}<br \/>\n1\\\\4\\\\-1\\\\3<br \/>\n\\end{array}\\right]<br \/>\n}_u=<br \/>\n\\underbrace{\\begin{bmatrix}<br \/>\n0\\\\<br \/>\n0<br \/>\n\\end{bmatrix}}_0\n<\/div>\n<p>\u4e58\u4ee5\u77e9\u9635A\u540e\uff0c\u5c06x\u53d8\u4e3ab,\u5c06u\u53d8\u4e3a\u96f6\u5411\u91cf<br \/>\n<a class=\"wp-editor-md-post-content-link\" href=\"http:\/\/www.wayln.com\/wp-content\/uploads\/2021\/04\/wp_editor_md_9ae932662f059d9b9f24e47eba1104ca.jpg\"><img decoding=\"async\" src=\"http:\/\/www.wayln.com\/wp-content\/uploads\/2021\/04\/wp_editor_md_9ae932662f059d9b9f24e47eba1104ca.jpg\" alt=\"\" \/><\/a><br \/>\n\u7531\u8fd9\u4e2a\u65b0\u89c2\u70b9\uff0c\u89e3\u65b9\u7a0b<span class=\"katex math inline\">Ax=b<\/span>\u5c31\u662f\u8981\u6c42\u51fa<span class=\"katex math inline\">R^4<\/span>\u4e2d\u6240\u6709\u7ecf\u8fc7\u4e58\u4ee5A\u7684\u201c\u4f5c\u7528\u201d\u540e\u53d8\u4e3ab\u7684\u5411\u91cfx.<br \/>\n\u7531x\u5230Ax\u7684\u5bf9\u5e94\u662f\u7531\u4e00\u4e2a\u5411\u91cf\u96c6\u5230\u53e6\u4e00\u4e2a\u5411\u91cf\u96c6\u7684\u51fd\u6570\uff0c\u8fd9\u4e2a\u6982\u5ff5\u63a8\u5e7f\u4e86\u901a\u5e38\u7684\u51fd\u6570\u6982\u5ff5\uff0c\u901a\u5e38\u7684\u51fd\u6570\u662f\u628a\u4e00\u4e2a\u5b9e\u6570\u53d8\u4e3a\u53e6\u4e00\u4e2a\u5b9e\u6570\u7684\u89c4\u5219<br \/>\n\u7531<span class=\"katex math inline\">R^n<\/span>\u5230<span class=\"katex math inline\">R^m<\/span>\u7684\u4e00\u4e2a<strong>\u53d8\u6362<\/strong>\uff08\u6216\u79f0<strong>\u51fd\u6570\u3001\u6620\u5c04<\/strong>\uff09T\u662f\u4e00\u4e2a\u89c4\u5219\uff0c\u5b83\u628a<span class=\"katex math inline\">R^n<\/span>\u4e2d\u6bcf\u4e2a\u5411\u91cfx\u5bf9\u5e94\u4ee5<span class=\"katex math inline\">R^m<\/span>\u4e2d\u7684\u4e00\u4e2a\u5411\u91cfT(x)\u3002\u96c6<span class=\"katex math inline\">R^n<\/span>\u79f0\u4e3aT\u7684<strong>\u5b9a\u4e49\u57df<\/strong>\uff0c\u800c<span class=\"katex math inline\">R^m<\/span>\u79f0\u4e3aT\u7684<strong>\u4f59\u5b9a\u4e49\u57df<\/strong>\uff08\u6216\u53d6\u503c\u7a7a\u95f4\uff09\u3002\u7b26\u53f7<span class=\"katex math inline\">T:R^n\\rightarrow R^m<\/span>\u8bf4\u660eT\u7684\u5b9a\u4e49\u57df\u662f<span class=\"katex math inline\">R^n<\/span>\u800c\u4f59\u5b9a\u4e49\u57df\u662f<span class=\"katex math inline\">R^m<\/span>\uff0c\u5bf9\u4e8e<span class=\"katex math inline\">R^n<\/span>\u4e2d\u5411\u91cfx,<span class=\"katex math inline\">R^m<\/span>\u4e2d\u5411\u91cf<span class=\"katex math inline\">T(x)<\/span>\u79f0\u4e3ax(\u5728T\u4f5c\u7528\u4e0b)\u7684\u50cf\uff0c\u6240\u6709\u50cfT(x)\u7684\u96c6\u5408\u79f0\u4e3aT\u7684<strong>\u503c\u57df<\/strong><br \/>\n<a class=\"wp-editor-md-post-content-link\" href=\"http:\/\/www.wayln.com\/wp-content\/uploads\/2021\/04\/wp_editor_md_73018eb1384329f2b77787e02babae3d.jpg\"><img decoding=\"async\" src=\"http:\/\/www.wayln.com\/wp-content\/uploads\/2021\/04\/wp_editor_md_73018eb1384329f2b77787e02babae3d.jpg\" alt=\"\" \/><\/a><\/p>\n<h2><span id=\"i\">\u77e9\u9635\u53d8\u6362<\/span><\/h2>\n<p>\u5bf9<span class=\"katex math inline\">R^n<\/span>\u4e2d\u6bcf\u4e2ax,T\uff08x\uff09\u7531Ax\u8ba1\u7b97\u5f97\u5230\uff0c\u5176\u4e2dA\u662fmxn\u77e9\u9635\uff0c\u4e3a\u4e86\u7b80\u5355\u8d77\u89c1\uff0c\u6709\u65f6\u5c06\u8fd9\u6837\u4e00\u4e2a<strong>\u77e9\u9635\u53d8<\/strong>\u6362\u8bb0\u4e3a<span class=\"katex math inline\">x\\rightarrow Ax<\/span>,\u6ce8\u610f<span class=\"katex math inline\">{\\color{red}\u5f53A\u6709n\u5217\u65f6}<\/span>\uff0cT\u7684\u5b9a\u4e49\u57df\u4e3a<span class=\"katex math inline\">R^n<\/span>,\u800c\u5f53<span class=\"katex math inline\">{\\color{red}A\u7684\u6bcf\u4e2a\u5217\u6709m\u4e2a\u5143\u7d20\u65f6}<\/span>\uff0cT\u7684\u4f59\u5b9a\u4e49\u57df\u4e3a<span class=\"katex math inline\">R^m<\/span>\u3002T\u7684\u503c\u57df\u4e3aA\u7684\u5217\u7684\u6240\u6709\u7ebf\u6027\u7ec4\u5408\u7684\u96c6\u5408\uff0c\u56e0\u4e3a\u6bcf\u4e2a\u50cfT(x)\u6709Ax\u7684\u5f62\u5f0f\u3002<\/p>\n<p>\u4f8b 3 \u8bbe<span class=\"katex math inline\">A=\\begin{bmatrix}1&amp;3&#92;&#92;0&amp;1&#92;&#92;\\end{bmatrix}<\/span>,\u53d8\u6362<span class=\"katex math inline\">T:R^2 \\rightarrow R^2<\/span> \u5b9a\u4e49\u4e3a<span class=\"katex math inline\">T(x)=Ax<\/span>\u79f0\u4e3a\u526a\u5207\u53d8\u6362\u3002<br \/>\n<a class=\"wp-editor-md-post-content-link\" href=\"http:\/\/www.wayln.com\/wp-content\/uploads\/2021\/04\/wp_editor_md_e965ddd595ec6a791394c7dd971c2fd0.jpg\"><img decoding=\"async\" src=\"http:\/\/www.wayln.com\/wp-content\/uploads\/2021\/04\/wp_editor_md_e965ddd595ec6a791394c7dd971c2fd0.jpg\" alt=\"\" \/><\/a><br \/>\n<a class=\"wp-editor-md-post-content-link\" href=\"http:\/\/www.wayln.com\/wp-content\/uploads\/2021\/04\/wp_editor_md_a70e9ec041b64d3393e77d757935f07a.jpg\"><img decoding=\"async\" src=\"http:\/\/www.wayln.com\/wp-content\/uploads\/2021\/04\/wp_editor_md_a70e9ec041b64d3393e77d757935f07a.jpg\" alt=\"\" \/><\/a><\/p>\n<h2><span id=\"i-2\">\u7ebf\u6027\u53d8\u6362<\/span><\/h2>\n<p>\u82e5A\u662fmxn\u77e9\u9635\uff0c\u5219\u53d8\u6362<span class=\"katex math inline\">x \\mapsto Ax<\/span>\u6709\u4ee5\u4e0b\u6027\u8d28\uff1a<\/p>\n<div class=\"katex math multi-line no-emojify\">A(u+v)=Au+Av\u548cA(cu)=cAu\n<\/div>\n<p><span class=\"katex math inline\">u,v\u662fR^n<\/span>\u4e2d\u4efb\u610f\u5411\u91cf\uff0cc\u662f\u4efb\u610f\u6570\u3002\u8fd9\u4e9b\u6027\u8d28\u82e5\u7528\u51fd\u6570\u8bb0\u53f7\u6765\u8868\u793a\uff0c\u5c31\u5f97\u5230\u7ebf\u6027\u4ee3\u6570\u91cd\u6700\u91cd\u8981\u7684\u4e00\u7c7b\u53d8\u6362\u3002<\/p>\n<blockquote><p>\n  \u5b9a\u4e49 \u53d8\u6362(\u6216\u6620\u5c04)T\u79f0\u4e3a\u7ebf\u6027\u7684\uff0c\u82e5<br \/>\n  1. \u5bf9T\u7684\u5b9a\u4e49\u57df\u4e2d\u4e00\u5207 <span class=\"katex math inline\">u,v,T(u+v)=T(u)+T(v)<\/span><br \/>\n  2. \u5bf9\u4e00\u5207u\u548c\u6807\u91cfc,<span class=\"katex math inline\">T(cu)=cT(u)<\/span><\/p>\n<p>  \u82e5T\u662f\u7ebf\u6027\u53d8\u6362\uff0c\u5219<\/p>\n<div class=\"katex math multi-line no-emojify\">T(0)=0 \\tag{3}\n  <\/div>\n<p>  \u4e14\u5bf9T\u7684\u5b9a\u4e49\u57df\u4e2d\u4e00\u5207\u5411\u91cfu\u548cv\u4ee5\u53ca\u6570c\u548cd\u6709\uff1a<\/p>\n<div class=\"katex math multi-line no-emojify\">T(cu+dv)=cT(u)+dT(v) \\tag{4}\n  <\/div>\n<p>  \u91cd\u590d\u5e94\u7528\uff084\uff09\u5f97\u51fa\u6709\u7528\u7684\u63a8\u5e7f\uff1a\n<\/p><\/blockquote>\n<div class=\"katex math multi-line no-emojify\">T(c_1v_1+\\dots+c_pv_p)=c_1T(v_1)+\\dots+c_pT(v_p) \\tag{5}\n<\/div>\n<p>\u5728\u5de5\u7a0b\u548c\u7269\u7406\u4e2d\uff0c\uff085\uff09\u5f0f\u79f0\u4e3a\u53e0\u52a0\u539f\u7406\u3002<\/p>\n<blockquote><p>\n  \u7ed9\u5b9a\u6807\u91cfr,\u5b9a\u4e49<span class=\"katex math inline\">T:R^2 \\mapsto R^2\u4e3aT(x)=rx<\/span>\u3002\u5f53<span class=\"katex math inline\">0 \\leq r\\leq1<\/span>\u65f6\uff0cT\u79f0\u4e3a<strong>\u538b\u7f29\u53d8\u6362<\/strong>,\u5f53<span class=\"katex math inline\">r > 1<\/span>\u65f6\uff0cT\u79f0\u4e3a<strong>\u62c9\u4f38\u53d8\u6362<\/strong>\u3002\n<\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"<p>Contents1 1.8 \u7ebf\u6027\u53d8\u6362\u4ecb\u7ecd1.1 \u77e9\u9635\u53d8\u63621.2 \u7ebf\u6027\u53d8\u6362 1.8 \u7ebf\u6027\u53d8\u6362\u4ecb\u7ecd \u901a\u5e38\u662f\u628a\u77e9\u9635A [&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-235","post","type-post","status-publish","format-standard","hentry","category-21","category-22","category-26"],"_links":{"self":[{"href":"http:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/235","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=235"}],"version-history":[{"count":9,"href":"http:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/235\/revisions"}],"predecessor-version":[{"id":245,"href":"http:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/235\/revisions\/245"}],"wp:attachment":[{"href":"http:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=235"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=235"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=235"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}