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復(fù)旦大學(xué)謝啟鴻高等代數(shù)每周一題[2021A08]參考解答

2021-11-20 18:08 作者:CharlesMa0606 0人讀過 | 我要投稿

本文是本人給出的2021年復(fù)旦大學(xué)謝啟鴻高等代數(shù)的每周一題[問題2021A08]的解答

題目來自于復(fù)旦大學(xué)謝啟鴻教授在他的博客提供的每周一題練習(xí)

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

本文僅供學(xué)習(xí)交流，如有錯誤懇請指正！

[問題2021A08]設(shè)K為數(shù)域，p個(gè)線性無關(guān)的m維列向量 $%5C%7B%5Calpha_1%2C%5Calpha_2%2C%5Ccdots%2C%5Calpha_p%5C%7D%5Csubseteq%20K%5Em%5C%20$ ,q個(gè)線性無關(guān)的n維列向量 $%5C%7B%5Cbeta_1%2C%5Cbeta_2%2C%5Ccdots%2C%5Cbeta_q%5C%7D%5Csubseteq%20K%5En$ .

證明： $%5C%7B%5Calpha_i%5Ccdot%5Cbeta_j%5E%5Cprime%2C1%5Cle%20i%5Cle%20p%2C1%5Cle%20j%5Cle%20q%5C%7D$ 是pq個(gè)線性無關(guān)的 $m%5Ctimes%20n$ 階矩陣.

解? 不妨設(shè)：

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

提出公因式 $%5Calpha_i$ ，有：

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

考慮左邊的第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 $%5Calpha_1%2C%5Calpha_2%2C%5Ccdots%2C%5Calpha_p$ 線性無關(guān)以及 $%5Cbeta_1%2C%5Cbeta_2%2C%5Ccdots%2C%5Cbeta_q$ 線性無關(guān)，可以得到，

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