设 [tex=0.857x1.286]ZpwhzmyivskaH5M1X7ozaQ==[/tex] 是数域 [tex=0.786x1.286]dSWbQCTjdbLxKy7q0ps2gg==[/tex] 上 4 阶幻方所构成的线性空间，求 [tex=2.5x1.286]+xUVRiAQe0xHEuYC2z8BFQ==[/tex] 与 [tex=0.857x1.286]ZpwhzmyivskaH5M1X7ozaQ==[/tex] 的基.

2022-05-31

设 [tex=0.857x1.286]ZpwhzmyivskaH5M1X7ozaQ==[/tex] 是数域 [tex=0.786x1.286]dSWbQCTjdbLxKy7q0ps2gg==[/tex] 上 4 阶幻方所构成的线性空间，求 [tex=2.5x1.286]+xUVRiAQe0xHEuYC2z8BFQ==[/tex] 与 [tex=0.857x1.286]ZpwhzmyivskaH5M1X7ozaQ==[/tex] 的基.

答案：

解 令[tex=18.214x4.5]M5jMZHlBYWtxDwaYVBilSLH/z77VH0RsR0fe+WLGW4YUq2lAds/G5edyUetUVCX58JeTh68vkR+6glg4flevEUG+IhxxcGiXDHpJAg3TedJ5TVeWxRKSVuc0FV5j62vEFqdk3QpSe5nI5izEirpXspztKwAkpqqvof4BP+kYWrhhD7UFFUM9FOMfWveJzUwXNLyhn/EO6+RIQD/ZlNXugA==[/tex]则 [tex=5.929x1.214]WaTRfLNS5p2ZjlW50XvGPCMkG4RNc1ajMD3tM8EWTCLDwnx9d6Bjm0Jb/P3tb3b/[/tex] 由 [tex=2.786x1.429]z7wqdKodFk3FiMxOsKm1cHwNB0O0tQn+JT7SGBl5vF4=[/tex]，于是有[p=align:center][tex=13.786x5.786]7EJHVCtO2IWq3KpdB+jQsnddFuE1KhDLt5KyTxKWSCu/f0qlsNXLGIaifsmgItyu+tNLCQxX3KTOn6h/DIY0fdRcdCpIOz81ktWU2Hm6rVOFMHPvXY78f9i5+FCoSzCum6KuYxwURxx395CthT5j+7cdBBNQCKreVc2wWCyGs1psN/jAb5eD7+fzQkTG2OxoDMTvdoM6o0lNyWNAoaElGAqzNFkG75QoqHD2GJig0I8=[/tex]因此 [tex=4.071x1.214]K58aKVfsWp92+LOxrfHz1g==[/tex]，其中 [tex=18.429x1.357]ajr7nVX521USJmbMGPvKO/rq9fhO3X3bpuD+oG/dpCygtEhJ37lKXkwZBV25oXsFla6ZQ+cRWbw07kmycFWnkr2LSRk83IE96iWXOM/Jr8Y=[/tex] 于是 [tex=5.571x1.357]E4/5bEM/N/AG9AX2WxmUlDhnqQEycvhCAwaJfly66r4=[/tex] 所以 [tex=4.286x1.357]/Q7wWS18KmlDcTDFcxhN7tlopLPaJB79y9lOgbA9G6A=[/tex] 于是问题归结为求 [tex=2.357x1.214]b+19PhVr4qu1uqfrbbodNg==[/tex] 的维数与基.[tex=1.0x1.214]hhEyiXsmUqGVtlGvWeNOYA==[/tex] 中元素为[p=align:center][tex=11.0x4.786]clWXPRVSJ+Gj8zywDe28/yQe1kWSCSf1BC1XvdAre7V/v/XpimOSdpBWmNqCrVoTJ8KPfUjkaW5H0rtonBQ55NogtSttYBh1TbdSoH5wtEfKZR7o7yg9jfy+SfY/0lzZBZHh6KzvyUS3e14aDYGwsqQ0piIIj4mMKUD0BzFfdbCe+CnQ4480IxR5TOfnKmtZ9TWYFtc6ZkP8CeZH4KgB3cfKHkXJ0uvoIDyQvfAHflbDS35kLcH78BweZIRZFEdV[/tex]，满足[p=align:center][tex=14.214x8.214]7EJHVCtO2IWq3KpdB+jQsiltifdm8Ik3YJF6NUgOLDXCLc49GWajrVwtWoKAPPL33yjdlDBcUotvmSxMHGFYVkN7Weg6yyS862KirJAz0lRWAYU4bWJKWK0SiBOPg2JfStGK8VQGj6hrgH+NTm84IiVN3WUyb4GtFdruptMkKh1E65L4kOksHWylwE7CT7gzID+BkrYZzUQrCpoMzozNjtwZC1Bx/JM/wPiCUp9vnZB+B6PfiqzrIifNJM59OdsieDklDie5jXOOs9SyguPgj/TLsPA3GHFnbAiSvjK1kvTOi6KLEKfC6xj+bPPZl5EUHog6El8ANAcgFFmluXCUqe6bR8VG22jWjYLjyvL2hzvFBSRuJxesDglCYNQYX2y6PdhM2VQp9Lqus5vy9J3WgFX7OSCibIvRkcjKuaRS1/8=[/tex]于是可得 [tex=2.0x1.286]3ZW0/tZMsKqhFAOeIdvWOQ==[/tex] 的齐次线性方程组，其系数矩阵为：[p=align:center][tex=20.0x7.214]jcCMHflCR8OS9TosV6N5vC5400geqN+fo7HEnochvu3I8wxy6PDl+0uBkKBjncIO3ffUwm3eglCgdJL0d9H69FBksgUyFNy5kqF6tVbkmTIEVDtvCfRACxyONIglKLxb4mvsTxA1cGKAP0t1WrG/SBUq9Vq0zoOazsGpwpKMzR2HVnPe8CeQBqZ9wR/f+q8LDseoQVr8qfmucJB4L9XsZjJxHcQtgPC6z+ecItaQsfMlR7FUY1OhmmOppGvO28CBSG0GJ+otql0fygY2PYtd1NmxVzegHSoZi5Y5yURqrgShGOw0acnFuFYncYPkAbbT9Szh/e24g/GsU3IEYgftFqcK2Wir88vAR4/fpQsWC6k=[/tex].第一行的 2 倍减第 3，4，5，6 行，得到第一行为 0， 第二行的 2 倍减第 4，5 行，得到[p=align:center][tex=20.0x7.071]jcCMHflCR8OS9TosV6N5vKjp+G5yyLQ2cZg+qyrLyehpwTRvZTx4koosGov/kH4yb1Fjb7I6dK09P+oN78eB+506xHf4lP37Dqkzx+EDEUqB4crc+3sLo4LUAZsnk654JiXEBETz8Vd1PUCTDuDhjIBQvzG8j8kjg0IxlCOo3+67nPEt+Mfx84yUdogvYVmiZIWN192YZpI+qmn691wpFR3/w3wlbaHcQmgE+2HzxAY6tXhknkfRkWM16JLz+WZ3Zz9IGVVvSlE3Yn0W03AvX4MowRCQA4KDNq3zZepjZH1OUfQUepJBqonWOWuHe06L/Wgsfcz/t0PZb2+35d42Eg==[/tex].注意第 4，5，6，7，8 列为此矩阵的列向量的极大线性无关组，于是 [tex=4.357x1.214]NovbxKl63Ey/milqTcbe/16sBGAqwZbl871itviyiQE=[/tex] 且可取 [tex=3.571x1.0]1OxXb/HtxwZSBmoxMs+T5CpPiru7ngyhOI3Tn6tJCR4=[/tex] 为自由未知量.当 [tex=6.429x1.214]A/beiWwDbduz5D/7FinyMTlZyC4zh67WMF4JUSq+rfg=[/tex] 时，有 [tex=13.786x1.214]o9tbzd0WeuyUFbuZy4D+41+vTlHPnki9/trIgZue5U7fNi/KgPU3izAzLdyqYmkR[/tex].当 [tex=6.429x1.214]ek23NbttNBMt44+01FlAw/dddLap7tAknmrWJSEcTxU=[/tex] 时，有 [tex=12.5x1.214]EnJPHOAhjCFR2gxdlXbdcDAoRnlEJnHMLkAbj9L7PKSPSiXDWs4P2yuxcnrSG/gO[/tex]当 [tex=6.429x1.214]tnTM8M+Kqsok46Fyewl1WTR1djkSfAysdoxypIvyUAQ=[/tex] 时，有 [tex=12.5x1.214]EnJPHOAhjCFR2gxdlXbdcLC/si8QeqdpOaQvqLAPhg6BvCLtp+M1XzpceR2vGGvG[/tex]由此 [tex=1.0x1.214]hhEyiXsmUqGVtlGvWeNOYA==[/tex] 有基：[tex=11.357x4.929]6sUbiybY3Dw4uQOmGHXZgWJPx7KtP76iQvJPwqVL/+ybS115EC+SaXuZQENNH+59P3/CUjp+r09UH8zxslYQn4dzn8I7VPglfVAxCj8dQ/xZCqOI+ilKx/aHtu7L3NRoeadsEoBx3sf6NhRMD36azRQN+eGPz/Hu4aDMY5mvCP4=[/tex]，[tex=9.857x4.5]PB8kmsqXUQve91GAff42OesIveesgB0cowquKKahtRj+VlpAaCaZkfWoLp2iUkaO9dOUHbxCD0ZCZE21rIlqMjZKpWCBxnGRDBhu0zarPEH0likeOj5bPY/1A7L+iYboJMEL+/9UOKRS37FeuB2KsC/v6tzPcA7vEX2zb3qJfy0=[/tex]，[tex=9.857x4.5]cKNUoSBvdxxXZvO6cVISl0y2nVz46odIvUht1rYnnTTl1i9EE81l0rjlxYm4eKyEBJprtekNjisbkxIBrY04JNUrKFGW28eD13L3kUHe72mhEFK1OixBUZLWQi+T++5OPxdBzUPYji1OwJwMsrQQtBGaoSSyM8Z31ymrZVddYH0=[/tex].[tex=1.0x1.214]++ZnQ9Yy0yDRqmUwKWQxMg==[/tex] 中元素为[p=align:center][tex=14.071x5.214]clWXPRVSJ+Gj8zywDe28/7YI1elnp0s04EbjLHyoA8hyDVPpfG/ZMlfHbdJAk4af7sIWals0ETyC0aB6Ug4Je3dtGpxVEiAZbjxYyqlZBTMfimSK7KqAF/r2wm216F/J9KCQObwdrbQYxqaLLhaWouEkhpSlx/onBqcNwh1tGeBb764oT537+aa9JArdZShNYNC39+xvfHn8BwZgi/EX156+qUIV6wSxsTQhHqvy+xe9LygzFEXxUNFebDHlYEQY[/tex]，满足[p=align:center][tex=9.071x4.071]7EJHVCtO2IWq3KpdB+jQsiltifdm8Ik3YJF6NUgOLDXCLc49GWajrVwtWoKAPPL3K19Cbb4fTnree44da1wJZ7YkRJf1aAMlT5WsiVRZP+wPwA2BBWLKVUHiC0/Fg7gRHZTBMK+7c28AhsTCQpKzG4ElBBEKh0pqsRi74hsEn24=[/tex]于是可得 [tex=2.0x1.286]3ZW0/tZMsKqhFAOeIdvWOQ==[/tex] 的齐次线性方程组，其系数矩阵为[p=align:center][tex=15.143x3.643]jcCMHflCR8OS9TosV6N5vC5400geqN+fo7HEnochvu14paC+tCuL0NPp+oRbXtgYMx3d/b/+xcvK1W+BCsLjyS5gulrVEpy7Z/BYOCULZSzTqxX/Ma3zultsET4mRlm6xYQbxh2iHDnFEsHRXkVMJOD9OFB3WZVpvRt/t/60Rlhz6PWo5sAM9guX94s/mHkAV5IRYDnT71LQX+5rvmUVUg==[/tex].注意第 4,5,6 列是列向量的极大线性无关组，故 [tex=4.071x1.214]NovbxKl63Ey/milqTcbe/+CH5f0+yLLp15wixXTORAY=[/tex]，可取 [tex=6.0x1.214]1OxXb/HtxwZSBmoxMs+T5P+dAx9oZ3OrAyE8mrJFOA2dw8UqUrar8deyhT5f1oJp[/tex] 为自由未知量，由此可得 [tex=1.0x1.214]++ZnQ9Yy0yDRqmUwKWQxMg==[/tex] 的基：[tex=12.143x4.929]k42qRIbdWkCHF2Ebs6EcQf1ePIIeB19pF3o7Mh8Ov8wvJM9Wm+30A86JMO145KO3it6SGqBJcfrJyPqQEsdlc8g7m/6d3oC+NCH7cZPfNwU9VAtnCGA5/xouDYSunxVMPq5MDmkgVpQKx7+0tHF6pw8WwWSYuePhBOV7TiSdO8o=[/tex]，[tex=12.143x4.929]UCiICwMC9RZG/j7PG7sNzMZWlqMhaLcApm4/ZBViVenfFAg7k8GAQKA8h49KkAXpdotloIpHyTtpn1mq5DEIhdBo0q6ITaQxacsKzOvT4cq2bjCuWfReQigYjEDev73e5Q2Oxt20tzUzACtOAcD30fRWB49swF9mD1Oas4rqrtk=[/tex]，[tex=12.929x4.929]oJMJD3h5tRGNSkSZZlXv6wINUzMw8cIUknlhgvRBup6QQsODGe4gAaaijVHzFJ9q3xomGBknvfrWTXjCU4oInfzObq9yI3tDhzydslu5hL+NtW7TArliOLNLYAEVvmrt8q5CpKgYDHEr7f1c44HScTKaUxNgWSHamnGfc1vXo18=[/tex]，[tex=12.143x4.786]KjIDv1V+9bGPC2BNtcgP2MLP+65/Xb6WQ8ZSI1HB7v9KuOTcGXQMRhxPFOHB5x2ifyQYQ05SytqzstHP/dUwMq9DHe6xsh39UdtoMS7i9IsZDIU6SwhHiYPmgbFqzpWO3PQcgZTAii0Q2DLunBDBM7+PRepXYC7hR2QY1/94EvQ=[/tex]，[tex=11.357x4.786]AiJ41Ud1sQRYQVy3RcCmDi80LKwg1X698jOwDRhCk3VeesvZ3vCkI1GjkFTysikHwdGUlsQzc6+UqpI/gPnfsujwVWUprbnFPO9Eu6phWgidJmVf3+INLTlWuoCzImLCQZ8h6dSryRkADXXtUJ2Y7PEJOTu4Vs+b+qV6viG4jJs=[/tex].于是 [tex=3.714x1.0]NovbxKl63Ey/milqTcbe/xE8Tf5hWJItSAJasmgzZ+k=[/tex]，[tex=11.929x1.214]R0t0sGRUjbY4+uMiMs64IRA+F4LqHoSvXym3yKtucb+JGSJrw0Vg48sH53rF2VuzdIJvsxnNaMZO7WAmgHAKX7u0NlBq6DZgXjFiaU40w2Y=[/tex] 为 [tex=0.857x1.286]ZpwhzmyivskaH5M1X7ozaQ==[/tex] 的基.

举一反三

内容

0
设[tex=0.857x1.286]ZpwhzmyivskaH5M1X7ozaQ==[/tex]是域[tex=0.786x1.286]BlkXDnmzWHxe4M6E9LlofQ==[/tex]上的线性空间，[tex=0.714x1.286]yQZEV57S9rHjYvgfJydTyg==[/tex]是[tex=0.857x1.286]ZpwhzmyivskaH5M1X7ozaQ==[/tex]的任一子集。[tex=0.857x1.286]ZpwhzmyivskaH5M1X7ozaQ==[/tex]中包含[tex=0.714x1.286]yQZEV57S9rHjYvgfJydTyg==[/tex]的所有子空间的交称为由[tex=0.714x1.286]yQZEV57S9rHjYvgfJydTyg==[/tex]生成的子空间，记作[tex=1.357x1.286]FP0/Kp7AEY7Jbzr8yeovuxGYZvgPzg2vzFQmD9y3FIA=[/tex]，即[tex=4.429x1.286]k0NPyIz9PsRYJ2KJDl5JHp2uYIPrA48oe7uK+f1PuLg=[/tex]，其中[tex=1.071x1.286]U4awQ74hGmTHJgQmKU0Jmg==[/tex]取遍[tex=0.857x1.286]ZpwhzmyivskaH5M1X7ozaQ==[/tex]中包含[tex=0.714x1.286]yQZEV57S9rHjYvgfJydTyg==[/tex]的所有子空间，证明：[tex=3.357x1.357]hlzyIv+AZbG9YXlFnOROTobPfwqjCcU2K3kUTR5lVM0=[/tex]。
1
设[tex=0.857x1.286]ZpwhzmyivskaH5M1X7ozaQ==[/tex]和[tex=0.786x1.286]sgM90Q/VISKeSqiI8AMXRw==[/tex]都是域[tex=0.786x1.286]BlkXDnmzWHxe4M6E9LlofQ==[/tex]上的线性空间（[tex=0.857x1.286]ZpwhzmyivskaH5M1X7ozaQ==[/tex]和[tex=0.786x1.286]sgM90Q/VISKeSqiI8AMXRw==[/tex]都不必是有限维的），[tex=0.929x1.0]9ZOFmxCSrFOtuQaSWCydPg==[/tex]是[tex=0.857x1.286]ZpwhzmyivskaH5M1X7ozaQ==[/tex]到[tex=0.786x1.286]sgM90Q/VISKeSqiI8AMXRw==[/tex]的一个线性映射，[tex=1.286x1.071]c5Cf4pRARaBipYntugL/3vKeBzcFZmpil4mkUJnj1jI=[/tex]是[tex=0.929x1.0]9ZOFmxCSrFOtuQaSWCydPg==[/tex]的对偶映射。证明：若[tex=0.929x1.0]9ZOFmxCSrFOtuQaSWCydPg==[/tex]是满射，则[tex=1.286x1.071]c5Cf4pRARaBipYntugL/3vKeBzcFZmpil4mkUJnj1jI=[/tex]是单射。
2
取集合[tex=0.857x1.286]ZpwhzmyivskaH5M1X7ozaQ==[/tex]为实数域[tex=0.786x1.0]czmpOvTmaMgRl7StPBE3ig==[/tex]，数域为有理数域[tex=0.857x1.214]ChdusW5rAupjge6v/DGHRA==[/tex]。集合[tex=0.857x1.286]ZpwhzmyivskaH5M1X7ozaQ==[/tex]的向量加法规定为实数的加法，纯量与向量的乘法规定为有理数与实数的乘法，则[tex=0.857x1.286]ZpwhzmyivskaH5M1X7ozaQ==[/tex]成为有理数域[tex=0.857x1.214]ChdusW5rAupjge6v/DGHRA==[/tex]上的线性空间。证明：在线性空间[tex=0.857x1.286]ZpwhzmyivskaH5M1X7ozaQ==[/tex]中，实数1与[tex=0.643x0.786]hlJJ6/DUY+n2/FE6M2JdRA==[/tex]线性无关的充分必要条件是，[tex=0.643x0.786]hlJJ6/DUY+n2/FE6M2JdRA==[/tex]为无理数。
3
设[tex=3.143x1.286]W9AF7fR1WqhMGTtsETMyY3weIJPad4SLTOq9KrvSIVc=[/tex]是向量空间[tex=0.857x1.286]ZpwhzmyivskaH5M1X7ozaQ==[/tex]的子空间，证明[tex=3.786x1.286]CodcfJC5l11u2QacfGTUTvhCqQJBmY2d8mdiZi97mAE=[/tex]也是[tex=0.857x1.286]ZpwhzmyivskaH5M1X7ozaQ==[/tex]的子空间 .
4
在[tex=1.357x1.214]vz7Ug0CxQp8PofK4xjPihA==[/tex]中，由下述两个函数，[tex=5.571x1.214]7QH5fTmEi4pE0smvFbemuBkHq53gXD81gdRpWpfSSvY=[/tex]，[tex=6.5x1.214]j9+rBXoe53vYyAolbX8Cf++uorkH57pftUmONcQ2eWiajisWOsocHvZfJYRKLT62[/tex]，生成的子空间记作[tex=0.857x1.286]ZpwhzmyivskaH5M1X7ozaQ==[/tex]，其中[tex=2.214x1.286]Cn/gGa/6RUdNxg+ytprZ1g==[/tex]。[tex=0.857x1.286]O3TwAlpSL8Dofwuk3GRMyA==[/tex]，[tex=0.929x1.286]r+MGZrdXs5F5eGzFcjuRAQ==[/tex]是不是[tex=0.857x1.286]ZpwhzmyivskaH5M1X7ozaQ==[/tex]的一个基？求导数[tex=0.857x1.0]m2DKAQtGuc1DyN3zyNlILg==[/tex]是不是[tex=0.857x1.286]ZpwhzmyivskaH5M1X7ozaQ==[/tex]上的一个线性变换？如果是，求[tex=0.857x1.0]m2DKAQtGuc1DyN3zyNlILg==[/tex]在[tex=0.857x1.286]ZpwhzmyivskaH5M1X7ozaQ==[/tex]的一个基[tex=0.857x1.286]O3TwAlpSL8Dofwuk3GRMyA==[/tex]，[tex=0.929x1.286]r+MGZrdXs5F5eGzFcjuRAQ==[/tex]下的矩阵。