{"id":415,"date":"2021-04-30T06:50:30","date_gmt":"2021-04-29T22:50:30","guid":{"rendered":"http:\/\/www.wayln.com\/?p=415"},"modified":"2021-04-30T07:12:09","modified_gmt":"2021-04-29T23:12:09","slug":"1-3-%e5%90%91%e9%87%8f%e6%96%b9%e7%a8%8b","status":"publish","type":"post","link":"https:\/\/www.wayln.com\/?p=415","title":{"rendered":"1.3 \u5411\u91cf\u65b9\u7a0b"},"content":{"rendered":"<div id=\"toc_container\" class=\"toc_transparent no_bullets\"><p class=\"toc_title\">Contents<\/p><ul class=\"toc_list\"><li><a href=\"#R2\"><span class=\"toc_number toc_depth_1\">1<\/span> R^2 \u4e2d\u7684\u5411\u91cf<\/a><\/li><li><a href=\"#R2-2\"><span class=\"toc_number toc_depth_1\">2<\/span> R^2 \u7684\u51e0\u4f55\u8868\u793a<\/a><\/li><li><a href=\"#Rn\"><span class=\"toc_number toc_depth_1\">3<\/span> R^n \u4e2d\u7684\u5411\u91cf<\/a><\/li><li><a href=\"#i\"><span class=\"toc_number toc_depth_1\">4<\/span> \u7ebf\u6027\u7ec4\u5408<\/a><\/li><\/ul><\/div>\n<p>\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u91cd\u8981\u6027\u8d28\u90fd\u53ef\u4ee5\u7528\u5411\u91cf\u6982\u5ff5\u4e0e\u7b26\u53f7\u6765\u63cf\u8ff0\u3002<\/p>\n<h1><span id=\"R2\">R^2 \u4e2d\u7684\u5411\u91cf<\/span><\/h1>\n<p>\u4ec5\u542b\u6709\u4e00\u5217\u7684\u77e9\u9635\u79f0\u4e3a\u5217\u5411\u91cf\uff0c\u6216\u7b80\u79f0\u5411\u91cf\uff1a<\/p>\n<div class=\"katex math multi-line no-emojify\">u=<br \/>\n\\begin{bmatrix}<br \/>\n3\\\\<br \/>\n-1<br \/>\n\\end{bmatrix}<br \/>\n,v=<br \/>\n\\begin{bmatrix}<br \/>\n0.2\\\\<br \/>\n0.3\\\\<br \/>\n\\end{bmatrix},<br \/>\nw=<br \/>\n\\begin{bmatrix}<br \/>\nw_1\\\\<br \/>\nw_2\\\\<br \/>\n\\end{bmatrix}\n<\/div>\n<p>\u8fd9\u91cc<span class=\"katex math inline\">w_1,w_2<\/span>\u662f\u4efb\u610f\u5b9e\u6570\uff0c\u6240\u6709\u4e24\u4e2a\u5143\u7d20\u7684\u5411\u91cf\u7684\u96c6\u8bb0\u4e3a<span class=\"katex math inline\">R^2<\/span>,R\u8868\u793a\u5411\u91cf\u4e2d\u7684\u5143\u7d20\u662f\u5b9e\u6570\uff0c\u800c\u6307\u65702\u8868\u793a\u6bcf\u4e2a\u5411\u91cf\u5305\u542b\u4e24\u4e2a\u5143\u7d20\u3002<br \/>\n\u6709\u65f6\u4e3a\u4e86\u65b9\u4fbf\uff0c\u6211\u4eec\u628a\u5217\u5411\u91cf<\/p>\n<div class=\"katex math multi-line no-emojify\">\\begin{bmatrix}3\\\\-1\\\\\\end{bmatrix}\n<\/div>\n<p>\u5199\u6210(3,-1)\u7684\u5f62\u5f0f\uff0c\u8fd9\u65f6\u5019\uff0c\u7528\u5706\u62ec\u5f27\u8868\u793a\u5411\u91cf\uff0c\u5e76\u5728\u4e24\u4e2a\u5143\u7d20\u4e4b\u95f4\u52a0\u4e0a\u9017\u53f7\uff0c\u4ee5\u533a\u522b1&#215;2\u884c\u77e9\u9635<\/p>\n<div class=\"katex math multi-line no-emojify\">\\begin{bmatrix}<br \/>\n3&amp;-1<br \/>\n\\end{bmatrix}\n<\/div>\n<p>\u4e8e\u662f\uff1a<\/p>\n<div class=\"katex math multi-line no-emojify\">\\begin{bmatrix}<br \/>\n3\\\\-1<br \/>\n\\end{bmatrix}\\neq<br \/>\n\\begin{bmatrix}<br \/>\n3&amp;-1<br \/>\n\\end{bmatrix}\n<\/div>\n<p>\u56e0\u4e3a\u8fd9\u4e24\u4e2a\u77e9\u9635\u7684\u7ef4\u5ea6\u4e0d\u540c\uff0c\u5c3d\u7ba1\u4ed6\u4eec\u6709\u76f8\u540c\u7684\u5143\u7d20<\/p>\n<h1><span id=\"R2-2\">R^2 \u7684\u51e0\u4f55\u8868\u793a<\/span><\/h1>\n<p>\u8003\u8651\u5e73\u9762\u4e0a\u7684\u76f4\u89d2\u5750\u6807\u7cfb\u3002\u56e0\u5e73\u9762\u4e0a\u6bcf\u4e2a\u70b9\u7531\u5b9e\u6570\u7684\u6709\u5e8f\u5bf9\u786e\u5b9a\uff0c\u6211\u4eec\u53ef\u628a\u51e0\u4f55\u70b9\uff08a,b\uff09\u4e0e\u5217\u5411\u91cf <span class=\"katex math inline\">\\begin{bmatrix}a&#92;&#92;b\\end{bmatrix}<\/span> \u7b49\u540c\uff0c\u6240\u4ee5\u53ef\u4ee5\u628a<span class=\"katex math inline\">R^2<\/span>\u770b\u505a\u5e73\u9762\u4e0a\u6240\u6709\u70b9\u7684\u96c6\u5408\u3002<\/p>\n<h1><span id=\"Rn\">R^n \u4e2d\u7684\u5411\u91cf<\/span><\/h1>\n<p>\u82e5n\u662f\u6b63\u6574\u6570\uff0c<span class=\"katex math inline\">R^n<\/span>\u8868\u793a\u6240\u6709n\u4e2a\u5b9e\u6570\u6570\u5217\uff08\u6216\u6709\u5e8fn\u5143\u7ec4\uff09\u7684\u96c6\u5408\uff0c\u901a\u5e38\u5199\u6210nx1\u5217\u77e9\u9635\u7684\u5f62\u5f0f\uff0c\u5982\uff1a<\/p>\n<div class=\"katex math multi-line no-emojify\">\\begin{bmatrix}<br \/>\nu_1\\\\<br \/>\nu_2\\\\<br \/>\n\\vdots\\\\<br \/>\nu_n<br \/>\n\\end{bmatrix}\n<\/div>\n<p>\u6240\u6709\u5143\u7d20\u90fd\u662f\u96f6\u7684\u5411\u91cf\u79f0\u4e3a\u96f6\u5411\u91cf\uff0c\u75280\u8868\u793a\uff080\u4e2d\u5143\u7d20\u7684\u4e2a\u6570\u53ef\u7531\u4e0a\u4e0b\u6587\u786e\u5b9a)<\/p>\n<blockquote><p>\n  <span class=\"katex math inline\">R^n<\/span>\u4e2d\u5411\u91cf\u7684\u4ee3\u6570\u6027\u8d28<br \/>\n  \u5bf9<span class=\"katex math inline\">R^n<\/span>\u4e2d\u7684\u4e00\u5207\u5411\u91cfu,v,w\u4ee5\u53ca\u6807\u91cfc\u548cd<br \/>\n  1.  <span class=\"katex math inline\">u+v=v+u<\/span><br \/>\n  2. <span class=\"katex math inline\">(u+v)+w=u+(v+w)<\/span><br \/>\n  3. <span class=\"katex math inline\">u+0=0+u=u<\/span><br \/>\n  4. <span class=\"katex math inline\">u+(-u)=-u+u=0<\/span><br \/>\n  5. <span class=\"katex math inline\">c(u+v)=cu+cv<\/span><br \/>\n  6. <span class=\"katex math inline\">(c+d)u=cu+du<\/span><br \/>\n  7. <span class=\"katex math inline\">1u=u<\/span><br \/>\n  \u5176\u4e2d<span class=\"katex math inline\">-u\u8868\u793a(-1)u<\/span>\n<\/p><\/blockquote>\n<h1><span id=\"i\">\u7ebf\u6027\u7ec4\u5408<\/span><\/h1>\n<p>\u7ed9\u5b9a<span class=\"katex math inline\">R^n<\/span>\u4e2d\u5411\u91cf<span class=\"katex math inline\">v_1,v_2,\\cdots,v_p<\/span>\u548c\u6807\u91cf<span class=\"katex math inline\">c_1,c_2,\\cdots,c_p<\/span>\uff0c\u5411\u91cf<br \/>\n<span class=\"katex math inline\">y=c_1v_1+\\cdots+c_pv_p<\/span><br \/>\n\u79f0\u4e3a\u5411\u91cf<span class=\"katex math inline\">v_1,v_2,\\cdots,v_p<\/span>\u4ee5<span class=\"katex math inline\">c_1,c_2,\\cdots,c_p<\/span>\u4e3a\u6743\u7684\u7ebf\u6027\u7ec4\u5408<\/p>\n<blockquote><p>\n  \u5411\u91cf\u65b9\u7a0b<br \/>\n  <span class=\"katex math inline\">x_1a_1+x_2a_2+\\cdots+x_na_n=b<\/span><br \/>\n  \u548c\u589e\u5e7f\u77e9\u9635\u4e3a<\/p>\n<div class=\"katex math multi-line no-emojify\">\\begin{bmatrix}<br \/>\n  a_1&amp;a_2&amp;\\cdots&amp;a_n&amp;b \\tag{5}<br \/>\n  \\end{bmatrix}\n  <\/div>\n<p>  \u7684\u7ebf\u6027\u65b9\u7a0b\u7ec4\u6709\u76f8\u540c\u7684\u89e3\u96c6\u3002\u7279\u522b\u5730\uff0c<span class=\"katex math inline\">b<\/span>\u53ef\u4ee5\u8868\u793a\u4e3a<span class=\"katex math inline\">a_1,a_2,\\cdots,a_n<\/span>\u7684\u7ebf\u6027\u7ec4\u5408\uff0c\u5f53\u4e14\u4ec5\u5f53\u5bf9\u5e94\u4e8e\uff085\uff09\u5f0f\u7684\u65b9\u7a0b\u7ec4\u6709\u89e3\u3002\n<\/p><\/blockquote>\n<p>\u7ebf\u6027\u4ee3\u6570\u7684\u4e00\u4e2a\u4e3b\u8981\u601d\u60f3\u662f\u7814\u7a76\u53ef\u4ee5\u8868\u793a\u4e3a\u67d0\u4e00\u56fa\u5b9a\u5411\u91cf\u96c6\u5408<\/p>\n<div class=\"katex math multi-line no-emojify\">\\{v_1,v_2,\\cdots,v_n\\}\n<\/div>\n<p>\u7684\u7ebf\u6027\u7ec4\u5408\u7684\u6240\u6709\u5411\u91cf<\/p>\n<blockquote><p>\n  \u5b9a\u4e49 \u82e5<span class=\"katex math inline\">v_1,v_2,\\dots,v_p<\/span>\u662f<span class=\"katex math inline\">R^n<\/span>\u4e2d\u7684\u5411\u91cf\uff0c\u5219<span class=\"katex math inline\">v_1,v_2,\\dots,v_p<\/span>\u7684\u6240\u6709\u7ebf\u6027\u7ec4\u5408\u6240\u6210\u7684\u96c6\u5408\u7528\u8bb0\u53f7<\/p>\n<div class=\"katex math multi-line no-emojify\">Span\\{v_1,v_2,\\dots,v_p\\}\n  <\/div>\n<p>  \u8868\u793a\uff0c\u79f0\u4e3a\u7531<span class=\"katex math inline\">v_1,v_2,\\dots,v_n<\/span>\u6240\u751f\u6210\uff08\u6216\u5f20\u6210\uff09\u7684<span class=\"katex math inline\">R^n<\/span>\u7684\u5b50\u96c6\uff0c\u4e5f\u5c31\u662f\u8bf4\uff0c<\/p>\n<div class=\"katex math multi-line no-emojify\">Span\\{v_1,v_2,\\dots,v_p\\}\n  <\/div>\n<p>  \u662f\u5f62\u5982\uff1a<br \/>\n  <span class=\"katex math inline\">c_1v_1+c_2v_2+\\dots+c_nv_n<\/span>\u7684\u5411\u91cf\u7684\u96c6\u5408\uff0c\u5176\u4e2d<span class=\"katex math inline\">c_1,c_2,\\dots,c_p<\/span>\u4e3a\u6807\u91cf\n<\/p><\/blockquote>\n<p>\u8981\u5224\u65ad\u5411\u91cfb\u662f\u5426\u5c5e\u4e8e<\/p>\n<div class=\"katex math multi-line no-emojify\">Span\\{v_1,v_2,\\dots,v_p\\}\n<\/div>\n<p>,\u5c31\u662f\u5224\u65ad\u5411\u91cf\u65b9\u7a0b<span class=\"katex math inline\">x_1v_1+x_2v_2+\\dots+x_pv_p=b<\/span>\u662f\u5426\u6709\u89e3\uff0c\u6216\u7b49\u4ef7\u5730\uff0c\u5224\u65ad\u589e\u5e7f\u77e9\u9635\u4e3a<span class=\"katex math inline\">\\begin{bmatrix}v_1&amp;v_2&amp;\\dots&amp;v_p&amp;b\\end{bmatrix}<\/span>\u7684\u7ebf\u6027\u65b9\u7a0b\u7ec4\u662f\u5426\u6709\u89e3\u3002<br \/>\n\u6ce8\u610f<span class=\"katex math inline\">Span&#92;{v_1,v_2,\\dots,v_p&#92;}<\/span>\u5305\u542b<span class=\"katex math inline\">v_1<\/span>\u7684\u6240\u6709\u500d\u6570\uff0c\u56e0<span class=\"katex math inline\">cv_1=cv_1+0v_2+\\dots+0v_p<\/span>,\u7279\u522b\u5730\uff0c\u5b83\u4e00\u5b9a\u5305\u542b\u96f6\u5411\u91cf\u3002<\/p>\n<p><span class=\"katex math inline\">Span&#92;{v&#92;}\u4e0eSpan&#92;{u,v&#92;}<\/span>\u7684\u96c6\u5408\u89e3\u91ca<br \/>\n\u8bbev\u662f<span class=\"katex math inline\">R^3<\/span>\u4e2d\u7684\u5411\u91cf\uff0c\u90a3\u4e48<span class=\"katex math inline\">Span&#92;{v&#92;}<\/span>\u5c31\u662fv\u7684\u6240\u6709\u6570\u91cf\u500d\u6570\u7684\u96c6\u5408\uff0c\u4e5f\u5c31\u662f\u901a\u8fc7v\u548c0\u7684\u76f4\u7ebf\u4e0a\u7684\u6240\u6709\u70b9\u7684\u96c6\u5408\uff0c<br \/>\n\u82e5u\u548cv\u662f<span class=\"katex math inline\">R^3<\/span>\u4e2d\u7684\u975e\u96f6\u5411\u91cf\uff0cv\u4e0d\u662fu\u7684\u500d\u6570\uff0c\u5219<span class=\"katex math inline\">Span&#92;{u,v&#92;}\u662fR^3<\/span>\u4e2d\u901a\u8fc7u,v\u548c0\u7684\u5e73\u9762\uff0c\u7279\u522b\u5730\uff0c<span class=\"katex math inline\">Span&#92;{u,v&#92;}<\/span>\u5305\u542b<span class=\"katex math inline\">R^3<\/span>\u4e2d\u901a\u8fc7u\u4e0e0\u7684\u76f4\u7ebf\uff0c\u4e5f\u5305\u542b\u901a\u8fc7v\u4e0e0\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_aedd4e18b53a973477ab494fa05e7318.jpg\"><img decoding=\"async\" src=\"http:\/\/www.wayln.com\/wp-content\/uploads\/2021\/04\/wp_editor_md_aedd4e18b53a973477ab494fa05e7318.jpg\" alt=\"\" \/><\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Contents1 R^2 \u4e2d\u7684\u5411\u91cf2 R^2 \u7684\u51e0\u4f55\u8868\u793a3 R^n \u4e2d\u7684\u5411\u91cf4 \u7ebf\u6027\u7ec4\u5408 \u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u91cd\u8981\u6027 [&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-415","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\/415","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=415"}],"version-history":[{"count":7,"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/415\/revisions"}],"predecessor-version":[{"id":429,"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/415\/revisions\/429"}],"wp:attachment":[{"href":"https:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=415"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=415"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=415"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}