{"id":374,"date":"2021-04-15T21:14:02","date_gmt":"2021-04-15T13:14:02","guid":{"rendered":"http:\/\/www.wayln.com\/?p=374"},"modified":"2021-04-22T21:48:53","modified_gmt":"2021-04-22T13:48:53","slug":"%e7%ac%ac%e5%9b%9b%e7%ab%a0-%e5%90%91%e9%87%8f%e7%a9%ba%e9%97%b4","status":"publish","type":"post","link":"https:\/\/www.wayln.com\/?p=374","title":{"rendered":"4.1 \u5411\u91cf\u7a7a\u95f4\u4e0e\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> \u5b50\u7a7a\u95f4<\/a><\/li><li><a href=\"#i-2\"><span class=\"toc_number toc_depth_1\">2<\/span> \u7531\u4e00\u4e2a\u96c6\u5408\u751f\u6210\u7684\u5b50\u7a7a\u95f4<\/a><\/li><\/ul><\/div>\n<blockquote><p>\n  <strong>\u5b9a\u4e49<\/strong>\u3000\u4e00\u4e2a\u5411\u91cf\u7a7a\u95f4\u662f\u7531\u4e00\u4e9b\u88ab\u79f0\u4e3a\u5411\u91cf\u7684\u5bf9\u8c61\u6784\u6210\u7684\u975e\u7a7a\u96c6\u5408<span class=\"katex math inline\">V<\/span>\uff0c\u5728\u8fd9\u4e2a\u96c6\u5408\u4e0a\u5b9a\u4e49\u4e24\u4e2a\u8fd0\u7b97\uff0c\u79f0\u4e3a\u52a0\u6cd5\u548c\u6807\u91cf\u4e58\u6cd5\uff08\u6807\u91cf\u53d6\u5b9e\u6570\uff09\uff0c\u670d\u4ece\u4ee5\u4e0b\u516c\u91cc\uff08\u6216\u6cd5\u5219\uff09\uff0c\u8fd9\u4e9b\u516c\u7406\u5fc5\u987b\u5bf9<span class=\"katex math inline\">V<\/span>\u4e2d\u6240\u6709\u5411\u91cf<span class=\"katex math inline\">u,v,w<\/span>\u53ca\u6240\u6709\u6807\u91cf<span class=\"katex math inline\">c<\/span>\u548c<span class=\"katex math inline\">d<\/span>\u5747\u6210\u7acb\u3002<br \/>\n  1.\u3000<span class=\"katex math inline\">u,v<\/span>\u4e4b\u548c\u8868\u793a\u4e3a<span class=\"katex math inline\">u+v<\/span>\uff0c\u4ecd\u5728<span class=\"katex math inline\">V<\/span>\u4e2d<br \/>\n  2.\u3000<span class=\"katex math inline\">u+v=v+u<\/span><br \/>\n  3.\u3000<span class=\"katex math inline\">(u+v)+w=u+(v+w)<\/span><br \/>\n  4.\u3000<span class=\"katex math inline\">V<\/span>\u4e2d\u5b58\u5728\u4e00\u4e2a\u96f6\u5411\u91cf<span class=\"katex math inline\">0<\/span>\uff0c\u4f7f\u5f97<span class=\"katex math inline\">u+0=u<\/span><br \/>\n  5.\u3000\u5bf9<span class=\"katex math inline\">V<\/span>\u4e2d\u6bcf\u4e2a\u5411\u91cfu\uff0c\u5b58\u5728<span class=\"katex math inline\">V<\/span>\u4e2d\u5411\u91cf<span class=\"katex math inline\">-u<\/span>\uff0c\u4f7f\u5f97<span class=\"katex math inline\">u+(-u)=0<\/span><br \/>\n  6.\u3000<span class=\"katex math inline\">u<\/span>\u4e0e\u6807\u91cf<span class=\"katex math inline\">c<\/span>\u7684\u6807\u91cf\u4e58\u6cd5\u8bb0\u4e3a<span class=\"katex math inline\">cu<\/span>,\u4ecd\u5728<span class=\"katex math inline\">V<\/span>\u4e2d<br \/>\n  7.\u3000<span class=\"katex math inline\">c(u+v)=cu+cv<\/span><br \/>\n  8.\u3000<span class=\"katex math inline\">(c+d)u=cu+du<\/span><br \/>\n  9.\u3000<span class=\"katex math inline\">c(du)=(cd)u<\/span><br \/>\n  10.\u3000<span class=\"katex math inline\">1u=u<\/span>\n<\/p><\/blockquote>\n<p>\u516c\u74065\u4e2d\u5411\u91cf<span class=\"katex math inline\">-u<\/span>\u79f0\u4e3a<span class=\"katex math inline\">u<\/span>\u7684<strong>\u8d1f\u5411\u91cf<\/strong>,\u4e5f\u662f\u552f\u4e00\u7684.<\/p>\n<p>\u5bf9<span class=\"katex math inline\">V<\/span>\u4e2d\u6ca1\u4e2a\u5411\u91cf<span class=\"katex math inline\">u<\/span>\u548c\u4efb\u610f\u6807\u91cf<span class=\"katex math inline\">c<\/span>,\u6709<\/p>\n<div class=\"katex math multi-line no-emojify\">0u=0 \\tag{1}\n<\/div>\n<div class=\"katex math multi-line no-emojify\">c0=0 \\tag{2}\n<\/div>\n<div class=\"katex math multi-line no-emojify\">-u=(-1)u \\tag{3}\n<\/div>\n<h1><span id=\"i\">\u5b50\u7a7a\u95f4<\/span><\/h1>\n<blockquote><p>\n  <strong>\u5b9a\u4e49<\/strong> \u5411\u91cf\u7a7a\u95f4V\u7684\u4e00\u4e2a\u5b50\u7a7a\u95f4\u662fV\u7684\u4e00\u4e2a\u6ee1\u8db3\u4ee5\u4e0b\u4e09\u4e2a\u6027\u8d28\u7684\u5b50\u96c6H\uff1a<br \/>\n  a.\u3000<span class=\"katex math inline\">V<\/span>\u4e2d\u7684\u96f6\u5411\u91cf\u5728<span class=\"katex math inline\">H<\/span>\u4e2d<br \/>\n  b.\u3000<span class=\"katex math inline\">H<\/span>\u5bf9\u5411\u91cf\u52a0\u6cd5\u5c01\u95ed\uff0c\u5373\u5bf9H\u4e2d\u4efb\u610f\u5411\u91cf<span class=\"katex math inline\">u<\/span>,<span class=\"katex math inline\">v<\/span>\u548c<span class=\"katex math inline\">u+v<\/span>\u4ecd\u5728<span class=\"katex math inline\">H<\/span>\u4e2d<br \/>\n  c.\u3000<span class=\"katex math inline\">H<\/span>\u5bf9\u6807\u91cf\u4e58\u6cd5\u5c01\u95ed\uff0c\u5373\u5bf9H\u4e2d\u4efb\u610f\u5411\u91cf<span class=\"katex math inline\">u<\/span>\u548c\u4efb\u610f\u6807\u91cf<span class=\"katex math inline\">c<\/span>,\u5411\u91cf<span class=\"katex math inline\">cu<\/span>\u4ecd\u5728<span class=\"katex math inline\">H<\/span>\u4e2d\n<\/p><\/blockquote>\n<p>\u4f8b8 \u5411\u91cf\u7a7a\u95f4<span class=\"katex math inline\">R^2<\/span>\u4e0d\u662f<span class=\"katex math inline\">R^3<\/span>\u7684\u5b50\u7a7a\u95f4\uff0c\u56e0\u4e3a<span class=\"katex math inline\">R^2<\/span>\u751a\u81f3\u4e0d\u662f<span class=\"katex math inline\">R^3<\/span>\u7684\u5b50\u96c6\uff08<span class=\"katex math inline\">R^3<\/span>\u4e2d\u7684\u5411\u91cf\u67093\u4e2a\u5206\u91cf\uff0c\u800c<span class=\"katex math inline\">R^2<\/span>\u4e2d\u7684\u5411\u91cf\u4ec5\u6709\u4e24\u4efd\u5206\u91cf\uff09<\/p>\n<h1><span id=\"i-2\">\u7531\u4e00\u4e2a\u96c6\u5408\u751f\u6210\u7684\u5b50\u7a7a\u95f4<\/span><\/h1>\n<p><strong>\u7ebf\u6027\u7ec4\u5408<\/strong>\u8fd9\u4e2a\u8bcd\u8868\u793a\u4e00\u4e9b\u5411\u91cf\u7684\u4efb\u610f\u6807\u91cf\u4e58\u6cd5\u4e4b\u548c\uff0c<span class=\"katex math inline\">Span&#92;{v_1,\\dots,v_p&#92;}<\/span>\u8868\u793a\u6240\u6709\u53ef\u4ee5\u8868\u793a\u6210<span class=\"katex math inline\">v_1,\\dots,v_p<\/span>\u7684\u7ebf\u6027\u7ec4\u5408\u7684\u5411\u91cf\u96c6\u5408<\/p>\n<p><strong>\u5b9a\u74061<\/strong> \u82e5<span class=\"katex math inline\">v_1,\\dots,v_p<\/span>\u5728\u5411\u91cf\u7a7a\u95f4V\u4e2d\uff0c\u5219<span class=\"katex math inline\">Span&#92;{v_1,\\dots,v_p&#92;}<\/span>\u662fV\u7684\u4e00\u4e2a\u5b50\u7a7a\u95f4<br \/>\n\u6211\u4eec\u79f0<span class=\"katex math inline\">Span&#92;{v_1,\\dots,v_p&#92;}<\/span>\u662f\u7531<span class=\"katex math inline\">v_1,\\dots,v_p<\/span>\u751f\u6210(\u6216\u5f20\u6210)\u7684\u5b50\u7a7a\u95f4\uff0c\u4efb\u7ed9V\u7684\u5b50\u7a7a\u95f4H\uff0cH\u7684\u751f\u6210(\u6216\u5f20\u6210)\u96c6\u662f\u96c6\u5408<span class=\"katex math inline\">v_1,\\dots,v_p \\subset H<\/span>,\u4f7f\u5f97<span class=\"katex math inline\">H=Span&#92;{v_1,\\dots,v_p&#92;}<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Contents1 \u5b50\u7a7a\u95f42 \u7531\u4e00\u4e2a\u96c6\u5408\u751f\u6210\u7684\u5b50\u7a7a\u95f4 \u5b9a\u4e49\u3000\u4e00\u4e2a\u5411\u91cf\u7a7a\u95f4\u662f\u7531\u4e00\u4e9b\u88ab\u79f0\u4e3a\u5411\u91cf\u7684\u5bf9\u8c61\u6784\u6210\u7684\u975e\u7a7a\u96c6\u5408 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":375,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,22,26],"tags":[],"class_list":["post-374","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\/374","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=374"}],"version-history":[{"count":8,"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/374\/revisions"}],"predecessor-version":[{"id":386,"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/374\/revisions\/386"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/media\/375"}],"wp:attachment":[{"href":"https:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=374"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=374"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=374"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}