• 2022-06-29
    设矩阵 [tex=12.0x3.643]3BT1BgBZQ5uJXxD5dg+w2zx15MAnC8507w6DKIIBomYmPiLryqg1pDYftwancKapCt0V0IcyFtNGRZSZdIjF87MrgQW4X7PnjLbj3bv8mmVy0u4hYCXLMnmBIsPqVpv8nDvKoeSwA22yFaMpSBNPFw==[/tex] 若齐次线性方程组[tex=2.643x1.0]Luk4dywqmDJgAqza1pE8oQ==[/tex]的基础解系中含有两个线性无关的解向量. 试求方程组 [tex=2.643x1.0]Luk4dywqmDJgAqza1pE8oQ==[/tex] 的全部解. 
  • 解: 因为 [tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex] 是[tex=2.286x1.143]Rt90nlird0F3OkJeoMVgsg==[/tex]矩阵,所以 [tex=2.643x1.0]Luk4dywqmDJgAqza1pE8oQ==[/tex] 是四元齐次线性方程组. 根据题设条件,方程组[tex=2.929x1.0]s5ChnUJhIxqFSdXmAN58D3RZhzrok/9Wutvdmol2bd328PF0aDrBHtOJFUG1HQPq[/tex]的基础解系中线性无关的解向量的个数为[tex=4.643x1.357]oA2bog/K6v8UG6HCSXJVoQS0pLNWHvAqXfd1sIAQXT0DS8sdjafs6SKMhRB1Atjm[/tex]由此可得 [tex=3.643x1.357]SMB0AC6IZNDjxg6K+6zWVp3CpGbA8AiP3llQ8qx5nGiEOfydR/lgoaFgZqaili07[/tex] 对矩阵[tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex] 施以初等行变换:[tex=23.5x3.643]ysPsVBYgue2sVzMz/Uq3uztS+sWCIHkAGhkbo6F1ClDi1ol7cxb1xuqxQdi1FwgnOuax8EPl/yR6hMmCkyx9xPwmrYdATLuGLIs8pO2ClCETFySzbxaC7lIYshqVm9/bD+guqfg3kRLnb2sJfHyqR2BkwgONJARZIw0fzXT8rBhAHCRtlysV5A+10hJE8p6pLDHkpAj7uq1wlzkIOPh/c6dKpp/H7LobojGJy3RgjSm27O3txzhV2queATL5QvT6b6s78IDQlkLdxC4X6Qr0qegkxU9lp24DGxcRRw4SPambxYakXBA02PCQ03cgUEGD[/tex][tex=13.286x3.643]SSohy12uIjks1YNZ0sUMrC2IiCnZD3ppPofRxjymxJgS5Zknbx1zrHQuBDnfO4n8+fWc3UWh2hqQxMNyn49J9DSG37sZOmbaGJ8PThtXfXqmIaIlkMp7sNiE1+8XAaJd+XzptSN0LanhB6tH9w9VGEjAi9W+oZIpXmKEHuxFxqc=[/tex]  [tex=1.286x1.357]rGbPpVrby9YXsv3eGNgEJw==[/tex]由上面最后一个矩阵可以看出,要使[tex=3.357x1.357]SMB0AC6IZNDjxg6K+6zWVnxCqjREfDvbwzJFHkfvyoY=[/tex], 只需[tex=1.929x1.0]tQp/GUl8n5ArAMn67lnWag==[/tex]此时,继续对矩阵[tex=1.286x1.357]rGbPpVrby9YXsv3eGNgEJw==[/tex]施以初等行变换,将[tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex] 化为简化的阶梯形矩阵:[tex=11.643x3.643]CgAResoOKlo52BbELcWuiUeC7YP81JUphgtUoxDBeoUo2NXh+u+xXisgk1dW/LNKDdkClIpxbxKP4YwZAKxZZImAVr4Iqf+QiJBkHjkMpz5RHRocwUdObsbD8ODZ8aYsAwM10oTP1aCDfhJvRSW0ZlP0H0kCihslle+HRsyw03k=[/tex]可得原方程组的同解方程组为[tex=6.857x2.786]GE56u9QCDTqcLxZ66HADyjJiEdC+WzN+E7rRFVulsHw6ei5LjHBmWKTPzjEbxvxJaMWT39CIHSU5NRBJJaaGySUCyz4963Z99xs/iEE23uLL6L8Faq4jycsQDuw0I/UU[/tex]令自由未知量[tex=3.143x2.357]jcCMHflCR8OS9TosV6N5vDyfDGBMSkq10p9kpuAv5SrcUpOVYFMvav37JGy3WzfrL0jbOB5tDp+9wGgLGK0YPQ==[/tex] 分别取 [tex=6.714x2.786]jcCMHflCR8OS9TosV6N5vBnggl6dHvC3qVk7cYRglSW0LoAYYFXdqEC6rPLsmcli2DWLdz9vj6TJ7si73rQqQfVATlLW5odDcMA3VVur8XelWlaVSrFUnJcujubUKJgE[/tex], 得方程组[tex=2.929x1.0]s5ChnUJhIxqFSdXmAN58D3RZhzrok/9Wutvdmol2bd328PF0aDrBHtOJFUG1HQPq[/tex] 的基础解系[tex=13.357x1.5]jMswEq2ewVek4yDHZiDErEMfU9TN90O8DOVbCOXCMv/9lRsQXYCZJqBaz54yePIfYIxumdTo5H47NwgDDwvFh2Ck5kTFiRLAf0CokjPFQkI=[/tex]所以,方程组 [tex=2.929x1.0]s5ChnUJhIxqFSdXmAN58D3RZhzrok/9Wutvdmol2bd328PF0aDrBHtOJFUG1HQPq[/tex] 的全部解为[tex=20.357x4.643]iPHJ1YxOhT0uzXvfDAUzvBr7IwhMhqaIJjRl3+ER+g7QXF8QgqjvQyJwOp6z7xHjSxThlEkd86h6te3OaiGQgCujobmrmDV9Wih02xLNCCp/xPDFcItV8n02cDgxJSiUKsONxpcAR1OzG/Uw30YE8QcdeipAG14IZnowBOVntXh8aCxWiH+37DYFW1YS28vRDcBC9TTTXXNC04Hm/HJC5QIU1284mHhkeOFtOvhtRD4LY+Np/C5V1gr1RzGquBTe[/tex]为任意常数 )

    举一反三

    内容

    • 0

      证明:设[tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex]元齐次线性方程组(1)的系数矩阵的秩为[tex=3.643x1.357]yBlNyz2xzn3Ca7e545goUg==[/tex],则方程组(1)的任意[tex=1.857x1.071]kw/I29OLYXCHVLVrD23+Ig==[/tex]个线性无关的解向量都是它的一个基础解系.

    • 1

      方程3x﹣1=8的解是 A: x=3 B: x=4 C: x=5 D: x=6

    • 2

      set1 = {x for x in range(10)} print(set1) 以上代码的运行结果为? A: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} B: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9,10} C: {1, 2, 3, 4, 5, 6, 7, 8, 9} D: {1, 2, 3, 4, 5, 6, 7, 8, 9,10}

    • 3

      设A=(α1,α2,α3,α4)是4阶矩阵,A*为A的伴随矩阵,若(1,0,1,0)T是方程组Ax=0的一个基础解系,则A*x=0的基础解系可为()。 A: α1,α3 B: α1,α2 C: α1,α2,α3 D: α2,α3,α4

    • 4

      设A是m×n矩阵,齐次线性方程组AX=0,r(A)=n-5,α1,α2,α3,α4,α5是该方程组5个线性无关的解向量,则方程组AX=0的一个基础解系是______. A: α1+α2,α2+α3,α3+α4,α4+α5,α5+α1 B: α1-α2,α2+α3,α3+α4,α4+α5,α5+α1 C: α1-α2,α2-α3,α3-α4,α4+α5,α5+α1 D: α1-α2,α2-α3,α3-α4,α4-α5,α5-α1