[tex=2.714x1.071]nCe3KjbN5N38t1r/7/3V+g==[/tex]矩阵[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的秩为[tex=0.5x0.786]Tg0I1PUwmDJ7uXa9+yiYMA==[/tex]，则有[tex=2.571x1.071]O8MXxCyH82iQBjE8tUx7+Q==[/tex]的列满秩矩阵[tex=0.643x1.0]Ft8KOBgb78fnKY0jEt4Rsg==[/tex]和[tex=2.286x1.071]zXLE9Sy0lPfi6rhDrfbNLg==[/tex]的行满秩矩阵[tex=0.857x1.214]ChdusW5rAupjge6v/DGHRA==[/tex]，使[tex=3.0x1.214]InSRQVNnaVoKAJCKaKaLlw==[/tex]．

2022-06-09

[tex=2.714x1.071]nCe3KjbN5N38t1r/7/3V+g==[/tex]矩阵[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的秩为[tex=0.5x0.786]Tg0I1PUwmDJ7uXa9+yiYMA==[/tex]，则有[tex=2.571x1.071]O8MXxCyH82iQBjE8tUx7+Q==[/tex]的列满秩矩阵[tex=0.643x1.0]Ft8KOBgb78fnKY0jEt4Rsg==[/tex]和[tex=2.286x1.071]zXLE9Sy0lPfi6rhDrfbNLg==[/tex]的行满秩矩阵[tex=0.857x1.214]ChdusW5rAupjge6v/DGHRA==[/tex]，使[tex=3.0x1.214]InSRQVNnaVoKAJCKaKaLlw==[/tex]．

答案：

证: 由秩[tex=3.571x1.357]J5v5rgot/ujghQtLg3sIVA==[/tex]，必存在[tex=3.0x1.071]cn3RMv8FmBk7eY3/E8VMLg==[/tex]可逆矩阵[tex=0.643x1.0]J+LW/0i6Fe+lWEmBUgT8zg==[/tex] 与[tex=2.429x1.071]jLyhB8GAUqIuDKvKM/p5zw==[/tex]可逆矩阵[tex=0.786x1.0]JTRtgqQ00R3dUQzwS4iwbg==[/tex]，使[tex=20.429x2.786]XSLibGAdw6XRuf53orEVSw+BFlFoMfWuYR3is3KpCwpq+qgGSxJag34jwSFlVTY8rEjS2ZP9lj4zw6JC0DgnY/2WlLhFPt4xFe69U7bmht2ncoiniWUfwUCohVUnwlByLEyL7deKTnfPDCJkXKyK59E5jiDB/K2EATTRFpbtHc0zdoJYiElHkvDPghSRZOpfYG7bSlWIv855jI6IqqIsfPCs4kdRxc60Lv8YBlBKRvc=[/tex]记，[tex=19.0x2.786]EqplGG4SyCdA5YxRY0Dkk6WTY3I80tG87JldN5iwFztDyJOjSiYRv+S7vdey+Ql9j3JcYqNfhpqIUuWrfKgp8l8GYUYGpbUjPGKoTpE/uyG+KvyNvrhwtCm6sv6z+rbgCLSpxzfpCOz7O/Gi8VM8MjHs6vsafwl33Ba6NH+xuPH8O/MNbizjy9pA5F3nBQGnwlGtEfQQBydZ80aOlI/DyawjrtluBnT5nrDYkmPhK0IEv5u0QZBioqcby9g8rNwh[/tex]其中[tex=3.286x1.214]bvbBqfE4C1c7dY3C5N2qEQ==[/tex]均为[tex=0.5x0.786]Tg0I1PUwmDJ7uXa9+yiYMA==[/tex]级方阵，则[tex=30.714x2.786]hB8sGfF8hpZRTKdvt1J/eJHkBpzR0eh6go4KeAVFpwE6dPDuknmWUu7RTriJaFCOuPYDZmJ+MtMys/cK69APN8Vkrb8ktM67dneAH+silnmyhUCYmPlt0cEkGTvqwniXX5h2fbN+tehM+6IoLO/90Jot1eOkINOjL65orBlQOPznlKo8F+7AuU9CN2+MPw+ByXtYlRZFDoBbTTTrwWIsNv7m9+yS8ylIOY+p7vlX7/OzEnJaTWhQj+JRBajlPR/JPX7iMM1nYYziGST9vw9A4YOOb1INLVS/SrrGZQvp63CgZeSOoVF7ZeZIIJa0kgfmcl4zC091x7VYJENiKnvMBMbff5wGmOPAsZYv/JVuVPxA3MGDI9VtwLOKDYaUN0HJJMylHkzF/jNQSCmtw0YhaaUBjkrIXVK1/LGmXUuC7+RlgUwbOTwg1sYYk4R4jmS31Xkxmw6YMxyIg0afMA2IfM3DPKCGQe2ktKUyukuwnig=[/tex]  [tex=22.857x2.786]ThIgXOUyLYK/6IHx89IH/u6M9ILbPXpLST78v2UPgTIz3C8TCqx1BhZARwWCwt+d4pdCzXAe6o1zbtaV+JLMR/DWPTmhAkz3f8qLDEk41rr7LVC6JeFVxzcaRasjpCrAZje3BNZREJknJWo5rnexPa2BT59/9XuLSxWO0tCKw0iMjLRJjUx5+/gr8Ag/DhawnJ0DtytDMeQOx3aUgAl6jddj8S1C7qTzFxAUvD+XoBtZ8t1PBSC2j4Y+23wQ7twO6e5AjEOaoynSty4gXbNYIU4cq1ZfJpYJ6fBrEvuvHrZIiq+XNeuHHDwojvLGI0qK[/tex]其中[tex=5.0x2.786]BJ01inipONhLpj7h0EHN8u/LRLtPSkUrXGSP2ozHHyrK8nZ/dOTakiFUva2jCcZ+qWLW+2uHyr+8DkjpyZgYzg==[/tex]为非奇异矩阵 [tex=1.714x1.214]VX2BWD5NU6xP9ml/sYb17w==[/tex]的前 [tex=0.5x0.786]Tg0I1PUwmDJ7uXa9+yiYMA==[/tex]列构成的列满秩矩阵，而 [tex=1.857x1.214]RhFHWM9PRQyA/oCbXEmFfg==[/tex][tex=3.786x1.357]0t3CZY4ZLFFG2PrE+ZnfUxmCBtzewkfwJ11hLd+Rm/g=[/tex]为非奇异矩阵[tex=1.786x1.214]ppUUX8qKX3+NEpxnDAjMVA==[/tex]的前[tex=0.5x0.786]Tg0I1PUwmDJ7uXa9+yiYMA==[/tex]行构成的行满秩矩阵，它们即是欲证结论 中的两个矩阵[tex=0.643x1.0]Ft8KOBgb78fnKY0jEt4Rsg==[/tex] 和[tex=0.857x1.214]ChdusW5rAupjge6v/DGHRA==[/tex] ，得证．

举一反三

内容

0
设[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是数域[tex=0.857x1.0]eMszuSG5by5UfRZVROYp5A==[/tex]上[tex=2.357x1.071]QArHY/B/HPaeI4OFb8f5sA==[/tex]矩阵，证明：[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的秩为[tex=0.5x0.786]Tg0I1PUwmDJ7uXa9+yiYMA==[/tex]当且仅当存在数域[tex=0.857x1.0]eMszuSG5by5UfRZVROYp5A==[/tex]上[tex=2.143x1.071]Rtxts52p4rdDLdHA6Amz2w==[/tex]列满秩矩阵[tex=0.786x1.0]ri6gmnf1+J9dGqG5/1sV6A==[/tex]与[tex=2.286x1.071]Ef3ubbPzNaK2nOISNr/qww==[/tex]行满秩矩阵[tex=0.714x1.0]J/aA9EEo0KmJFnWWfX7LmQ==[/tex]，使得[tex=3.071x1.0]Mb/+JoHmaaZzuBXR7KsjSg==[/tex].
1
设[tex=2.357x1.071]0nq0b1fEFW/AV6tuzNPMsA==[/tex]矩阵[tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex]的秩为[tex=3.857x1.357]qXpz7Vckr31FXDcDaN2KhQ==[/tex]证明：存在[tex=2.143x1.071]rEyV9COcOO6bGHvQoT8WZA==[/tex]列满秩矩阵[tex=1.0x1.214]szVnMPaRHLo99rUmmmexUw==[/tex]与[tex=2.286x1.071]qxUBJkw5pHPFqpR4rHoDwQ==[/tex]行满秩矩阵[tex=1.143x1.214]33XDFahjdy1KHCYObeGaBg==[/tex]，使得[tex=3.714x1.214]M4XIxclCkO36b8kKPZybyg==[/tex].
2
设[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是[tex=2.357x1.071]QArHY/B/HPaeI4OFb8f5sA==[/tex]矩阵.证明：[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是行满秩矩阵当且仅当存在[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]级可逆矩阵[tex=0.857x1.214]ChdusW5rAupjge6v/DGHRA==[/tex]，使得[tex=4.643x1.357]RqsCL9/WuCBwDvrRjd44OA==[/tex].
3
设 [tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex] 是 [tex=2.714x1.071]Xa6YzCV9VTlW9p4lLOpktw==[/tex] 矩阵, 求证:(1) 若 [tex=3.357x1.357]a7qAbmiLBFc3iSK33Jqg/g==[/tex], 即 [tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex] 是列满秩阵, 则必存在秩等于 [tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex] 的 [tex=2.714x1.071]/nWgWZWXmeNCPcwAggrwNg==[/tex]矩阵 [tex=0.786x1.0]ri6gmnf1+J9dGqG5/1sV6A==[/tex]，使 [tex=3.214x1.214]qFuOqB/J5YwAsAHomJYPyw==[/tex];(2) 若 [tex=3.643x1.357]NrKc/6u1O1LFs1JAil+zeg==[/tex], 即 [tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex] 是行满秩阵, 则必存在秩等于 [tex=0.929x0.786]VF0GLe2VBE/4VKNzpyOfFg==[/tex] 的 [tex=2.714x1.071]/nWgWZWXmeNCPcwAggrwNg==[/tex] 矩阵 [tex=0.714x1.0]YiLkHgl7MlxE+QjUplQUKA==[/tex], 使 [tex=3.643x1.214]zyEHVZjYzQ8SDWBlfQFbZA==[/tex]
4
设[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是[tex=2.357x1.071]QArHY/B/HPaeI4OFb8f5sA==[/tex]矩阵.证明：[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是列满秩矩阵当且仅当存在[tex=0.5x0.786]ICKY+F5VdoSQrRn/wUUOyw==[/tex]级可逆矩阵[tex=0.643x1.0]Ft8KOBgb78fnKY0jEt4Rsg==[/tex]，使得[tex=5.714x2.786]GOgGvXf8fpWrP7XGIdsj8/NUqPJl/9cX6USGFSTASsQOKHl4580Mpt6T6Dn85xpMOGmjKwp5eZciA0U/RdGbPg==[/tex].