\u5411\u91cf\u7a7a\u95f4(vector space)<\/strong>\u5305\u542b\u4e86\u5411\u91cf\u548c\u6807\u91cf\u7684\u96c6\u5408\u3002\u5728\u672c\u7bc7\u6587\u7ae0\u4e2d\uff0c\u6240\u8003\u5bdf\u7684\u662f\u5411\u91cf\u662f\u5217\u5411\u91cf\uff0c\u6807\u91cf\u5747\u4e3a\u5b9e\u6570\u3002<\/p>\n \u5411\u91cf\u7a7a\u95f4\u8981\u6c42\u8be5\u5411\u91cf\u7a7a\u95f4\u4e2d\u7684\u5411\u91cf\u548c\u6807\u91cf\u2014\u2014\u5728\u6807\u91cf\u4e58\u6cd5\u548c\u5411\u91cf\u52a0\u6cd5\u7684\u60c5\u51b5\u4e0b\u662f\u5c01\u95ed\u7684(closed)<\/strong>\u3002\u8fd9\u610f\u5473\u7740\uff0c\u4efb\u610f\u51e0\u4e2a\u6807\u91cf\u6216\u8005\u5411\u91cf\u7ec4\u5408\u51fa\u6765\u7684\u5411\u91cf\uff0c\u4ecd\u7136\u4f4d\u4e8e\u8fd9\u4e2a\u5411\u91cf\u7a7a\u95f4\u3002\u6211\u4eec\u770b\u4e00\u4e2a\u4f8b\u5b50\uff1a<\/p>\n \u6211\u4eec\u6709\uff1a<\/p>\n $$ \u4ee4${\\rm w} = a{\\rm u} + b{\\rm v}$\uff0c\u4e8e\u662f\uff1a<\/p>\n $$ \u8fd9\u4e2a\u5411\u91cf\u4e5f\u662f$3-{\\rm by}-1$\u7684\uff0c\u56e0\u6b64\u4e5f\u5c5e\u4e8e$\\rm u, v$\u6240\u5728\u7684\u5411\u91cf\u7a7a\u95f4\u3002\u8fd9\u4e2a\u5411\u91cf\u7a7a\u95f4\u88ab\u79f0\u4e3a\uff1a$\\mathbb{R}^{3}$\u3002<\/p>\n \u6211\u4eec\u7814\u7a76\u5411\u91cf\u7a7a\u95f4\u4e3b\u8981\u662f\u4e3a\u4e86\u786e\u5b9a\u4e0e\u77e9\u9635\u76f8\u5173\u7684\u5411\u91cf\u7a7a\u95f4\u3002\u4e00\u4e2a$m-{\\rm by}-n$\u7684\u77e9\u9635\u6709\u56db\u4e2a\u57fa\u672c\u7684\u5411\u91cf\u7a7a\u95f4\uff1a<\/p>\n \u5177\u4f53\u7684\uff0c\u5411\u91cf\u96c6\u5408\u6240\u6784\u6210\u7684\u5411\u91cf\u7a7a\u95f4\u9700\u8981\u6ee1\u8db3\u4ee5\u4e0b\u51e0\u4e2a\u6761\u4ef6\uff1a<\/p>\n \u56e0\u6b64\uff0c\u5224\u65ad\u4e00\u4e2a\u5411\u91cf\u96c6\u5408(\u77e9\u9635)\u662f\u5426\u662f\u4e00\u4e2a\u5411\u91cf\u7a7a\u95f4\uff0c\u53ef\u4ee5\u9075\u5faa\u5982\u4e0b\u6b65\u9aa4\uff1a<\/p>\n \u5047\u8bbe\u6211\u4eec\u73b0\u5728\u6709\u5411\u91cf$\\rm u$\u548c$\\rm v$\uff0c\u4ee5\u53ca\u6807\u91cf$k$\u3002<\/p>\n \u4f8b1\uff0c\u5047\u8bbe\u6211\u4eec\u6709\u201c\u7b2c\u4e8c\u884c\u4e3a\u96f6\u7684\u6240\u6709$3-{\\rm by}-1$\u77e9\u9635\u201d\uff1a<\/p>\n \u4ee4:<\/p>\n $$ \u9996\u5148\u9a8c\u8bc1\u77e9\u9635\u52a0\u6cd5\uff1a<\/p>\n $$ \u4ecd\u7136\u6ee1\u8db3\u7b2c\u4e8c\u884c\u4e3a\u96f6\u3002\u63a5\u7740\u9a8c\u8bc1\u6807\u91cf\u4e58\u6cd5\uff1a<\/p>\n $$ \u4ecd\u7136\u6ee1\u8db3\u7b2c\u4e8c\u884c\u4e3a\u96f6\u3002\u8fd9\u4e24\u4e2a\u5411\u91cf\u6ee1\u8db3\u4e86\u5c01\u95ed\u6027\uff0c\u56e0\u6b64\u5b83\u4eec\u6784\u6210\u5411\u91cf\u7a7a\u95f4\u3002<\/p>\n \u4f8b2\uff0c\u5047\u8bbe\u6211\u4eec\u6709\u201c\u6240\u6709\u884c\u5143\u7d20\u4e4b\u548c\u7b49\u4e8e1\u7684$3-{\\rm by}-1$\u77e9\u9635\u201d\uff1a<\/p>\n \u4ee4\uff1a<\/p>\n $$ \u5176\u6ee1\u8db3\uff1a<\/p>\n $$ \u9996\u5148\u9a8c\u8bc1\u77e9\u9635\u52a0\u6cd5\uff1a<\/p>\n $$ \u5176\u4e2d\uff0c$a_1 + b_1 + a_2 + b_2 + a_3 + b_3 = 2$\uff0c\u4e0d\u6ee1\u8db3\u201c\u6240\u6709\u884c\u4e4b\u548c\u4e3a1\u201d\uff0c\u4e0d\u5c01\u95ed\u3002\u56e0\u6b64\u4e0d\u6784\u6210\u5411\u91cf\u7a7a\u95f4\u3002<\/p>\n \u5047\u8bbe\u6211\u4eec\u7684\u5411\u91cf\u7a7a\u95f4\u4e2d\u6709\u5411\u91cf\u96c6\u5408$u_1, u_2, \\dots , u_n$\u4ee5\u53ca\u4efb\u610f\u6807\u91cf$c_1, c_2, \\dots , c_n$\uff0c\u5982\u679c\u4e0b\u9762\u7b49\u5f0f\uff1a<\/p>\n $$ \u5982\u679c\u5b58\u5728\u552f\u4e00\u89e3$c_1 = c_2 = \\dots = c_n = 0$\uff0c\u8fd9\u610f\u5473\u7740\u6211\u4eec\u65e0\u6cd5\u7528\u5411\u91cf\u96c6\u5408$\\rm U$\u4e2d\u7684\u4efb\u4f55\u51e0\u4e2a\u5411\u91cf\u6765\u8868\u793a\u53e6\u4e00\u4e2a\u5411\u91cf\uff0c\u90a3\u4e48\u6211\u4eec\u53ef\u4ee5\u8bf4\u8fd9\u4e2a\u5411\u91cf\u96c6\u5408\u662f\u7ebf\u6027\u65e0\u5173(linear independent)<\/strong>\u7684\u3002<\/p>\n \u6211\u4eec\u4ece\u4e0b\u9762\u4e00\u4e2a\u4f8b\u5b50\u6765\u770b\uff1a<\/p>\n $$ \u6211\u4eec\u53ef\u4ee5\u7528$\\rm w = 2u + 3v$\u6765\u8868\u793a\uff0c\u56e0\u6b64\u8fd9\u4e2a\u5411\u91cf\u7ec4\u662f\u7ebf\u6027\u76f8\u5173\u7684\u3002\u6211\u4eec\u4fdd\u6301$\\rm u,v$\u4e0d\u53d8\uff0c\u66f4\u6539$\\rm w$\u6765\u770b\u4e00\u4e0b\uff1a<\/p>\n $$ \u6b64\u65f6\uff0c\u6211\u4eec\u65e0\u6cd5\u7528\u53e6\u5916\u4e24\u4e2a\u5411\u91cf\u6765\u8868\u793a\u7b2c\u4e09\u4e2a\u5411\u91cf\uff0c\u56e0\u6b64\u73b0\u5728\u8fd9\u4e2a\u5411\u91cf\u7ec4\u662f\u7ebf\u6027\u65e0\u5173\u7684\u3002\u540c\u65f6\u6211\u4eec\u4e5f\u53ef\u4ee5\u7528\u7ebf\u6027\u65e0\u5173\u7684\u5b9a\u4e49\u6765\u770b\uff1a<\/p>\n $$ \u56e0\u4e3a$a = b = c = 0$\uff0c\u6240\u4ee5\u8be5\u5411\u91cf\u7ec4\u662f\u7ebf\u6027\u65e0\u5173\u7684\u3002<\/p>\n \u5bf9\u4e8e\u7b80\u5355\u7684\u4f8b\u5b50\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u76ee\u89c6\u89e3\u51b3\u3002\u5bf9\u4e8e\u590d\u6742\u7684\u77e9\u9635\uff0c\u901a\u5e38\u91c7\u53d6\u5c06\u5176\u5316\u7b80\u4e3a\u884c\u6700\u7b80\u7684\u5f62\u5f0f\uff1a\u5982\u679c\u6700\u540e\u4e00\u884c\u5168\u4e3a\u96f6\u5219\u662f\u7ebf\u6027\u76f8\u5173\u7684\uff1b\u5982\u679c\u6700\u540e\u4e00\u884c\u4e0d\u5168\u4e3a\u96f6\uff0c\u5219\u662f\u7ebf\u6027\u65e0\u5173\u7684\u3002<\/p>\n \u7ed9\u5b9a\u4e00\u4e2a\u5411\u91cf\u7ec4\uff0c\u53ef\u4ee5\u7531\u8be5\u5411\u91cf\u7ec4\u7684\u7ec4\u5408\u751f\u6210\u5411\u91cf\u7a7a\u95f4\u3002\u6211\u4eec\u8bf4\u8be5\u5411\u91cf\u96c6\u5408span<\/strong>\u8fd9\u4e2a\u5411\u91cf\u7a7a\u95f4\u3002<\/p>\n \u4e0b\u9762\u662f\u4e00\u4e2a\u5411\u91cf\u96c6\u5408\uff1a<\/p>\n $$ \u8be5\u96c6\u5408\uff0c\u8de8\u8d8a\u7b2c\u4e09\u884c\u5305\u542b\u96f6\u7684\u6240\u6709$3-{\\rm by}-1$\u77e9\u9635\u7684\u5411\u91cf\u7a7a\u95f4\u3002\u6b64\u5411\u91cf\u7a7a\u95f4\u662f\u6240\u6709\u6ee1\u8db3\u7b2c\u4e09\u884c\u4e3a\u96f6\u7684$3-{\\rm by}-1$\u77e9\u9635\u7684\u5411\u91cf\u5b50\u7a7a\u95f4\u3002<\/p>\n \u4e0d\u9700\u8981\u6240\u6709\u8fd9\u4e09\u4e2a\u5411\u91cf\u6765\u751f\u6210\u8fd9\u4e2a\u5411\u91cf\u5b50\u7a7a\u95f4\uff0c\u56e0\u4e3a\u4efb\u4f55\u4e00\u4e2a\u5411\u91cf\u90fd\u7ebf\u6027\u4f9d\u8d56\u4e8e\u5176\u4ed6\u4e24\u4e2a\u5411\u91cf\u3002span\u4e00\u4e2a\u5411\u91cf\u7a7a\u95f4\u6240\u9700\u7684\u6700\u5c0f\u5411\u91cf\u96c6\u6784\u6210\u4e86\u8be5\u5411\u91cf\u7a7a\u95f4\u7684\u57fa(basis)<\/strong>\uff1a<\/p>\n $$ \u867d\u7136\u8fd9\u4e09\u79cd\u7ec4\u5408\u90fd\u6784\u6210\u4e86\u5411\u91cf\u5b50\u7a7a\u95f4\u7684\u57fa\uff0c\u4f46\u901a\u5e38\u9996\u9009\u7b2c\u4e00\u79cd\u7ec4\u5408\uff0c\u56e0\u4e3a\u8fd9\u662f\u6b63\u4ea4\u57fa\u3002\u6b64\u57fa\u4e2d\u7684\u5411\u91cf\u76f8\u4e92\u6b63\u4ea4\uff0c\u4e14\u5177\u6709\u5355\u4f4d\u8303\u6570\u3002\u57fa\u7684\u6570\u91cf\u51b3\u5b9a\u4e86\u5411\u91cf\u7684\u7ef4\u5ea6\uff0c\u5728\u7b2c\u4e09\u884c\u5168\u4e3a\u96f6\u7684$3-{\\rm by}-1$\u5411\u91cf\u7a7a\u95f4\u4e2d\uff0c\u5176\u7ef4\u5ea6\u662f2\u3002<\/p>\n \u5bf9\u4e8e\u4e00\u4e2a\u5411\u91cf\u7a7a\u95f4\u7684\u6b63\u4ea4\u57fa\uff0c\u5176\u6ee1\u8db3\uff1a<\/p>\n \u5047\u8bbe\u73b0\u5728\u6211\u4eec\u6709\u4e24\u4e2a\u5411\u91cf\uff1a$\\rm v$\u548c$\\rm u$\uff0c\u5176\u5939\u89d2\u4e3a$\\theta$\uff1b$\\rm u$\u5728$\\rm v$\u4e0a\u7684\u6295\u5f71\u4e3a$\\rm u'$\u3002\u73b0\u5728\u6211\u4eec\u60f3\u8981\u7528$\\rm u$\u548c$\\rm v$\u6765\u8868\u793a\u6295\u5f71$\\rm u'$\u3002\u4e8e\u662f\u6709\uff1a<\/p>\n $$ \u6a21\u957f${\\left| u' \\right|}$\u53ef\u4ee5\u8868\u793a\u4e3a\uff1a<\/p>\n $$ \u5939\u89d2$\\theta$\u7531\u5411\u91cf\u7684\u5185\u79ef\u7ed9\u51fa\uff1a<\/p>\n $$ \u56e0\u6b64\uff1a<\/p>\n $$ \u7ed9\u5b9a\u4e00\u4e2a\u5411\u91cf\u7a7a\u95f4\u7684\u4efb\u610f\u57fa\uff0c\u6211\u4eec\u53ef\u4ee5\u7528\u683c\u62c9\u59c6-\u65bd\u5bc6\u7279\u6b63\u4ea4\u5316(Gram-Schmidt process)<\/strong>\u6765\u6784\u9020\u5176\u6b63\u4ea4\u57fa\u3002<\/p>\n \u5047\u8bbe\u6211\u4eec\u542b\u6709\u57fa\uff1a<\/p>\n $$ \u6211\u4eec\u60f3\u8981\u6784\u9020\u8fd9\u4e2a\u5411\u91cf\u7a7a\u95f4\u7684\u6b63\u4ea4\u57fa\uff1a<\/p>\n $$ \u6784\u9020\u8fc7\u7a0b\u5982\u4e0b\uff1a<\/p>\n \u6211\u4eec\u5148\u770b\u4e00\u4e2a\u4ece\u666e\u901a\u57fa$\\rm v$\u6784\u9020\u6b63\u4ea4\u57fa$\\rm u$\u7684\u4e00\u822c\u4f8b\u5b50\uff1a<\/p>\n \u6211\u4eec\u9996\u5148\u9009\u62e9\u7b2c\u4e00\u4e2a\u6b63\u4ea4\u57fa$\\rm u_1$\uff0c\u4ee4$\\rm u_1 = v_1$\u3002\u7136\u540e\u6784\u9020\u4e0b\u4e00\u4e2a\u6b63\u4ea4\u57fa$\\rm u_2$\uff1a<\/p>\n $$ \u663e\u7136\uff0c$\\rm u_2$\u7b49\u4e8e$\\rm v_2$\u51cf\u53bb$\\rm v_2$\u4e2d\u5e73\u884c\u4e8e$\\rm u_1$\u7684\u90e8\u5206\u3002\u4e0b\u9762\u6211\u4eec\u6765\u9a8c\u8bc1\u4e00\u4e0b$\\rm u_1$\u548c$\\rm u_2$\u662f\u5426\u662f\u6b63\u4ea4\u7684\uff1a<\/p>\n $$ \u8bc1\u660e$\\rm u_1$\u548c$\\rm u_2$\u662f\u6b63\u4ea4\u7684\uff0c\u63a5\u7740\u6211\u4eec\u6765\u6784\u9020\u4e0b\u4e00\u4e2a\u6b63\u4ea4\u57fa$\\rm u_3$\u3002\u6309\u7167\u7ecf\u9a8c\uff0c\u8fd9\u4e2a\u6b63\u4ea4\u57fa\u5e94\u8be5\u662f\u201c$\\rm v_3$\uff0c\u51cf\u53bb$\\rm v_3$\u4e0e$\\rm u_1$\u5e73\u884c\u7684\u90e8\u5206\uff0c\u51cf\u53bb$\\rm v_3$\u4e0e$\\rm u_2$\u5e73\u884c\u7684\u90e8\u5206\u201d\u3002\u56e0\u6b64\uff1a<\/p>\n $$ \u4e0a\u9762\u7684\u6b63\u4ea4\u5411\u91cf\u8fd8\u672a\u8fdb\u884c\u5f52\u4e00\u5316\u3002\u5411\u91cf\u7684\u5f52\u4e00\u5316\u53ef\u4ee5\u901a\u8fc7\u5982\u4e0b\u516c\u5f0f\u5b8c\u6210\uff1a<\/p>\n $$ \u56e0\u4e3a$\\rm u_k$\u662f$\\rm v_1, v_2, \\dots , v_k$\u7684\u7ebf\u6027\u7ec4\u5408\uff0c\u56e0\u6b64\u539f\u59cb\u5411\u91cf\u7a7a\u95f4\u7684\u524d$k$\u4e2a\u57fa\u5411\u91cf\u6240\u6784\u6210\u7684\u5411\u91cf\u5b50\u7a7a\u95f4\u4e0e\u901a\u8fc7\u683c\u62c9\u59c6-\u65bd\u5bc6\u7279\u6b63\u4ea4\u5316\u751f\u6210\u7684\u524d$k$\u4e2a\u6b63\u4ea4\u5411\u91cf\u6240\u6784\u6210\u7684\u5b50\u7a7a\u95f4\u76f8\u540c\u3002\u53ef\u4ee5\u8868\u793a\u4e3a\uff1a<\/p>\n $$ \u77e9\u9635$\\rm A$\u7684null space\u88ab\u8bb0\u4f5c$\\rm Null(A)$\uff0c\u5b83\u662f\u4e00\u4e2a\u88ab\u6240\u6709\u6ee1\u8db3\u5982\u4e0b\u6761\u4ef6\u7684\u5217\u5411\u91cf\u6a2a\u8de8\u7684\u5411\u91cf\u7a7a\u95f4\uff1a<\/p>\n $$ \u663e\u7136\uff0c\u5982\u679c$\\rm x$\u548c$\\rm y$\u5c5e\u4e8enull space\uff0c\u90a3\u4e48$a{\\rm x} + b{\\rm y}$\u4e5f\u5c5e\u4e8enull space\uff0c\u5176\u7b26\u5411\u91cf\u7a7a\u95f4\u7684\u5408\u5c01\u95ed\u539f\u5219\u3002\u5982\u679c\u77e9\u9635$\\rm A$\u5927\u5c0f\u662f$m-{\\rm by}-n$\uff0c\u90a3\u4e48$\\rm Null(A)$\u662f\u6240\u6709$n-{\\rm by}-1$\u7684\u5217\u77e9\u9635\u7684\u5411\u91cf\u5b50\u7a7a\u95f4\u3002\u5982\u679c$\\rm A$\u662f\u4e00\u4e2a\u53ef\u9006\u65b9\u9635\uff0c\u90a3\u4e48$\\rm Null(A)$\u4ec5\u6709\u96f6\u5411\u91cf\u7ec4\u6210\u3002<\/p>\n \u6211\u4eec\u4ece\u4e0b\u9762\u4e00\u4e2a\u4f8b\u5b50\u6765\u8003\u5bdf\u5982\u4f55\u5bfb\u627e\u5230\u4e00\u4e2a\u4e0d\u53ef\u9006\u77e9\u9635\u7684null space\u3002\u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a$3-{\\rm by}-5$\u7684\u77e9\u9635\uff1a<\/p>\n $$ \u6211\u4eec\u5c06\u5176\u5316\u7b80\u4e3a\u884c\u6700\u7b80\u7684\u5f62\u5f0f\uff1a<\/p>\n $$ \u56e0\u4e3a$\\rm Ax = 0$\uff0c\u6211\u4eec\u5c06\u77e9\u9635$A$\u4e2d\u4e3b\u5143\u5217(pivot columns)\u5bf9\u5e94\u7684\u7684$x_1$\u548c$x_3$\u79f0\u4e3a\u57fa\u7840\u53d8\u91cf(basic variables)<\/strong>\uff0c\u5c06\u975e\u4e3b\u5143\u5217(non-pivot columns)\u5bf9\u5e94\u7684$x_2$\u3001$x_4$\u548c$x_5$\u79f0\u4e3a\u81ea\u7531\u53d8\u91cf(free variables)<\/strong>\u3002\u6211\u4eec\u7528\u81ea\u7531\u53d8\u91cf\u6765\u8868\u793a\u57fa\u7840\u53d8\u91cf\u6709\uff1a<\/p>\n $$ \u901a\u8fc7\u6d88\u9664$x_1$\u548c$x_3$\uff0c\u6211\u4eec\u5f97\u5230\uff1a<\/p>\n $$ \u5176\u4e2d$x_2$\u3001$x_4$\u548c$x_5$\u53ef\u4ee5\u53d6\u4efb\u610f\u503c\uff0c\u901a\u8fc7\u5c06null space\u5199\u4e3a\u5982\u6b64\u5f62\u5f0f\uff0c$\\rm Null(A)$\u663e\u800c\u6613\u89c1\u5730\u53ef\u4ee5\u5199\u4e3a\uff1a<\/p>\n $$ $\\rm A$\u7684null space\u662f\u6240\u6709$5-{\\rm by}-1$\u7684\u5217\u77e9\u9635\u7684\u4e09\u7ef4\u5b50\u7a7a\u95f4\u3002\u603b\u7684\u6765\u8bf4\uff0c$\\rm Null(A)$\u7684\u7ef4\u5ea6\u7b49\u4e8e$\\rm rref(A)$\u7684\u975e\u4e3b\u5143\u5217\u7684\u6570\u91cf\u3002<\/p>\n Null space\u7684\u4e00\u4e2a\u5e94\u7528\u662f\u7528\u4e8e\u6c42underdetermined equation\u7684\u901a\u89e3\u3002Underdetermined equation\u6307\u7684\u662f\u65b9\u7a0b\u6570\u91cf\u5c0f\u4e8e\u672a\u77e5\u6570\u7684\u65b9\u7a0b\u7ec4\u3002<\/p>\n \u5047\u8bbe\u6211\u4eec\u8981\u6c42\u89e3$\\rm Ax = b$\uff0c\u5982\u679c$\\rm u$\u662f\u77e9\u9635$\\rm A$\u7684null space\u7684\u4e00\u822c\u5f62\u5f0f\uff0c$\\rm v$\u662f\u4efb\u4f55\u6ee1\u8db3$\\rm Av = b$\u7684\u5411\u91cf\uff0c\u90a3\u4e48$\\rm x = u + v$\u6ee1\u8db3${\\rm Ax} = {\\rm A(u+v)} = {\\rm Au} + {\\rm Av} = 0 + b = b$\u3002\u56e0\u6b64\uff0c$\\rm Ax=b$\u7684\u901a\u89e3\u53ef\u4ee5\u8868\u793a\u4e3a\uff1a$\\rm Null(A)$\u7684\u901a\u89e3\u5411\u91cf\uff0c\u4e0e\u6ee1\u8db3\u8be5\u65b9\u7a0b\u7ec4\u7684\u4efb\u610f\u7279\u89e3\u5411\u91cf\u3002<\/p>\n \u6211\u4eec\u901a\u8fc7\u4e0b\u9762\u4e00\u4e2a\u4f8b\u5b50\u6765\u8003\u5bdf\u5982\u4f55\u6c42\u89e3\u65b9\u7a0b\u7ec4\uff1a<\/p>\n $$ \u7528\u77e9\u9635\u7684\u5f62\u5f0f\u8868\u793a\u4e3a\uff1a<\/p>\n $$ \u6211\u4eec\u5c06\u5176\u8f6c\u5316\u4e3a\u589e\u5e7f\u77e9\u9635\u7684\u5f62\u5f0f\uff0c\u5e76\u5316\u7b80\u4e3a\u884c\u6700\u7b80\u3002<\/p>\n $$ \u8fd9\u4e2anull space\u6ee1\u8db3$\\rm Au = 0$\uff0c\u56e0\u6b64\u6211\u4eec\u6709\uff1a$u_1 = 0$\u548c$u_2 = -u_3 \/ 2$\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5176\u5199\u6210\uff1a<\/p>\n $$ \u6211\u4eec\u4ee5\u53ca\u5f97\u5230\u4e86\u901a\u89e3\uff0c\u4e0b\u9762\u6765\u6c42\u7279\u89e3\u3002\u7279\u89e3\u6ee1\u8db3${\\rm Av} = b$\uff0c\u56e0\u6b64\u6709\uff1a$v_1 = 1\/4$\u548c$v_2 + v_3 = -1\/4$\u3002\u56e0\u4e3a\u5b58\u57283\u4e2a\u672a\u77e5\u6570\u548c2\u4e2a\u65b9\u7a0b\uff0c\u6211\u4eec\u5c06\u5176\u4e2d\u4e00\u4e2a\u672a\u77e5\u6570\u53d6\u4efb\u610f\u503c\uff0c\u7136\u540e\u8868\u793a\u53e6\u5916\u4e24\u4e2a\u53d8\u91cf\u3002\u4e3a\u4e86\u65b9\u4fbf\uff0c\u6211\u4eec\u5c06$v_3$\u53d6$0$\uff0c\u4e8e\u662f$v_1 = 1\/4$\u3001$v_2 = -1\/4$\u3002\u6211\u4eec\u5c06\u7279\u89e3\u5199\u4e3a\u5411\u91cf\u7684\u5f62\u5f0f\uff1a<\/p>\n $$ \u6211\u4eec\u5c06\u901a\u89e3\u548c\u7279\u89e3\u5199\u5230\u4e00\u8d77\u6709\uff1a<\/p>\n $$ \u8be5\u5f62\u5f0f\u6784\u6210\u4e86$\\rm Ax=b$\u7684\u901a\u89e3\uff0c\u5176\u4e2d$a$\u53d6\u4efb\u610f\u503c\u3002<\/p>\n \u77e9\u9635\u7684column space\u662f\u6a2a\u8de8\u77e9\u9635\u5217\u7684\u5411\u91cf\u7a7a\u95f4\u3002\u5f53\u77e9\u9635\u4e58\u4ee5\u4e00\u4e2a\u5217\u5411\u91cf\u662f\uff0c\u7ed3\u679c\u4ecd\u7136\u5728\u8fd9\u4e2a\u5411\u91cf\u7a7a\u95f4\u4e2d\uff0c\u53ef\u4ee5\u8868\u793a\u4e3a\uff1a<\/p>\n $$ \u603b\u7684\u6765\u8bf4\uff0c$\\rm Ax$\u662f\u77e9\u9635$\\rm A$\u5217\u7684\u7ebf\u6027\u7ec4\u5408\u3002\u7ed9\u5b9a\u4e00\u4e2a$m-{\\rm by}-n$\u7684\u77e9\u9635$\\rm A$\uff0c$\\rm A$\u7684\u5217\u7a7a\u95f4\u7684\u7ef4\u5ea6\u662f\u591a\u5c11\uff1f\u6211\u4eec\u8981\u5982\u4f55\u627e\u5230\u57fa\uff1f\u7531\u4e8e$\\rm A$\u6709$m$\u884c\uff0c\u56e0\u6b64$\\rm A$\u7684\u5217\u7a7a\u95f4\u662f\u6240\u6709$m-{\\rm by}-1$\u5217\u77e9\u9635\u7684\u5b50\u7a7a\u95f4\u3002<\/p>\n \u6211\u4eec\u53ef\u4ee5\u6309\u7167\u5982\u4e0b\u65b9\u6cd5\u6765\u627e\u5230\u5217\u7a7a\u95f4\u7684\u57fa\uff1a<\/p>\n \u5047\u8bbe\u6211\u4eec\u6709\u5982\u4e0b\u77e9\u9635\uff1a<\/p>\n $$ $\\rm Ax=0$\u8868\u793a\u77e9\u9635$\\rm A$\u7684\u7ebf\u6027\u76f8\u5173\uff0c\u5e76\u4e14\u884c\u64cd\u4f5c\u5e76\u4e0d\u4f1a\u6539\u53d8\u5176\u7ebf\u6027\u5173\u7cfb(\u53ef\u4ee5\u7406\u89e3\u4e3a\u884c\u64cd\u4f5c\u662f\u5728\u89e3\u65b9\u7a0b)\u3002\u4ece\u4e0a\u9762\u7684\u884c\u6700\u7b80\u77e9\u9635\u4e2d\u663e\u7136\u53ef\u4ee5\u770b\u51fa\uff0c\u53ea\u6709\u4e3b\u5143\u5217\u662f\u7ebf\u6027\u65e0\u5173\u7684\uff0c\u5e76\u4e14$\\rm A$\u7684column space\u7684\u7ef4\u5ea6\u7b49\u4e8e\u4e3b\u5143\u7684\u6570\u91cf\u3002\u540c\u65f6\uff0c\u57fa\u5411\u91cf\u7531\u884c\u6700\u7b80\u77e9\u9635$\\rm rref(A)$\u6240\u5728\u4e3b\u5143\u5217\u7684\u4f4d\u7f6e\u5bf9\u5e94\u7684\u539f\u77e9\u9635$\\rm A$\u7ed9\u51fa\uff1a<\/p>\n $$ \u4e4b\u524d\u6211\u4eec\u63d0\u5230\uff0cnull space\u7684\u7ef4\u5ea6\u7b49\u4e8e\u77e9\u9635\u7684\u975e\u4e3b\u5143\u5217\u7684\u6570\u91cf\uff0c\u56e0\u6b64null space\u7684\u7ef4\u5ea6\u52a0\u4e0acolumn space\u7684\u7ef4\u5ea6\u7b49\u4e8e\u77e9\u9635\u7684\u5217\u6570\u3002\u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a$m-{\\rm by}-n$\u7684\u77e9\u9635\uff0c\u90a3\u4e48\uff1a<\/p>\n $$ \u7b80\u5355\u5730\u6765\u8bf4\uff0c\u5c06column space\u8f6c\u7f6e\u4e4b\u540e\u5c31\u5f97\u5230\u4e86row space\uff1a<\/p>\n $$ column space\u7684\u5927\u5c0f\u662f$m-{\\rm by}-1$\uff0c\u800crow space\u7684\u5927\u5c0f\u662f$n-{\\rm by}-1$\u3002<\/p>\n \u540c\u6837\u7684\uff0c\u5982\u679c$\\rm Ax = 0$\u8868\u793anull space\uff0c\u90a3\u4e48\u53ef\u4ee5\u7528$\\rm x^T A = 0$\u6765\u8868\u793aleft null space\uff1a<\/p>\n $$ null space\u7684\u5927\u5c0f\u662f$n-{\\rm by}-1$\uff0c\u800cleft null space\u7684\u5927\u5c0f\u662f$m-{\\rm by}-1$\u3002<\/p>\n \u63a5\u4e0b\u6765\u6211\u4eec\u6765\u8003\u5bdf\u4e00\u4e0b\u8fd9\u51e0\u4e2a\u5411\u91cf\u7a7a\u95f4\u4e4b\u95f4\u7684\u5173\u7cfb\u3002<\/p>\n Null space\u5305\u542b\u6240\u6709\u6ee1\u8db3$\\rm Ax=0$\u7684\u5411\u91cf$\\rm x$\uff0c\u6362\u8a00\u4e4bnull space\u662f\u6240\u6709\u4e0e$\\rm A$\u7684\u884c\u6b63\u4ea4\u7684\u5411\u91cf\u7684\u96c6\u5408\uff0c\u5373null space\u4e0erow space\u6b63\u4ea4\u3002<\/p>\n \u524d\u9762\u63d0\u5230column space\u901a\u8fc7$\\rm rref(A)$\u5bfb\u627e\u4e3b\u5143\u5217<\/strong>\u6765\u786e\u5b9a\u5176\u7ef4\u5ea6\u548c\u57fa\uff0crow space\u540c\u6837\u901a\u8fc7$\\rm rref(A)$\u7684\u5e26\u6709\u4e3b\u5143\u5217\u7684\u884c<\/strong>\u6765\u786e\u5b9a\u5176\u7ef4\u5ea6\u548c\u57fa\u3002\u800cnull space\u7684\u7ef4\u5ea6\u662f\u975e\u4e3b\u5143\u5217<\/strong>\u7684\u6570\u91cf\uff0c\u8fd9\u4e24\u4e2a\u5b50\u7a7a\u95f4\u7684\u5e76\u96c6(union)\u6784\u6210\u4e86\u6240\u6709$n-{\\rm by}-1$\u77e9\u9635\u7684\u5411\u91cf\u7a7a\u95f4\uff0c\u540c\u65f6\u6211\u4eec\u8bf4\u8fd9\u4e24\u4e2a\u5b50\u7a7a\u95f4\u6b63\u4ea4\u4e92\u8865(orthogonal complements)<\/strong>\u3002<\/p>\n \u8fdb\u4e00\u6b65\u7684\uff0c\u4e3b\u5143\u5217\u7684\u6570\u91cf<\/strong>\u4e0e\u5e26\u6709\u4e3b\u5143\u5217\u7684\u884c<\/strong>\u7684\u6570\u91cf\u662f\u76f8\u7b49\u7684\uff0c\u56e0\u6b64\uff1a<\/p>\n $$ \u6211\u4eec\u5c06\u8fd9\u4e2a\u7ef4\u5ea6\u53eb\u505a\u77e9\u9635\u7684\u79e9(Rank)<\/strong>\u3002\u8fd9\u662f\u4e00\u4e2a\u5f88\u4e0d\u5bfb\u5e38\u7684\u7ed3\u679c\uff0c\u56e0\u4e3acolumn space\u548crow space\u662f\u4e24\u4e2a\u4e0d\u540c\u7684\u5411\u91cf\u7a7a\u95f4\u3002\u603b\u4e4b\uff0c\u6211\u4eec\u6709\uff1a<\/p>\n $$ \u5f53\u7b49\u5f0f\u6210\u7acb\u7684\u65f6\u5019\uff0c\u6211\u4eec\u8bf4\u8fd9\u4e2a\u77e9\u9635\u6ee1\u79e9(full rank)<\/strong>\u3002\u5f53\u4e00\u4e2a\u65b9\u9635\u6ee1\u79e9\u7684\u65f6\u5019\uff0c\u56e0\u4e3a\u5176null space\u7684\u7ef4\u5ea6\u4e3a0(\u5373\uff0c\u4e0d\u5b58\u5728)\uff0c\u6240\u4ee5\u8be5\u65b9\u9635\u53ef\u9006\u3002<\/p>\n \u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a\u5173\u4e8e\u6240\u6709$n-{\\rm by}-1$\u77e9\u9635\u7684$n$\u7ef4\u5411\u91cf\u7a7a\u95f4$V$\uff0c\u4ee5\u53ca\u4e00\u4e2a$p$\u7ef4\u7684\u5b50\u7a7a\u95f4$W$\u3002\u4ee4$\\{ s_1, s_2, \\dots ,s_p\\}$\u4e3a$W$\u7684\u6b63\u4ea4\u57fa\uff0c\u6211\u4eec\u5c06\u5176\u6269\u5c55\uff0c\u4ee4$\\{ s_1, s_2, \\dots ,s_p, t_1, t_2, \\dots ,t_{n-p}\\}$\u4e3a$V$\u7684\u6b63\u4ea4\u57fa\u3002<\/p>\n \u5bf9\u4e8e\u4efb\u4f55$V$\u4e2d\u7684\u5411\u91cf$\\rm v$\u6765\u8bf4\uff0c\u5176\u53ef\u4ee5\u7528\u6b63\u4ea4\u57fa\u7684\u5f62\u5f0f\u6765\u8868\u793a\uff1a<\/p>\n $$ \u5176\u4e2d$a$\u548c$b$\u662f\u6807\u91cf\u7cfb\u6570\u3002\u5c06\u5411\u91cf$\\rm v$\u6295\u5f71\u5230\u5411\u91cf\u7a7a\u95f4$W$\u53ef\u4ee5\u5b9a\u4e49\u4e3a\uff1a<\/p>\n $$ \u5982\u679c\u6211\u4eec\u77e5\u9053\u5411\u91cf$\\rm v$\u548c\u5411\u91cf\u7a7a\u95f4$W$\u7684\u6b63\u4ea4\u57fa\uff0c\u6211\u4eec\u53ef\u4ee5\u7528$\\rm v^T s$\u6765\u8868\u793a\u6807\u91cf\u7cfb\u6570\uff1a<\/p>\n $$ \u53e6\u5916\uff0c\u6211\u4eec\u9700\u8981\u4ecb\u7ecd\u4e00\u4e2a\u7279\u6b8a\u7684\u6027\u8d28\uff1a\u5728\u5411\u91cf\u7a7a\u95f4$W$\u4e2d\uff0c${\\rm v_{proj}}_{W}$\u662f\u6700\u63a5\u8fd1$\\rm v$\u7684\u5411\u91cf\u3002\u7b80\u8981\u8bc1\u660e\u5982\u4e0b\u3002<\/p>\n \u4ee4$\\rm w$\u8868\u793a\u5411\u91cf\u7a7a\u95f4$W$\u4e2d\u4efb\u610f\u4e00\u4e2a\u5411\u91cf\uff1a<\/p>\n $$ $\\rm v$\u548c$\\rm w$\u4e4b\u95f4\u7684\u8ddd\u79bb\u53ef\u4ee5\u7531\u8303\u6570$\\left \\Vert {\\rm v - w} \\right \\Vert$\u7ed9\u51fa\uff0c\u4e8e\u662f\u6709\uff1a<\/p>\n $$ \u8fd9\u4e2a\u91cd\u8981\u7684\u6027\u8d28\u5c06\u5728\u4e0b\u4e00\u7ae0\u7684\u6700\u5c0f\u4e8c\u4e58\u6cd5\u4e2d\u8ba8\u8bba\u3002<\/p>\n \u5047\u8bbe\u6211\u4eec\u73b0\u5728\u6709\u4e00\u4e2a\u6570\u636e\u96c6\u9700\u8981\u62df\u5408\uff0c\u5b83\u6709\u4e00\u4e2a\u7ef4\u5ea6\u7684label\uff0c\u548c\u4e00\u4e2a\u7ef4\u5ea6\u7684output\uff0c\u5373\uff0c\u5b83\u662f\u4e00\u4e2a\u4e8c\u7ef4\u6570\u636e\u96c6\u3002\u6211\u4eec\u8bd5\u56fe\u7528\u4e00\u6761\u76f4\u7ebf\u6765\u62df\u5408\u8fd9\u4e2a\u6570\u636e\u96c6\uff0c\u6211\u4eec\u5c06label\u7528$x$\u6765\u8868\u793a\uff0c\u8fd9\u662f\u786e\u5207\u7684\u503c\uff1b\u6211\u4eec\u7528$y$\u6765\u8868\u793aoutput\uff0c\u5b83\u662f\u542b\u6709\u566a\u97f3\u7684\u503c(\u76f8\u5bf9\u4e8e\u6211\u4eec\u7684\u62df\u5408\u6765\u8bf4)\u3002\u90a3\u4e48\uff0c\u6bcf\u4e00\u4e2a\u6570\u636e\u90fd\u53ef\u4ee5\u8868\u793a\u4e3a$(x_k, y_k)$\uff0c\u62df\u5408\u7684\u76f4\u7ebf\u5219\u662f$y={\\beta}_0 + {\\beta}_1 x$\uff0c\u4e8e\u662f\u6709\uff1a<\/p>\n $$ \u6211\u4eec\u5c06\u5176\u6539\u5199\u4e3a$\\rm Ax=b$\u7684\u5f62\u5f0f\uff1a<\/p>\n $$ \u56e0\u4e3a\u8be5\u65b9\u7a0b\u7ec4\u662f\u4e00\u4e2aoverdetermined system\uff0c\u56e0\u6b64\u5b83\u6ca1\u6709\u89e3\u3002\u4f46\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u6700\u5c0f\u4e8c\u4e58\u6cd5\u6765\u5bfb\u627e\u5b83\u7684\u6700\u4f18\u7684\u89e3\u3002<\/p>\n \u6211\u4eec\u53ef\u4ee5\u5982\u4e0b\u63cf\u8ff0\u8fd9\u4e00\u8fc7\u7a0b\u3002\u5047\u8bbe\u6211\u4eec\u6709$\\rm Ax=b$\uff0c\u4f46\u56e0\u4e3a$\\rm b$\u4e0d\u5728$\\rm A$\u7684column space\u4e2d\uff0c\u56e0\u6b64\u65b9\u7a0b\u65e0\u89e3\u3002\u4f46\u662f\u6211\u4eec\u53ef\u4ee5\u5c06$\\rm b$\u6295\u5f71\u5230$\\rm {b_{proj}}_{Col(A)}$\u6765\u89e3\u51b3$\\rm Ax = {b_{proj}}_{Col(A)}$\u3002\u8be5\u65b9\u6cd5\u88ab\u79f0\u4e3a\u6700\u5c0f\u4e8c\u4e58\u6cd5(least-squares)<\/strong>\u3002\u4e0b\u9762\u6765\u770b\u4e00\u4e0b\u6700\u5c0f\u4e8c\u4e58\u6cd5\u7684\u4e00\u822c\u6d41\u7a0b\u3002<\/p>\n \u5047\u8bbe\u6211\u4eec\u6709$\\rm Ax=b$\uff0c\u6211\u4eec\u5c06$\\rm b$\u6295\u5f71\u5230$\\rm {b_{proj}}_{Col(A)}$\uff1a<\/p>\n $$ \u5176\u4e2d$\\rm {b_{proj}}_{Col(A)}$\u662f$\\rm b$\u5728$\\rm A$column space\u4e0a\u7684\u6295\u5f71\uff1b$\\rm (b - {b_{proj}}_{Col(A)})$\u662f\u4e0e$\\rm A$\u7684column space\u6b63\u4ea4\u7684\u9879\uff0c\u5b83\u540c\u65f6\u6b63\u4ea4\u4e8e$\\rm Row(A^T)$\uff0c\u4ea6\u4f4d\u4e8e$\\rm Null(A^T)$\u3002\u5c06overdetermined matrix equation\u4e24\u8fb9\u540c\u65f6\u4e58\u4ee5$\\rm A^T$\uff0c\u5f97\u5230\u4e00\u4e2a\u53ef\u89e3\u7684\u7b49\u5f0f\uff0c\u8fd9\u4e2a\u7b49\u5f0f\u88ab\u79f0\u4e3a$\\rm Ax=b$\u7684\u6b63\u89c4\u5316\u65b9\u7a0b(normal equations)<\/strong>\uff1a<\/p>\n $$ \u5f53$\\rm A$\u7684\u5217\u7ebf\u6027\u65e0\u5173\u7684\u65f6\u5019(\u5173\u4e8e\u7ebf\u6027\u65e0\u5173\uff0c\u6211\u4eec\u5c06\u5728\u540e\u9762\u8ba8\u8bba)\uff0c\u5b58\u5728\u552f\u4e00\u89e3\u3002\u5c06\u6b63\u89c4\u65b9\u7a0b\u7684\u4e24\u8fb9\u540c\u65f6\u4e58\u4ee5$\\rm A(A^T A)^{-1}$\uff0c\u6211\u4eec\u5f97\u5230\uff1a<\/p>\n $$ \u5176\u4e2d\uff0c\u6295\u5f71\u77e9\u9635$\\rm P = A(A^T A)^{-1}A^T$\u6ee1\u8db3$\\rm P^2 = P$\u3002\u5982\u679c$\\rm A$\u672c\u8eab\u5c31\u662f\u4e00\u4e2a\u53ef\u9006\u65b9\u6b63\uff0c\u90a3\u4e48$\\rm P = I$\uff0c\u5e76\u4e14$\\rm b$\u672c\u8eab\u5df2\u7ecf\u4f4d\u4e8e$A$\u7684column space\u4e2d\u3002<\/p>\n \u5173\u4e8e\u6295\u5f71\u77e9\u9635$\\rm P = A(A^T A)^{-1}A^T$\uff0c\u6211\u4eec\u989d\u5916\u8ba8\u8bba\u4e00\u4e0b\u5b83\u662f\u600e\u4e48\u5f97\u51fa\u7684\u3002<\/p>\n \u7ed3\u5408\u6211\u4eec\u4e4b\u524d\u8bf4\u7684\uff0c\u7531\u4e8e$\\rm b_{proj}$\u662f$\\rm b$\u5728$\\rm Col(A)$\u4e0a\u7684\u6295\u5f71\uff0c\u8bef\u5dee$\\rm b - b_{proj}$\u5e94\u8be5\u4e0e$\\rm Col(A)$\u6b63\u4ea4\u3002\u8fd9\u610f\u5473\u7740\u5bf9\u4e8e$\\rm Col(A)$\u4e2d\u7684\u4efb\u610f\u5217\u5411\u91cf${\\rm a}_i$\u90fd\u6709\uff1a<\/p>\n $$ \u56e0\u4e3a\u6211\u4eec\u6b64\u65f6\u4e0d\u5728\u6c42\u89e3$\\rm Ax = b$\uff0c\u800c\u662f\u6c42\u89e3$\\rm Ax = b_{proj}$\uff0c\u5c06\u5176\u4ee5\u77e9\u9635\u5f62\u5f0f\u8868\u8fbe\u5219\u662f\uff1a<\/p>\n $$ \u6211\u4eec\u5c06\u5176\u5316\u7b80\u5f97\u5230\uff1a<\/p>\n $$ \u7531\u4e8e$\\rm A^T A$\u662f\u53ef\u9006\u7684(\u56e0\u4e3a\u6211\u4eec\u5047\u8bbe\u4e86$\\rm A$\u7684\u5217\u7ebf\u6027\u65e0\u5173)\uff0c\u6211\u4eec\u6709\uff1a<\/p>\n $$ \u73b0\u5728\u6211\u4eec\u56de\u5230$\\rm b_{proj}$\uff1a<\/p>\n $$ \u4e8e\u662f\u6211\u4eec\u5f97\u5230\u4e86\u6295\u5f71\u77e9\u9635$\\rm P = A(A^T A)^{-1}A^T$\u3002<\/p>\n \u5047\u8bbe\u6211\u4eec\u73b0\u5728\u6709\u4e09\u4e2a\u70b9\uff0c\u6211\u4eec\u8bd5\u56fe\u8fdb\u884c\u62df\u5408\uff0c\u8fd9\u4e09\u4e2a\u70b9\u662f\uff1a$(1, 1)$\u3001$(2, 3)$\u548c$(3, 2)$\u3002\u6211\u4eec\u5c06\u62df\u5408\u7684\u6267\u884c\u8bbe\u4e3a$y = {\\beta}_0 + {\\beta}_1 x$\uff0c\u8fd9\u4e2a\u65b9\u7a0b\u7ec4\u53ef\u4ee5\u8868\u793a\u4e3a\uff1a<\/p>\n $$ \u6211\u4eec\u7528\u6700\u5c0f\u4e8c\u4e58\u6cd5\u5c06\u5176\u8868\u793a\u4e3a\uff1a<\/p>\n $$ \u5373\uff1a<\/p>\n $$ \u53ef\u4ee5\u89e3\u5f97\uff1a${\\beta}_{0} = 1$\u3001${\\beta}_{1} = \\frac{1}{2}$\u3002<\/p>\n \u6211\u4eec\u8fd8\u9700\u8981\u63d0\u4e00\u4e0b\u4e3a\u4ec0\u4e48\u9700\u8981\u5047\u8bbe$\\rm A$\u7684\u5217\u5411\u91cf\u662f\u7ebf\u6027\u65e0\u5173\u7684\u3002<\/p>\n \u56e0\u6b64\uff0c\u5982\u679c\u662f\u7ebf\u6027\u76f8\u5173\u7684\u60c5\u51b5\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u5e7f\u4e49\u9006\u548c\u6700\u5c0f\u8303\u6570\u89e3\u6765\u6c42\u6700\u5c0f\u4e8c\u4e58\u6cd5\u3002\u5176\u4e2d\uff0c\u5e7f\u4e49\u9006$A^{+}$\u6ee1\u8db3\u5982\u4e0b\u6761\u4ef6\uff1a<\/p>\n $$ \u5229\u7528\u5e7f\u4e49\u9006\uff0c\u6211\u4eec\u53ef\u4ee5\u627e\u5230\u4e00\u4e2a\u6700\u5c0f\u4e8c\u4e58\u89e3\uff1a<\/p>\n $$ \u5177\u4f53\u6765\u8bf4\uff0c\u6211\u4eec\u53ef\u4ee5\u7528\u5947\u5f02\u503c\u5206\u89e3\u6765\u6c42\u5e7f\u4e49\u9006\uff0c\u5047\u8bbe$\\rm A$\u7684\u5947\u5f02\u503c\u5206\u89e3\u4e3a\uff1a<\/p>\n $$ \u90a3\u4e48:<\/p>\n $$ \u4e00\u3001\u5411\u91cf\u7a7a\u95f4 1.1 \u7b80\u4ecb \u5411\u91cf\u7a7a\u95f4(vector space)\u5305\u542b\u4e86\u5411\u91cf\u548c\u6807\u91cf\u7684\u96c6\u5408\u3002\u5728\u672c\u7bc7\u6587\u7ae0\u4e2d\uff0c\u6240\u8003 …<\/p>","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[293],"tags":[],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/swordofmorning.com\/index.php\/wp-json\/wp\/v2\/posts\/1370"}],"collection":[{"href":"https:\/\/swordofmorning.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/swordofmorning.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/swordofmorning.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/swordofmorning.com\/index.php\/wp-json\/wp\/v2\/comments?post=1370"}],"version-history":[{"count":4,"href":"https:\/\/swordofmorning.com\/index.php\/wp-json\/wp\/v2\/posts\/1370\/revisions"}],"predecessor-version":[{"id":1374,"href":"https:\/\/swordofmorning.com\/index.php\/wp-json\/wp\/v2\/posts\/1370\/revisions\/1374"}],"wp:attachment":[{"href":"https:\/\/swordofmorning.com\/index.php\/wp-json\/wp\/v2\/media?parent=1370"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/swordofmorning.com\/index.php\/wp-json\/wp\/v2\/categories?post=1370"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/swordofmorning.com\/index.php\/wp-json\/wp\/v2\/tags?post=1370"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n{\\rm u} =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{u_1} \\newline
\n{u_2} \\newline
\n{u_3}
\n\\end{array}
\n\\right),
\n{\\rm v} =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{v_1} \\newline
\n{v_2} \\newline
\n{v_3}
\n\\end{array}
\n\\right)
\n$$<\/p>\n
\n{\\rm w} = a{\\rm u} + b{\\rm v} =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{au_1 + bv_1} \\newline
\n{au_2 + bv_2} \\newline
\n{au_3 + bv_3}
\n\\end{array}
\n\\right)
\n$$<\/p>\n\n
1.2 \u5982\u4f55\u5224\u65ad\u5411\u91cf\u7a7a\u95f4<\/h3>\n
\n
\n
1.3 \u4f8b\u5b50<\/h3>\n
\n{\\rm A} =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{a_1} \\newline
\n{0} \\newline
\n{a_3}
\n\\end{array}
\n\\right),
\n{\\rm B} =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{b_1} \\newline
\n{0} \\newline
\n{b_3}
\n\\end{array}
\n\\right)
\n$$<\/p>\n
\n{\\rm A + B} = \\left(
\n\\begin{array}{*{20}{c}}
\n{a_1 + b_1} \\newline
\n{0} \\newline
\n{a_3 + b_3}
\n\\end{array}
\n\\right)
\n$$<\/p>\n
\nk {\\rm A} = \\left(
\n\\begin{array}{*{20}{c}}
\n{k a_1} \\newline
\n{0} \\newline
\n{k a_3}
\n\\end{array}
\n\\right)
\n$$<\/p>\n
\n{\\rm A} =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{a_1} \\newline
\n{a_2} \\newline
\n{a_3}
\n\\end{array}
\n\\right),
\n{\\rm B} =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{b_1} \\newline
\n{b_2} \\newline
\n{b_3}
\n\\end{array}
\n\\right)
\n$$<\/p>\n
\na_1 + a_2 + a_3 = 1, b_1 + b_2 + b_3 = 1
\n$$<\/p>\n
\n{\\rm A + B} =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{a_1 + b_1} \\newline
\n{a_2 + b_2} \\newline
\n{a_3 + b_3}
\n\\end{array}
\n\\right)
\n$$<\/p>\n\u4e8c\u3001\u7ebf\u6027\u65e0\u5173<\/h2>\n
\n{c}_{1}{u}_{1} + {c}_{2}{u}_{2} + \\dots + {c}_{n}{u}_{n} = 0
\n$$<\/p>\n
\n{\\rm u} =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{1} \\newline
\n{0} \\newline
\n{0}
\n\\end{array}
\n\\right),
\n{\\rm v} =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{0} \\newline
\n{1} \\newline
\n{0}
\n\\end{array}
\n\\right),
\n{\\rm w} =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{2} \\newline
\n{3} \\newline
\n{0}
\n\\end{array}
\n\\right)
\n$$<\/p>\n
\n{\\rm u} =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{1} \\newline
\n{0} \\newline
\n{0}
\n\\end{array}
\n\\right),
\n{\\rm v} =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{0} \\newline
\n{1} \\newline
\n{0}
\n\\end{array}
\n\\right),
\n{\\rm w} =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{0} \\newline
\n{0} \\newline
\n{1}
\n\\end{array}
\n\\right)
\n$$<\/p>\n
\na{\\rm u} + b{\\rm v} + c{\\rm w} =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{a} \\newline
\n{b} \\newline
\n{c}
\n\\end{array}
\n\\right)=
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{0} \\newline
\n{0} \\newline
\n{0}
\n\\end{array}
\n\\right)
\n$$<\/p>\n\u4e09\u3001Span, Basis, Dimension<\/h2>\n
\n\\left\\{
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{1} \\newline
\n{0} \\newline
\n{0}
\n\\end{array}
\n\\right),
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{0} \\newline
\n{1} \\newline
\n{0}
\n\\end{array}
\n\\right),
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{2} \\newline
\n{3} \\newline
\n{0}
\n\\end{array}
\n\\right)
\n\\right\\}
\n$$<\/p>\n
\n\\left\\{
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{1} \\newline
\n{0} \\newline
\n{0}
\n\\end{array}
\n\\right),
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{0} \\newline
\n{1} \\newline
\n{0}
\n\\end{array}
\n\\right)
\n\\right\\}
\n,
\n\\left\\{
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{1} \\newline
\n{0} \\newline
\n{0}
\n\\end{array}
\n\\right),
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{2} \\newline
\n{3} \\newline
\n{0}
\n\\end{array}
\n\\right)
\n\\right\\}
\n,
\n\\left\\{
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{0} \\newline
\n{1} \\newline
\n{0}
\n\\end{array}
\n\\right),
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{2} \\newline
\n{3} \\newline
\n{0}
\n\\end{array}
\n\\right)
\n\\right\\}
\n$$<\/p>\n\n
\u56db\u3001\u683c\u62c9\u59c6-\u65bd\u5bc6\u7279\u6b63\u4ea4\u5316<\/h2>\n
4.1 \u5411\u91cf\u6295\u5f71<\/h3>\n
\n\\rm u' = {\\left| u' \\right|} \\times \\frac{v}{\\left| v \\right|}
\n$$<\/p>\n
\n{\\rm \\left| u' \\right|} = {\\rm \\left| u \\right|} \\times {\\rm cos}{\\theta}
\n$$<\/p>\n
\n\\begin{eqnarray}
\n{\\rm v_1}{\\rm v_2} &=& {\\rm \\left| v_1 \\right|} \\times {\\rm \\left| v_2 \\right|} \\times {\\rm cos}{\\theta} \\newline
\n{\\rm cos}{\\theta} &=& \\frac{\\rm v_1 v_2}{{\\rm \\left| v_1 \\right|}{\\rm \\left| v_2 \\right|}}
\n\\end{eqnarray}
\n$$<\/p>\n
\n\\begin{eqnarray}
\n{\\rm \\left| u' \\right|} &=& {\\rm \\left| u \\right|} \\times \\frac{\\rm u v}{{\\rm \\left| u \\right|}{\\rm \\left| v \\right|}} = \\frac{\\rm u v}{{\\rm \\left| v \\right|}} \\newline
\n{\\rm u'} &=& \\frac{\\rm u v^2}{{\\rm \\left| v \\right|}^2}
\n\\end{eqnarray}
\n$$<\/p>\n4.2 \u683c\u62c9\u59c6-\u65bd\u5bc6\u7279\u6b63\u4ea4\u5316<\/h3>\n
\n\\rm \\left\\{ v_1, v_2, \\dots , v_n \\right\\}
\n$$<\/p>\n
\n\\rm \\left\\{ u_1, u_2, \\dots , u_n \\right\\}
\n$$<\/p>\n\n
\n\\rm u_2 = v_2 - \\frac{(u_1^T v_2)u_1}{(u_1^T u_1)}
\n$$<\/p>\n
\n\\begin{eqnarray}
\n\\rm u_2 &=& \\rm v_2 - \\frac{(u_1^T v_2)u_1}{u_1^T u_1} \\newline
\n\\rm u_1^T u_2 &=& \\rm u_1^T v_2 - u_1^T \\frac{(u_1^T v_2)u_1}{(u_1^T u_1)} \\newline
\n\\rm u_1^T u_2 &=& \\rm v_1^T v_2 - v_1^T \\frac{(v_1^T v_2)v_1}{(v_1^T v_1)} \\newline
\n\\rm u_1^T u_2 &=& \\rm v_1^T v_2 - v_1^T v_2 = 0
\n\\end{eqnarray}
\n$$<\/p>\n
\n\\rm u_3 = v_3 - \\frac{(u_1^T v_3)u_1}{(u_1^T u_1)} - \\frac{(u_2^T v_3)u_2}{(u_2^T u_2)}
\n$$<\/p>\n
\n\\hat{\\rm u}_1 = \\frac{\\rm u_1}{{(\\rm u_1^T u_1)}^{\\frac{1}{2}}}
\n$$<\/p>\n
\n\\rm span\\{ u_1, u_2, \\dots, u_k \\} = \\rm span\\{ v_1, v_2, \\dots, v_k \\}
\n$$<\/p>\n\u4e94\u3001\u77e9\u9635\u57fa\u672c\u5b50\u7a7a\u95f4<\/h2>\n
5.1 Null Space<\/h3>\n
\n\\rm Ax = 0
\n$$<\/p>\n
\n{\\rm A} =
\n\\left(
\n\\begin{array}{*{20}{r}}
\n{-3} & {6} & {-1} & {1} & {-7} \\newline
\n{1} & {-2} & {2} & {3} & {-1} \\newline
\n{2} & {-4} & {5} & {8} & {-4} \\newline
\n\\end{array}
\n\\right)
\n$$<\/p>\n
\n{\\rm A} =
\n\\left(
\n\\begin{array}{*{20}{r}}
\n{1} & {-2} & {0} & {-1} & {3} \\newline
\n{0} & {0} & {1} & {2} & {-2} \\newline
\n{0} & {0} & {0} & {0} & {0} \\newline
\n\\end{array}
\n\\right)
\n$$<\/p>\n
\n\\begin{eqnarray}
\nx_1 &=& 2x_2 + x_4 - 3_x5 \\newline
\nx_3 &=& -2x_4 + 2x_5
\n\\end{eqnarray}
\n$$<\/p>\n
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{2x_2 + x_4 - 3x_5} \\newline
\n{x_2} \\newline
\n{-2x_4 + 2x_5} \\newline
\n{x_4} \\newline
\n{x_5}
\n\\end{array}
\n\\right)=
\nx_2
\n\\left(
\n\\begin{array}{*{20}{r}}
\n{2} \\newline
\n{1} \\newline
\n{0} \\newline
\n{0} \\newline
\n{0}
\n\\end{array}
\n\\right)+
\nx_4
\n\\left(
\n\\begin{array}{*{20}{r}}
\n{1} \\newline
\n{0} \\newline
\n{-2} \\newline
\n{1} \\newline
\n{0}
\n\\end{array}
\n\\right)+
\nx_5
\n\\left(
\n\\begin{array}{*{20}{r}}
\n{-3} \\newline
\n{0} \\newline
\n{2} \\newline
\n{0} \\newline
\n{1}
\n\\end{array}
\n\\right)
\n$$<\/p>\n
\n\\left\\{
\n\\left(
\n\\begin{array}{*{20}{r}}
\n{2} \\newline
\n{1} \\newline
\n{0} \\newline
\n{0} \\newline
\n{0}
\n\\end{array}
\n\\right),
\n\\left(
\n\\begin{array}{*{20}{r}}
\n{1} \\newline
\n{0} \\newline
\n{-2} \\newline
\n{1} \\newline
\n{0}
\n\\end{array}
\n\\right),
\n\\left(
\n\\begin{array}{*{20}{r}}
\n{-3} \\newline
\n{0} \\newline
\n{2} \\newline
\n{0} \\newline
\n{1}
\n\\end{array}
\n\\right)
\n\\right\\}
\n$$<\/p>\n5.2 Null Space\u7684\u5e94\u7528<\/h3>\n
\n\\begin{eqnarray}
\n2x_1 + 2x_2 + x_3 &=& 0 \\newline
\n2x_1 - 2x_2 - x_3 &=& 1
\n\\end{eqnarray}
\n$$<\/p>\n
\n\\left(
\n\\begin{array}{*{20}{r}}
\n{2} & {2} & {1} \\newline
\n{2} & {-2} & {-1}
\n\\end{array}
\n\\right)
\n\\left(
\n\\begin{array}{*{20}{r}}
\n{x_1} \\newline
\n{x_2} \\newline
\n{x_3}
\n\\end{array}
\n\\right)=
\n\\left(
\n\\begin{array}{*{20}{r}}
\n{0} \\newline
\n{1}
\n\\end{array}
\n\\right)
\n$$<\/p>\n
\n\\left(
\n\\begin{array}{*{20}{r}}
\n{2} & {2} & {1} & {0} \\newline
\n{2} & {-2} & {-1} & {1}
\n\\end{array}
\n\\right) \\to
\n\\left(
\n\\begin{array}{*{20}{r}}
\n{1} & {0} & {0} & {1\/4} \\newline
\n{0} & {1} & {1\/2} & {-1\/4}
\n\\end{array}
\n\\right)
\n$$<\/p>\n
\n\\rm Null(A) = span
\n\\left\\{
\n\\left(
\n\\begin{array}{*{20}{r}}
\n{0} \\newline
\n{-1} \\newline
\n{2}
\n\\end{array}
\n\\right)
\n\\right\\}
\n$$<\/p>\n
\n\\frac{1}{4}
\n\\left(
\n\\begin{array}{*{20}{r}}
\n{1} \\newline
\n{-1} \\newline
\n{0}
\n\\end{array}
\n\\right)
\n$$<\/p>\n
\n\\left(
\n\\begin{array}{*{20}{r}}
\n{x_1} \\newline
\n{x_2} \\newline
\n{x_3}
\n\\end{array}
\n\\right) = a
\n\\left(
\n\\begin{array}{*{20}{r}}
\n{0} \\newline
\n{-1} \\newline
\n{2}
\n\\end{array}
\n\\right)+
\n\\frac{1}{4}
\n\\left(
\n\\begin{array}{*{20}{r}}
\n{1} \\newline
\n{-1} \\newline
\n{0}
\n\\end{array}
\n\\right)
\n$$<\/p>\n5.3 Column Space<\/h3>\n
\n\\left(
\n\\begin{array}{*{20}{r}}
\n{a} & {b} \\newline
\n{c} & {d}
\n\\end{array}
\n\\right)
\n\\left(
\n\\begin{array}{*{20}{r}}
\n{x} \\newline
\n{y}
\n\\end{array}
\n\\right) =
\n\\left(
\n\\begin{array}{*{20}{r}}
\n{ax + by} \\newline
\n{cx + dy}
\n\\end{array}
\n\\right)= x
\n\\left(
\n\\begin{array}{*{20}{r}}
\n{a} \\newline
\n{c}
\n\\end{array}
\n\\right) + y
\n\\left(
\n\\begin{array}{*{20}{r}}
\n{b} \\newline
\n{d}
\n\\end{array}
\n\\right)
\n$$<\/p>\n
\n{\\rm A} =
\n\\left(
\n\\begin{array}{*{20}{r}}
\n{-3} & {6} & {-1} & {1} & {-7} \\newline
\n{1} & {-2} & {2} & {3} & {-1} \\newline
\n{2} & {-4} & {5} & {8} & {-4} \\newline
\n\\end{array}
\n\\right),
\n{\\rm rref(A)} =
\n\\left(
\n\\begin{array}{*{20}{r}}
\n{1} & {-2} & {0} & {-1} & {3} \\newline
\n{0} & {0} & {1} & {2} & {-2} \\newline
\n{0} & {0} & {0} & {0} & {0} \\newline
\n\\end{array}
\n\\right)
\n$$<\/p>\n
\n\\left\\{
\n\\left(
\n\\begin{array}{*{20}{r}}
\n{-3} \\newline
\n{1} \\newline
\n{2}
\n\\end{array}
\n\\right) ,
\n\\left(
\n\\begin{array}{*{20}{r}}
\n{-1} \\newline
\n{2} \\newline
\n{5}
\n\\end{array}
\n\\right),
\n\\right\\}
\n$$<\/p>\n
\n{\\rm dim(Col(A)) + dim(Null(A))} = n
\n$$<\/p>\n5.4 Row Space, Left Null Space, Rank<\/h3>\n
\n\\rm Row(A) = Col(A^T)
\n$$<\/p>\n
\n\\rm LeftNull(A) = Null(A^T)
\n$$<\/p>\n
\n\\rm dim(Col(A)) = dim(Row(A))
\n$$<\/p>\n
\n{\\rm rank(A)} \\le {\\rm min}(m, n)
\n$$<\/p>\n\u516d\u3001\u6b63\u4ea4\u6295\u5f71<\/h2>\n
\n{\\rm v} = a_1 {\\rm s}_1 + a_2 {\\rm s}_2 + \\dots + a_p {\\rm s}_p + b_1 {\\rm t}_1 + b_2 {\\rm t}_2 + \\dots + b_{n-p} {\\rm t}_{n-p}
\n$$<\/p>\n
\n{\\rm v_{proj}}_{W} = a_1 {\\rm s_1} + a_2 {\\rm s_2} + \\dots + a_p {\\rm s}_p
\n$$<\/p>\n
\n{\\rm v_{proj}}_{W} = ({\\rm v^T}{\\rm s}_1){\\rm s}_1 + ({\\rm v^T}{\\rm s}_2){\\rm s}_2 + \\dots + ({\\rm v^T}{\\rm s}_p){\\rm s}_p
\n$$<\/p>\n
\n{\\rm w} = c_1 {\\rm s_1} + c_2 {\\rm s_2} + \\dots + c_p {\\rm s}_p
\n$$<\/p>\n
\n\\begin{eqnarray}
\n{\\left \\Vert {\\rm v - w} \\right \\Vert}^2 &=& (a_1 - c_1)^2 + \\dots + (a_p - c_p)^2 + b_1^2 + \\dots + b_{n-p}^2 \\newline
\n&\\ge& b_1^2 + \\dots + b_{n-p}^2 \\newline
\n&=& {\\left \\Vert {\\rm v - {\\rm v_{proj}}_{W}} \\right \\Vert}^2
\n\\end{eqnarray}
\n$$<\/p>\n\u4e03\u3001\u6700\u5c0f\u4e8c\u4e58\u6cd5<\/h2>\n
7.1 \u57fa\u7840\u6982\u5ff5<\/h3>\n
\n\\begin{eqnarray}
\ny_1 &=& {\\beta}_0 + {\\beta}_1 x_1 \\newline
\ny_2 &=& {\\beta}_0 + {\\beta}_1 x_2 \\newline
\n&\\vdots& \\newline
\ny_n &=& {\\beta}_0 + {\\beta}_1 x_n
\n\\end{eqnarray}
\n$$<\/p>\n
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{1} & {x_1} \\newline
\n{1} & {x_2} \\newline
\n{\\vdots} & {\\vdots} \\newline
\n{1} & {x_n}
\n\\end{array}
\n\\right)
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{{\\beta}_{0}} \\newline
\n{{\\beta}_{1}} \\newline
\n\\end{array}
\n\\right) =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{y_1} \\newline
\n{y_2} \\newline
\n{\\vdots} \\newline
\n{y_n}
\n\\end{array}
\n\\right)
\n$$<\/p>\n
\n\\rm b = {b_{proj}}_{Col(A)} + (b - {b_{proj}}_{Col(A)})
\n$$<\/p>\n
\n\\rm A^T A x = A^T b
\n$$<\/p>\n
\n\\rm Ax = A(A^T A)^{-1}A^T b = {b_{proj}}_{Col(A)}
\n$$<\/p>\n7.2 \u6295\u5f71\u77e9\u9635P<\/h3>\n
\n{\\rm a}_{i}^{\\rm T}(\\rm b - b_{proj}) = 0
\n$$<\/p>\n
\n\\rm A^T (b - Ax) = 0
\n$$<\/p>\n
\n\\rm A^T b = A^T Ax
\n$$<\/p>\n
\n\\rm x = (A^T A)^{-1} A^T b
\n$$<\/p>\n
\n\\rm b_{proj} = Ax = A(A^T A)^{-1} A^T b
\n$$<\/p>\n7.3 \u6570\u503c\u5316\u7684\u4f8b\u5b50<\/h3>\n
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{1} & {1} \\newline
\n{1} & {2} \\newline
\n{1} & {3}
\n\\end{array}
\n\\right)
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{{\\beta}_{0}} \\newline
\n{{\\beta}_{1}} \\newline
\n\\end{array}
\n\\right) =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{1} \\newline
\n{3} \\newline
\n{2}
\n\\end{array}
\n\\right)
\n$$<\/p>\n
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{1} & {1} & {1} \\newline
\n{1} & {2} & {3}
\n\\end{array}
\n\\right)
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{1} & {1} \\newline
\n{1} & {2} \\newline
\n{1} & {3}
\n\\end{array}
\n\\right)
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{{\\beta}_{0}} \\newline
\n{{\\beta}_{1}} \\newline
\n\\end{array}
\n\\right) =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{1} & {1} & {1} \\newline
\n{1} & {2} & {3}
\n\\end{array}
\n\\right)
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{1} \\newline
\n{3} \\newline
\n{2}
\n\\end{array}
\n\\right)
\n$$<\/p>\n
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{3} & {6} \\newline
\n{6} & {14}
\n\\end{array}
\n\\right)
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{{\\beta}_{0}} \\newline
\n{{\\beta}_{1}} \\newline
\n\\end{array}
\n\\right) =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{6} \\newline
\n{13}
\n\\end{array}
\n\\right)
\n$$<\/p>\n7.4 A\u7684\u7ebf\u6027\u65e0\u5173<\/h3>\n
\n
\n\\rm {A}^{+} = ({A}^{T}{A})^{+} {A}^{T}
\n$$<\/p>\n
\n\\rm x_{min-norm} = {A}^{+}b
\n$$<\/p>\n
\n\\rm A = U \\Sigma V^T
\n$$<\/p>\n
\n\\rm A^{+} = V {\\Sigma}^{+} U^T
\n$$<\/p>\n","protected":false},"excerpt":{"rendered":"