复旦大学谢启鸿高等代数每周一题[2021A08]参考解答

2021-11-20 18:08 作者:CharlesMa0606 0人读过 | 我要投稿

本文是本人给出的2021年复旦大学谢启鸿高等代数的每周一题[问题2021A08]的解答

题目来自于复旦大学谢启鸿教授在他的博客提供的每周一题练习

（链接：https://www.cnblogs.com/torsor/p/15329047.html）

本文仅供学习交流，如有错误恳请指正！

[问题2021A08]设K为数域，p个线性无关的m维列向量 $%5C%7B%5Calpha_1%2C%5Calpha_2%2C%5Ccdots%2C%5Calpha_p%5C%7D%5Csubseteq%20K%5Em%5C%20$ ,q个线性无关的n维列向量 $%5C%7B%5Cbeta_1%2C%5Cbeta_2%2C%5Ccdots%2C%5Cbeta_q%5C%7D%5Csubseteq%20K%5En$ .

证明： $%5C%7B%5Calpha_i%5Ccdot%5Cbeta_j%5E%5Cprime%2C1%5Cle%20i%5Cle%20p%2C1%5Cle%20j%5Cle%20q%5C%7D$ 是pq个线性无关的 $m%5Ctimes%20n$ 阶矩阵.

解不妨设：

$c_%7B11%7D%5Calpha_1%5Cbeta_1%5E%5Cprime%2Bc_%7B12%7D%5Calpha_1%5Cbeta_2%5E%5Cprime%2B%5Ccdots%2Bc_%7Bpq%7D%5Calpha_p%5Cbeta_q%5E%5Cprime%3DO$

提出公因式 $%5Calpha_i$ ，有：

$%5Calpha_1%5Cleft(c_%7B11%7D%5Cbeta_1%5E%5Cprime%2B%5Ccdots%2Bc_%7B1q%7D%5Cbeta_q%5E%5Cprime%5Cright)%2B%5Calpha_2%5Cleft(c_%7B21%7D%5Cbeta_1%5E%5Cprime%2B%5Ccdots%2Bc_%7B2q%7D%5Cbeta_q%5E%5Cprime%5Cright)%2B%5Ccdots%2B%5Calpha_p%5Cleft(c_%7Bp1%7D%5Cbeta_1%5E%5Cprime%2B%5Ccdots%2Bc_%7Bpq%7D%5Cbeta_q%5E%5Cprime%5Cright)%3DO$

考虑左边的第k列，有：

$%5Calpha_1%5Cleft(c_%7B11%7D%5Cbeta_1%5E%5Cprime%2B%5Ccdots%2Bc_%7B1q%7D%5Cbeta_q%5E%5Cprime%5Cright)_k%2B%5Calpha_2%5Cleft(c_%7B21%7D%5Cbeta_1%5E%5Cprime%2B%5Ccdots%2Bc_%7B2q%7D%5Cbeta_q%5E%5Cprime%5Cright)_k%2B%5Ccdots%2B%5Calpha_p%5Cleft(c_%7Bp1%7D%5Cbeta_1%5E%5Cprime%2B%5Ccdots%2Bc_%7Bpq%7D%5Cbeta_q%5E%5Cprime%5Cright)_k%3D0$

由 $%5Calpha_1%2C%5Calpha_2%2C%5Ccdots%2C%5Calpha_p$ 线性无关以及 $%5Cbeta_1%2C%5Cbeta_2%2C%5Ccdots%2C%5Cbeta_q$ 线性无关，可以得到，

$%5Cleft(c_%7B11%7D%5Cbeta_1%5E%5Cprime%2B%5Ccdots%2Bc_%7B1q%7D%5Cbeta_q%5E%5Cprime%5Cright)_k%3D%5Ccdots%3D%5Cleft(c_%7Bp1%7D%5Cbeta_1%5E%5Cprime%2B%5Ccdots%2Bc_%7Bpq%7D%5Cbeta_q%5E%5Cprime%5Cright)_k%3D0$