{"id":8962,"date":"2024-05-19T20:01:01","date_gmt":"2024-05-19T12:01:01","guid":{"rendered":""},"modified":"2024-05-19T20:01:01","modified_gmt":"2024-05-19T12:01:01","slug":"\u7ebf\u6027\u65b9\u7a0b\u7ec4\uff08\u4e94\uff09- \u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u89e3\u96c6","status":"publish","type":"post","link":"https:\/\/mushiming.com\/8962.html","title":{"rendered":"\u7ebf\u6027\u65b9\u7a0b\u7ec4\uff08\u4e94\uff09- \u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u89e3\u96c6"},"content":{"rendered":"

\n <\/path> \n<\/svg> <\/p>\n

\u5c0f\u7ed3<\/h3>\n
    \n
  1. \u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u5b9a\u4e49\u3002<\/li>\n
  2. \u89e3\u96c6\u7684\u53c2\u6570\u5411\u91cf\u5f62\u5f0f\u3002<\/li>\n
  3. \u975e\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u89e3\u3002<\/li>\n<\/ol>\n
    \n

    \u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4<\/h3>\n

    \u7ebf\u6027\u65b9\u7a0b\u7ec4\u79f0\u4e3a\u9f50\u6b21\u7684<\/strong>\uff0c\u82e5\u5b83\u53ef\u5199\u6210 A x = 0 \\boldsymbol{Ax}=\\boldsymbol{0} <\/span><\/span>A<\/span>x<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u5f62\u5f0f\uff0c\u5176\u4e2d A \\boldsymbol{A} <\/span><\/span>A<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u662f m \u00d7 n m{\\times}n <\/span><\/span>m<\/span>\u00d7<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span>\u77e9\u9635\u800c 0 \\boldsymbol{0} <\/span><\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u662f R m \\mathbb{R}^{m} <\/span><\/span>R<\/span><\/span><\/span>m<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4e2d\u7684\u96f6\u5411\u91cf\u3002\u8fd9\u6837\u7684\u65b9\u7a0b\u7ec4\u81f3\u5c11\u6709\u4e00\u4e2a\u89e3\uff0c\u5373 x = 0 \\boldsymbol{x}=\\boldsymbol{0} <\/span><\/span>x<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff08 R n \\mathbb{R}^{n} <\/span><\/span>R<\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4e2d\u7684\u96f6\u5411\u91cf\uff09\uff0c\u8fd9\u4e2a\u89e3\u79f0\u4e3a\u5b83\u7684\u5e73\u51e1\u89e3<\/strong>\u3002\u5bf9\u7ed9\u5b9a\u65b9\u7a0b A x = 0 \\boldsymbol{Ax}=\\boldsymbol{0} <\/span><\/span>A<\/span>x<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff0c\u91cd\u8981\u7684\u662f\u5b83\u662f\u5426\u6709\u975e\u5e73\u51e1\u89e3<\/strong>\uff0c\u5373\u6ee1\u8db3 A x = 0 \\boldsymbol{Ax}=\\boldsymbol{0} <\/span><\/span>A<\/span>x<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u975e\u96f6\u5411\u91cf x \\boldsymbol{x} <\/span><\/span>x<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

    \u9f50\u6b21\u65b9\u7a0b A x = 0 \\boldsymbol{Ax}=\\boldsymbol{0} <\/span><\/span>A<\/span>x<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u6709\u975e\u5e73\u51e1\u89e3\u5f53\u4e14\u4ec5\u5f53\u65b9\u7a0b\u81f3\u5c11\u6709\u4e00\u4e2a\u81ea\u7531\u53d8\u91cf\u3002<\/p>\n

    \u786e\u5b9a\u9f50\u6b21\u65b9\u7a0b\u7ec4 { 3 x 1 + 5 x 2 \u2212 4 x 3 = 0 \u2212 3 x 1 \u2212 2 x 2 + 4 x 3 = 0 6 x 1 + x 2 \u2212 8 x 3 = 0 \\begin{cases} 3x_1 + 5x_2 - 4x_3 = 0 \\\\ -3x_1 - 2x_2 + 4x_3 = 0 \\\\ 6x_1 + x_2 - 8x_3 = 0 \\\\ \\end{cases} <\/span><\/span><\/span>\u23a9<\/span><\/span><\/span><\/span>\u23aa<\/span><\/span><\/span><\/span>\u23a8<\/span><\/span><\/span><\/span>\u23aa<\/span><\/span><\/span><\/span>\u23a7<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>5<\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>4<\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>0<\/span><\/span><\/span><\/span>\u2212<\/span>3<\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>2<\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>4<\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>0<\/span><\/span><\/span><\/span>6<\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>8<\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u662f\u5426\u6709\u5e73\u51e1\u89e3\uff0c\u5e76\u63cf\u8ff0\u5b83\u7684\u89e3\u96c6\u3002
    \u89e3\uff1a\u4ee4 A \\boldsymbol{A} <\/span><\/span>A<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4e3a\u8be5\u65b9\u7a0b\u7ec4\u7684\u7cfb\u6570\u77e9\u9635\uff0c\u7528\u884c\u5316\u7b80\u7b97\u6cd5\u628a\u589e\u5e7f\u77e9\u9635 [ A 0 ] \\begin{bmatrix}\\boldsymbol{A} &amp; \\boldsymbol{0} \\end{bmatrix} <\/span><\/span>[<\/span><\/span><\/span>A<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5316\u4e3a\u9636\u68af\u5f62
    [ 3 5 \u2212 4 0 \u2212 3 \u2212 2 4 0 6 1 \u2212 8 0 ] \\begin{bmatrix}3 &amp; 5 &amp; -4 &amp; 0 \\\\ -3 &amp; -2 &amp; 4 &amp; 0 \\\\ 6 &amp; 1 &amp; -8 &amp; 0\\end{bmatrix} <\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span>\u2212<\/span>3<\/span><\/span><\/span><\/span>6<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>5<\/span><\/span><\/span><\/span>\u2212<\/span>2<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>4<\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span>\u2212<\/span>8<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff5e [ 3 5 \u2212 4 0 0 3 0 0 0 9 0 0 ] \\begin{bmatrix} 3 &amp; 5 &amp; -4 &amp; 0 \\\\ 0 &amp; 3 &amp; 0 &amp; 0 \\\\ 0 &amp; 9 &amp; 0 &amp; 0\\end{bmatrix} <\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>5<\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span>9<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>4<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff5e [ 3 5 \u2212 4 0 0 3 0 0 0 0 0 0 ] \\begin{bmatrix} 3 &amp; 5 &amp; -4 &amp; 0 \\\\ 0 &amp; 3 &amp; 0 &amp; 0 \\\\ 0 &amp; 0 &amp; 0 &amp; 0\\end{bmatrix} <\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>5<\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>4<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
    \u56e0\u4e3a x 3 x_3 <\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u662f\u81ea\u7531\u53d8\u91cf\uff0c\u6545 A x = 0 \\boldsymbol{Ax}=\\boldsymbol{0} <\/span><\/span>A<\/span>x<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u6709\u5e73\u51e1\u89e3\uff08\u5bf9 x 3 x_3 <\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u6bcf\u4e00\u4e2a\u9009\u62e9\u90fd\u6709\u4e00\u4e2a\u89e3\uff09\u3002\u4e3a\u63cf\u8ff0\u89e3\u96c6\uff0c\u7ee7\u7eed\u628a [ A 0 ] \\begin{bmatrix}\\boldsymbol{A} &amp; \\boldsymbol{0} \\end{bmatrix} <\/span><\/span>[<\/span><\/span><\/span>A<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5316\u4e3a\u7b80\u5316\u9636\u68af\u5f62\uff1a
    [ 1 0 \u2212 4 3 0 0 1 0 0 0 0 0 0 ] { x 1 = f r a c 43 x 3 x 2 = 0 x 3 \u4e3a \u81ea \u7531 \u53d8 \u91cf \\begin{bmatrix} 1 &amp; 0 &amp; -\\frac{4}{3} &amp; 0 \\\\ 0 &amp; 1 &amp; 0 &amp; 0 \\\\ 0 &amp; 0 &amp; 0 &amp; 0 \\end{bmatrix} \\qquad \\begin{cases} x_1 = frac{4}{3}x_3 \\\\ x_2 = 0 \\\\ x_3\u4e3a\u81ea\u7531\u53d8\u91cf \\\\ \\end{cases} <\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a9<\/span><\/span><\/span><\/span>\u23aa<\/span><\/span><\/span><\/span>\u23a8<\/span><\/span><\/span><\/span>\u23aa<\/span><\/span><\/span><\/span>\u23a7<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>f<\/span>r<\/span>a<\/span>c<\/span>4<\/span><\/span>3<\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>0<\/span><\/span><\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4e3a<\/span>\u81ea<\/span>\u7531<\/span>\u53d8<\/span>\u91cf<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
    A x = 0 \\boldsymbol{Ax}=\\boldsymbol{0} <\/span><\/span>A<\/span>x<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u901a\u89e3\u6709\u5411\u91cf\u5f62\u5f0f
    x = [ x 1 x 2 x 3 ] = [ 4 3 x 3 0 x 3 ] = x 3 [ 4 3 0 1 ] = x 3 v \uff0c \u5176 \u4e2d v = [ 4 3 0 1 ] \\boldsymbol{x}=\\begin{bmatrix}x_1 \\\\ x_2 \\\\ x_3\\end{bmatrix}=\\begin{bmatrix}\\frac{4}{3}x_3 \\\\ 0 \\\\ x_3\\end{bmatrix}=x_3\\begin{bmatrix}\\frac{4}{3} \\\\ 0 \\\\ 1\\end{bmatrix}=x_3\\boldsymbol{v}\uff0c\u5176\u4e2d\\boldsymbol{v}=\\begin{bmatrix}\\frac{4}{3} \\\\ 0 \\\\ 1\\end{bmatrix} <\/span><\/span>x<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>v<\/span><\/span><\/span>\uff0c<\/span>\u5176<\/span>\u4e2d<\/span>v<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
    \u6ce8\u610f\uff0c\u975e\u5e73\u51e1\u89e3\u5411\u91cf x \\boldsymbol{x} <\/span><\/span>x<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u53ef\u80fd\u6709\u4e9b\u96f6\u5143\u7d20\uff0c\u53ea\u8981\u4e0d\u662f\u6240\u6709\u5143\u7d20\u90fd\u662f0\u5373\u53ef\u3002<\/p>\n

    \u63cf\u8ff0\u9f50\u6b21\u65b9\u7a0b\u7ec4 { 10 x 1 \u2212 3 x 2 \u2212 2 x 3 = 0 \\begin{cases} 10x_1 - 3x_2 - 2x_3 = 0 \\\\ \\end{cases} <\/span><\/span>{
    \n <\/span><\/span><\/span>1<\/span>0<\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>3<\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>2<\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u89e3\u96c6\u3002
    \u89e3\uff1a\u8fd9\u91cc\u65e0\u987b\u77e9\u9635\u8bb0\u53f7\u3002\u7528\u81ea\u7531\u53d8\u91cf x 2 x_2 <\/span><\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u548c x 3 x_3 <\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u8868\u793a\u57fa\u672c\u53d8\u91cf x 1 x_1 <\/span><\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002\u901a\u89e3\u4e3a:
    x = [ x 1 x 2 x 3 ] = [ 0.3 x 2 + 0.2 x 3 0 x 3 ] = [ 0.3 x 2 0 0 ] + [ 0.2 x 3 0 0 ] = x 2 [ 0.3 0 0 ] + x 3 [ 0.2 0 0 ] = x 2 u + x 3 v \uff0c \u5176 \u4e2d u = [ 0.3 0 0 ] \uff0c v = [ 0.2 0 0 ] \\boldsymbol{x}=\\begin{bmatrix}x_1 \\\\ x_2 \\\\ x_3\\end{bmatrix}=\\begin{bmatrix}0.3x_2 + 0.2x_3 \\\\ 0 \\\\ x_3\\end{bmatrix}=\\begin{bmatrix}0.3x_2 \\\\ 0 \\\\ 0\\end{bmatrix} + \\begin{bmatrix}0.2x_3 \\\\ 0 \\\\ 0\\end{bmatrix} \\\\ \\quad=x_2\\begin{bmatrix}0.3 \\\\ 0 \\\\ 0\\end{bmatrix} + x_3\\begin{bmatrix}0.2 \\\\ 0 \\\\ 0\\end{bmatrix} \\\\ \\quad=x_2\\boldsymbol{u}+x_3\\boldsymbol{v}\uff0c\u5176\u4e2d\\boldsymbol{u}=\\begin{bmatrix}0.3 \\\\ 0 \\\\ 0\\end{bmatrix}\uff0c\\boldsymbol{v}=\\begin{bmatrix}0.2 \\\\ 0 \\\\ 0\\end{bmatrix} <\/span><\/span>x<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span>.<\/span>3<\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>0<\/span>.<\/span>2<\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span>.<\/span>3<\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span>.<\/span>2<\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span>.<\/span>3<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span>.<\/span>2<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>v<\/span><\/span><\/span>\uff0c<\/span>\u5176<\/span>\u4e2d<\/span>u<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span>.<\/span>3<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff0c<\/span>v<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span>.<\/span>2<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

    \u9f50\u6b21\u65b9\u7a0b A x = 0 \\boldsymbol{Ax}=\\boldsymbol{0} <\/span><\/span>A<\/span>x<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u603b\u53ef\u8868\u793a\u4e3aSpan{
    \n <\/strong> v 1 , \u22ef &ThinSpace; , v p \\boldsymbol{v_1},\\cdots,\\boldsymbol{v_p} <\/span><\/span>v<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u22ef<\/span><\/span><\/span>,<\/span><\/span>v<\/span><\/span>p<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>}<\/strong>\uff0c\u5176\u4e2d v 1 , \u22ef &ThinSpace; , v p \\boldsymbol{v_1},\\cdots,\\boldsymbol{v_p} <\/span><\/span>v<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u22ef<\/span><\/span><\/span>,<\/span><\/span>v<\/span><\/span>p<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u662f\u9002\u5f53\u7684\u89e3\u5411\u91cf\u3002\u82e5\u552f\u4e00\u89e3\u662f\u96f6\u5411\u91cf\uff0c\u5219\u89e3\u96c6\u5c31\u662fSpan{
    \n <\/strong> 0 \\boldsymbol{0} <\/span><\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/span>}<\/strong>\uff1b\u82e5\u65b9\u7a0b A x = 0 \\boldsymbol{Ax}=\\boldsymbol{0} <\/span><\/span>A<\/span>x<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4ec5\u6709\u4e00\u4e2a\u81ea\u7531\u53d8\u91cf\uff0c\u5219\u89e3\u96c6\u662f\u901a\u8fc7\u539f\u70b9\u7684\u4e00\u6761\u76f4\u7ebf\u3002\u82e5\u6709\u4e24\u4e2a\u6216\u66f4\u591a\u4e2a\u81ea\u7531\u53d8\u91cf\uff0c\u5219\u89e3\u96c6\u662f\u901a\u8fc7\u539f\u70b9\u7684\u5e73\u9762\u3002<\/p>\n

    \u4e0a\u8ff0\u65b9\u7a0b { 10 x 1 \u2212 3 x 2 \u2212 2 x 3 = 0 \\begin{cases}{ 10x_1 - 3x_2 - 2x_3 = 0 }\\end{cases} <\/span><\/span>{
    \n <\/span><\/span><\/span>1<\/span>0<\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>3<\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>2<\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u662f\u5e73\u9762\u7684\u9690\u5f0f\u63cf\u8ff0<\/strong>\uff0c\u89e3\u6b64\u65b9\u7a0b\u5c31\u662f\u8981\u627e\u8fd9\u4e2a\u5e73\u9762\u7684\u663e\u793a\u63cf\u8ff0<\/strong>\uff08 x = x 2 u + x 3 v \\boldsymbol{x}=x_2\\boldsymbol{u}+x_3\\boldsymbol{v} <\/span><\/span>x<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>v<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff09\uff0c\u5c31\u662f\u8bf4\u5c06\u5b83\u4f5c\u4e3a u \\boldsymbol{u} <\/span><\/span>u<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u548c v \\boldsymbol{v} <\/span><\/span>v<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u5b50\u96c6\u3002
    \u663e\u793a\u63cf\u8ff0\u79f0\u4e3a\u5e73\u9762\u7684\u53c2\u6570\u5411\u91cf\u65b9\u7a0b<\/strong>\uff0c\u8bb0\u4e3a x = s u + t v ( s , t \u4e3a \u5b9e \u6570 ) \\boldsymbol{x}=s\\boldsymbol{u}+t\\boldsymbol{v}\\quad(s,t\u4e3a\u5b9e\u6570) <\/span><\/span>x<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>s<\/span>u<\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>t<\/span>v<\/span><\/span><\/span><\/span>(<\/span>s<\/span>,<\/span><\/span>t<\/span>\u4e3a<\/span>\u5b9e<\/span>\u6570<\/span>)<\/span><\/span><\/span><\/span><\/span>\u3002\u5f53\u89e3\u96c6\u7528\u5411\u91cf\u663e\u793a\u8868\u793a\uff0c\u6211\u4eec\u79f0\u4e4b\u4e3a\u89e3\u7684\u53c2\u6570\u5411\u91cf\u5f62\u5f0f<\/strong>\u3002<\/p>\n

    \u975e\u9f50\u6b21\u65b9\u7a0b\u7ec4\u7684\u89e3<\/h3>\n

    \u63cf\u8ff0 A x = b \\boldsymbol{Ax}=\\boldsymbol{b} <\/span><\/span>A<\/span>x<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u89e3\uff0c\u5176\u4e2d A = [ 3 5 \u2212 4 \u2212 3 \u2212 2 4 6 1 \u2212 8 ] \\boldsymbol{A}=\\begin{bmatrix}3 &amp; 5 &amp; -4 \\\\ -3 &amp; -2 &amp; 4 \\\\ 6 &amp; 1 &amp; -8\\end{bmatrix} <\/span><\/span>A<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span>\u2212<\/span>3<\/span><\/span><\/span><\/span>6<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>5<\/span><\/span><\/span><\/span>\u2212<\/span>2<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>4<\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span>\u2212<\/span>8<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff0c b = [ 7 \u2212 1 \u2212 4 ] \\boldsymbol{b}=\\begin{bmatrix} 7 &amp; -1 &amp; -4 \\end{bmatrix} <\/span><\/span>b<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>[<\/span><\/span><\/span>7<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span>
    \u89e3\uff1a\u5bf9 [ A b ] \\begin{bmatrix}\\boldsymbol{A} &amp; \\boldsymbol{b}\\end{bmatrix} <\/span><\/span>[<\/span><\/span><\/span>A<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4f5c\u884c\u53d8\u6362\u5f97
    [ 3 5 \u2212 4 7 \u2212 3 \u2212 2 4 \u2212 1 6 1 \u2212 8 \u2212 4 ] \\begin{bmatrix}3 &amp; 5 &amp; -4 &amp; 7 \\\\ -3 &amp; -2 &amp; 4 &amp; -1 \\\\ 6 &amp; 1 &amp; -8 &amp; -4\\end{bmatrix} <\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span>\u2212<\/span>3<\/span><\/span><\/span><\/span>6<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>5<\/span><\/span><\/span><\/span>\u2212<\/span>2<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>4<\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span>\u2212<\/span>8<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>7<\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span>\u2212<\/span>4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff5e [ 1 0 \u2212 4 3 \u2212 1 0 1 0 2 0 0 0 0 ] \\begin{bmatrix}1 &amp; 0 &amp; -\\frac{4}{3} &amp; -1 \\\\ 0 &amp; 1 &amp; 0 &amp; 2 \\\\ 0 &amp; 0 &amp; 0 &amp; 0\\end{bmatrix} <\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
    \u628a\u6bcf\u4e2a\u57fa\u672c\u53d8\u91cf\u7528\u81ea\u7531\u53d8\u91cf\u8868\u793a\uff1a { x 1 \u2212 4 3 = \u2212 1 x 2 = 2 0 = 0 \\begin{cases} x_1 - \\frac{4}{3} = -1 \\\\ x_2 = 2 \\\\ 0 = 0 \\\\ \\end{cases} <\/span><\/span><\/span>\u23a9<\/span><\/span><\/span><\/span>\u23aa<\/span><\/span><\/span><\/span>\u23a8<\/span><\/span><\/span><\/span>\u23aa<\/span><\/span><\/span><\/span>\u23a7<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>2<\/span><\/span><\/span><\/span>0<\/span><\/span>=<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
    A x = b \\boldsymbol{Ax}=\\boldsymbol{b} <\/span><\/span>A<\/span>x<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u901a\u89e3\u53ef\u5199\u6210\u5411\u91cf\u5f62\u5f0f
    x = [ x 1 x 2 x 3 ] = [ \u2212 1 + 4 3 x 3 2 x 3 ] = [ \u2212 1 2 0 ] + [ 4 3 x 3 0 x 3 ] = [ \u2212 1 2 0 ] + x 3 [ 4 3 0 1 ] \\boldsymbol{x}=\\begin{bmatrix}x_1 \\\\ x_2 \\\\ x_3\\end{bmatrix}=\\begin{bmatrix}-1+\\frac{4}{3}x_3 \\\\ 2 \\\\ x_3\\end{bmatrix}=\\begin{bmatrix}-1 \\\\ 2 \\\\ 0\\end{bmatrix}+\\begin{bmatrix}\\frac{4}{3}x_3 \\\\ 0 \\\\ x_3\\end{bmatrix}\\\\ \\quad=\\begin{bmatrix}-1 \\\\ 2 \\\\ 0\\end{bmatrix} + x_3\\begin{bmatrix}\\frac{4}{3} \\\\ 0 \\\\ 1\\end{bmatrix} <\/span><\/span>x<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span>+<\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
    \u65b9\u7a0b x = p + x 3 v \uff0c \u5176 \u4e2d p = [ \u2212 1 2 0 ] , v = [ 4 3 0 1 ] \\boldsymbol{x}=\\boldsymbol{p} + x_3\\boldsymbol{v}\uff0c\u5176\u4e2d\\boldsymbol{p}=\\begin{bmatrix}-1 \\\\ 2 \\\\ 0\\end{bmatrix},\\boldsymbol{v}=\\begin{bmatrix}\\frac{4}{3} \\\\ 0 \\\\ 1\\end{bmatrix} <\/span><\/span>x<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>p<\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>v<\/span><\/span><\/span>\uff0c<\/span>\u5176<\/span>\u4e2d<\/span>p<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>v<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff0c\u6216\u7528 t t <\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u8868\u793a\u81ea\u7531\u53d8\u91cf\uff0c x = p + t v \\boldsymbol{x}=\\boldsymbol{p} + t\\boldsymbol{v} <\/span><\/span>x<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>p<\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>t<\/span>v<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5c31\u662f\u7528\u53c2\u6570\u53d8\u91cf\u5f62\u5f0f\u8868\u793a\u7684 A x = b \\boldsymbol{Ax}=\\boldsymbol{b} <\/span><\/span>A<\/span>x<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u89e3\u96c6\u3002<\/p>\n

    \u6ce8\u610f\uff1a\u7b2c\u4e00\u4e2a\u4f8b\u5b50\u9f50\u6b21\u65b9\u7a0b\u7ec4 { 3 x 1 + 5 x 2 \u2212 4 x 3 = 0 \u2212 3 x 1 \u2212 2 x 2 + 4 x 3 = 0 6 x 1 + x 2 \u2212 8 x 3 = 0 \\begin{cases} 3x_1 + 5x_2 - 4x_3 = 0 \\\\ -3x_1 - 2x_2 + 4x_3 = 0 \\\\ 6x_1 + x_2 - 8x_3 = 0 \\\\ \\end{cases} <\/span><\/span><\/span>\u23a9<\/span><\/span><\/span><\/span>\u23aa<\/span><\/span><\/span><\/span>\u23a8<\/span><\/span><\/span><\/span>\u23aa<\/span><\/span><\/span><\/span>\u23a7<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>5<\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>4<\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>0<\/span><\/span><\/span><\/span>\u2212<\/span>3<\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>2<\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>4<\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>0<\/span><\/span><\/span><\/span>6<\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>8<\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u7cfb\u6570\u77e9\u9635\u548c\u4e0a\u8bc9\u4f8b\u5b50\u7684\u7cfb\u6570\u77e9\u9635\u662f\u540c\u4e00\u77e9\u9635\uff1a [ 3 5 \u2212 4 \u2212 3 \u2212 2 4 6 1 \u2212 8 ] \\begin{bmatrix}3 &amp; 5 &amp; -4 \\\\ -3 &amp; -2 &amp; 4 \\\\ 6 &amp; 1 &amp; -8\\end{bmatrix} <\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span>\u2212<\/span>3<\/span><\/span><\/span><\/span>6<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>5<\/span><\/span><\/span><\/span>\u2212<\/span>2<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>4<\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span>\u2212<\/span>8<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002\u4e24\u4e2a\u65b9\u7a0b\u7684\u53c2\u6570\u5f62\u5f0f\u7684 v \\boldsymbol{v} <\/span><\/span>v<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u662f\u76f8\u540c\u7684\u3002\u6545 A x = b \\boldsymbol{Ax}=\\boldsymbol{b} <\/span><\/span>A<\/span>x<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u89e3\u53ef\u7531\u5411\u91cf p \\boldsymbol{p} <\/span><\/span>p<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u52a0\u4e0a A x = 0 \\boldsymbol{Ax}=\\boldsymbol{0} <\/span><\/span>A<\/span>x<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u89e3\u5f97\u5230\uff0c\u5411\u91cf p \\boldsymbol{p} <\/span><\/span>p<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u672c\u8eab\u4e5f\u662f A x = b \\boldsymbol{Ax}=\\boldsymbol{b} <\/span><\/span>A<\/span>x<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u4e00\u4e2a\u7279\u89e3\u3002<\/p>\n

    \u4e3a\u4e86\u4ece\u51e0\u4f55\u4e0a\u63cf\u8ff0 A x = b \\boldsymbol{Ax}=\\boldsymbol{b} <\/span><\/span>A<\/span>x<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u89e3\u96c6\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u5411\u91cf\u52a0\u6cd5\u89e3\u91ca\u4e3a\u5e73\u79fb<\/strong>\u3002
    \u8bbe L \\boldsymbol{L} <\/span><\/span>L<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u662f\u901a\u8fc7 0 \\boldsymbol{0} <\/span><\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4e0e v \\boldsymbol{v} <\/span><\/span>v<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u76f4\u7ebf\u3002 L \\boldsymbol{L} <\/span><\/span>L<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u6bcf\u4e2a\u70b9\u52a0\u4e0a p \\boldsymbol{p} <\/span><\/span>p<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5f97\u5230 x = p + t v \\boldsymbol{x}=\\boldsymbol{p} + t\\boldsymbol{v} <\/span><\/span>x<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>p<\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>t<\/span>v<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u8868\u793a\u7684\u5e73\u79fb\u540e\u7684\u76f4\u7ebf\u3002\u6ce8\u610f p \\boldsymbol{p} <\/span><\/span>p<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4e5f\u5728\u5e73\u79fb\u540e\u7684\u76f4\u7ebf\u4e0a\u3002\u79f0 x = p + t v \\boldsymbol{x}=\\boldsymbol{p} + t\\boldsymbol{v} <\/span><\/span>x<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>p<\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>t<\/span>v<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4e3a\u901a\u8fc7<\/strong> p \\boldsymbol{p} <\/span><\/span>p<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5e73\u884c\u4e8e<\/strong> v \\boldsymbol{v} <\/span><\/span>v<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u76f4\u7ebf\u65b9\u7a0b<\/strong>\u3002\u7efc\u4e0a\uff0c A x = b \\boldsymbol{Ax}=\\boldsymbol{b} <\/span><\/span>A<\/span>x<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u89e3\u96c6\u662f\u4e00\u6761\u901a\u8fc7<\/strong> p \\boldsymbol{p} <\/span><\/span>p<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u800c\u5e73\u884c\u4e8e<\/strong> A x = 0 \\boldsymbol{Ax}=\\boldsymbol{0} <\/span><\/span>A<\/span>x<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u89e3\u96c6\u7684\u76f4\u7ebf<\/strong>\u3002<\/p>\n

    \u8bbe\u65b9\u7a0b A x = b \\boldsymbol{Ax}=\\boldsymbol{b} <\/span><\/span>A<\/span>x<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5bf9\u67d0\u4e2a b \\boldsymbol{b} <\/span><\/span>b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u662f\u76f8\u5bb9\u7684\uff0c p \\boldsymbol{p} <\/span><\/span>p<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4e3a\u4e00\u4e2a\u7279\u89e3\uff0c\u5219 A x = b \\boldsymbol{Ax}=\\boldsymbol{b} <\/span><\/span>A<\/span>x<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u89e3\u96c6\u662f\u6240\u6709\u5f62\u5982 w = p + v h \\boldsymbol{w}=\\boldsymbol{p} + \\boldsymbol{v_h} <\/span><\/span>w<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>p<\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>v<\/span><\/span>h<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u5411\u91cf\u7684\u96c6\uff0c\u5176\u4e2d v h \\boldsymbol{v_h} <\/span><\/span>v<\/span><\/span>h<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u662f\u9f50\u6b21\u65b9\u7a0b A x = 0 \\boldsymbol{Ax}=\\boldsymbol{0} <\/span><\/span>A<\/span>x<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u4efb\u610f\u4e00\u4e2a\u89e3\u3002
    \u6ce8\u610f\uff1a\u4ec5\u9002\u7528\u4e8e\u65b9\u7a0b A x = b \\boldsymbol{Ax}=\\boldsymbol{b} <\/span><\/span>A<\/span>x<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u81f3\u5c11\u6709\u4e00\u4e2a\u975e\u96f6\u89e3 p \\boldsymbol{p} <\/span><\/span>p<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u524d\u63d0\u4e0b\u3002\u5f53 A x = b \\boldsymbol{Ax}=\\boldsymbol{b} <\/span><\/span>A<\/span>x<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u65e0\u89e3\u65f6\uff0c\u89e3\u96c6\u662f\u7a7a\u96c6\u3002<\/p>\n

    \u628a\uff08\u76f8\u5bb9\u65b9\u7a0b\u7ec4\u7684\uff09\u89e3\u96c6\u8868\u793a\u79f0\u53c2\u6570\u5411\u91cf\u5f62\u5f0f\uff1a<\/p>\n

      \n
    1. \u628a\u589e\u5e7f\u77e9\u9635\u884c\u5316\u7b80\u4e3a\u7b80\u5316\u9636\u68af\u5f62\u77e9\u9635\u3002<\/li>\n
    2. \u628a\u6bcf\u4e2a\u57fa\u672c\u53d8\u91cf\u7528\u81ea\u7531\u53d8\u91cf\u8868\u793a\u3002<\/li>\n
    3. \u628a\u4e00\u822c\u89e3 x \\boldsymbol{x} <\/span><\/span>x<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u8868\u793a\u79f0\u5411\u91cf\uff0c\u5982\u679c\u6709\u81ea\u7531\u53d8\u91cf\uff0c\u5176\u5143\u7d20\u4f9d\u8d56\u4e8e\u81ea\u7531\u53d8\u91cf\u3002<\/li>\n
    4. \u628a x \\boldsymbol{x} <\/span><\/span>x<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5206\u89e3\u4e3a\u5411\u91cf\uff08\u5143\u7d20\u4e3a\u5e38\u6570\uff09\u7684\u7ebf\u6027\u7ec4\u5408\uff0c\u7528\u81ea\u7531\u53d8\u91cf\u4f5c\u4e3a\u53c2\u6570\u3002<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"\u7ebf\u6027\u65b9\u7a0b\u7ec4\uff08\u4e94\uff09- \u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u89e3\u96c6\u5c0f\u7ed3\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u5b9a\u4e49","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[],"tags":[],"_links":{"self":[{"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/posts\/8962"}],"collection":[{"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/comments?post=8962"}],"version-history":[{"count":0,"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/posts\/8962\/revisions"}],"wp:attachment":[{"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/media?parent=8962"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/categories?post=8962"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/tags?post=8962"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}