设 [tex=5.5x1.0]xJr2ny42kcAcTeyzkoXuGjF5Eh4v7S3Fd052Z/6/FnJsMrtWjHMkU+h8EqXqjCNU[/tex]是一组 [tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]维向量,证明它们线性无关的充分必要条件是任一[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]维向 量组都可由它们线性表示.
设 [tex=5.429x1.0]A4jSygN0882R6SV3eve5dyhKA/5f6aU7CkpCJuZGXtnX3JN5UgYMY49TyYJqmicpE4fKknnRFRUXSuEpXtwHwA==[/tex] 为一组 [tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex] 维单位向量,对于任意 [tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex] 维向量 [tex=8.214x1.571]qulE2au0sCsC2RUF6/a3J4K4vFhtBtwyIL/WXwRKgD35Z0S8+bOENBhul0XlCT1JlFeKgbY0a4RXmQxSBn0Pek5HBme3NwUaMbNQ9d2Fkgk=[/tex] 则 有 [tex=10.786x1.214]qulE2au0sCsC2RUF6/a3J3OWQKcPZzhKGdtZnJnk64k3Qaxu7walamxXkvmpThYuRCtUDfteaOCp7gbax+BhllE8o0O4a2ov2gKzXKyYqRmuQZRh8ZI4K1GHoyVyQaXHId8ctFaEceHyZB5+8rxNtg==[/tex], 即任意[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex] 维向量都可由单位向量线性表示. 先证必要性. 因 [tex=5.857x1.0]qulE2au0sCsC2RUF6/a3Jzvc72/Ojlqjke3MkB3mi/ocVm1zbQVoiZ3n4s0Tg+DcUYb3pwbmP3EXwTF2glwNookvPqLjZ8WM2JwV9JFjusoGV3okOYIXPKl6YqwjmmTB[/tex]线性无关 且 [tex=5.786x1.0]t2ggWTTv2XPLQkBXnhzUm5zvgFR1nlU7rFaEFtE3qjh/Cvezs7G66sP9fHqHuGgAk8xpMlXc7070IEywyVhpKEbw5eIbn4QkpYho8uCbCA0=[/tex] 能由单位向量线性表示,所以有 [tex=14.286x5.214]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[/tex][tex=4.071x6.357]075gCzZzsMRb6HYXYk9X91EhOb660MC0k9pqJ/fAvMzkki0oCVi04iTth2YlblksKdZHVeGPDaPqF/k3jwKuXPjWhxjhyvndKM45771qsAv9OJgnUEupwTAS9qTXYA1c2ga/FUidG5K/mE4qelU8UWoo/Kz00P91ltatRxmt6crQoyESpOeSJTHLLVp/oOKeNSoffI10AOkEa0jQR0Sk18zIK6zM3gjgLIzvORQsBOo=[/tex][tex=11.857x5.786]ZGTtUfVHiJDAU4i+qE4bMu3ve1zbxkNBcWYXhn1Wy7JDgdUdCuVos8g/v6zWweZKd+YkOxUKP2xelaM1glBTpnYG0hZEWL3Hwj/1Bn0svNeNT+6E8p+XCcc8GQ/xAEtmarqDfmZIVxp7Js5QLsOnM3PL078Y9BTpu3fuKx9Ap5Rjrml8lB0OiI0C+8jNOjfjk/qik/aDsGHUGdx9pvIc0FSmYVObBoA8Zeh/uAm9a9Z0KxQL+EQTRXHnqUJ48xxKeVyGS71VT4jtRIJV4xllfw==[/tex][tex=3.857x6.357]075gCzZzsMRb6HYXYk9X91EhOb660MC0k9pqJ/fAvMyS4FzKN+M4RiAXIzQ5Lnu3+v3nZZihYTaVAOpbW7owBZqNOkT1j7c6KJn1aSFWrBw4XJzvG/7DgSBw/zqxGFOQjKFsfjL4Ke9gbftg5IYsiweYhX3CCV953ohOrKz81Y5oml7VkBlNfiomN9SsfnzRxSomHmvSjybg0rsyNklT31uU8S8hiK4OWkOI8swFB7ljmBNYTmrB8w2NS2drkjFU[/tex] 两边取行列式,得 [tex=16.857x6.357]Uyz5s0rmQIddjb5Jc2T/YSnM9gnJHvPQR+KdltAeaSTt+X3Kc5hfysVOFXFeP2lj9wdSPn03Ai3364hII6ibZg1tN3lfODLtva2EvOnz09HtYebeMfnnwmFav2bavEXe4XP+bGhrEalT5ghgjl+a5c5mjeHiob+GOXzjq03qx1BPPBfFAZDF3opBPGPrEyCauGU7lIatg5XVlTCl8pVCoxpn+cBAq9jwmOORS2ZYesuTEAb8zJkej27ySEH+A6uCSuVts8WtRdF2ZdwXIlbUp2lhspLtB7Jljj2oZW+AitHWf4uYWwPGVvqOZWeON7kHt1L3tUMQlnxCPM4UJuXMPDlLl3gKogO+9xmAldcngkWGnaGOeghzGAeQ5RVjCjfS7VzBo6MtPbLDDtvXtBjhbcagS3TYgCughcDzSUwm/JQeNCjQgSt64IQJbJqcu1HO1SO7m0YVhR3b5VZJCuAP0AuIooEdmwk3W71Eo9KasuFMM8fXxQIXs3FYchBuJ8M/2A+AZg8ndxaCKFDF5YHfz7YBPBUJbsvsnnYFIYBF4CN4CdZsxpeKZr7ZvgqJfrMR3oXZtUQUtDuiFeESw+pLXvHKjaMS9zgzMYvPABW7SXyUa79Xgyq7LDtUoMcpq1BZRXF0fdPhRAKbf8NmvTnNl6GOxJvgLbxzi/9v3IqA3sCLNkDo5XFfrPDmI2AZ48foNLy9dLNkfKtCFM3tVYimqQ==[/tex][tex=4.857x6.357]Uyz5s0rmQIddjb5Jc2T/YSnM9gnJHvPQR+KdltAeaSTt+X3Kc5hfysVOFXFeP2lj9wdSPn03Ai3364hII6ibZg1tN3lfODLtva2EvOnz09HtYebeMfnnwmFav2bavEXe4XP+bGhrEalT5ghgjl+a5c5mjeHiob+GOXzjq03qx1BPPBfFAZDF3opBPGPrEyCauGU7lIatg5XVlTCl8pVCo52BWPnqCEN8AqxSrOYbwUo=[/tex], 得 [tex=11.714x5.357]Uyz5s0rmQIddjb5Jc2T/YY+qj2JqfG81d64SeetpT3DHJFlnaDYKdAeOCtnb9RHS+Baedb5eGVmo5LO4McYKmmEtv3OrTKtIqWblpdhZCfYn027cvYDzAcWndqpu74ZzHb6gL1phwNzvD04aeo9XOwCQcXlHJXMN67ZwD/jt8Et0h2qJCUmFlE7iKp4N3ZZxb0ICdXXko9gpTbUiD6BtAH5ZPCUcL7QX6hIo24oIDUWWa1Vd7haJhwaYGNhlqxoN1irJKS+dzcBUA07p9H4m2w==[/tex][tex=14.571x5.357]GkYFepGm/q/vepJY89S2Y6oJJ7bxTKAGihPmMYZCs29URRt6x1By35zgWy3ZZW3Yh+Hu9uSnLiaOFg4McB2Hx88m63i+gGdco9XIfmIvNe8LI0/vTQoW8ZGJHYoaADQ/rhDRo4OZoT6PeUcy3ESEyHlzALUjJuP/7UaWPf6+9FdqdnhQgIFMifXMM4laybcEqtEn/1KcbS6/4+Lp3kPXqS9hX/vD5WCwkqZhEGrdsdgDXJTBLE3eeWZMjoU/8iW9uWsxZQckcNIWiqEGRMY3gJbCGVkAelb7UIVUfSRlVf8=[/tex]由 [tex=9.857x6.357]075gCzZzsMRb6HYXYk9X91EhOb660MC0k9pqJ/fAvMzkki0oCVi04iTth2YlblksKdZHVeGPDaPqF/k3jwKuXPjWhxjhyvndKM45771qsAv9OJgnUEupwTAS9qTXYA1c2ga/FUidG5K/mE4qelU8UWoo/Kz00P91ltatRxmt6crQoyESpOeSJTHLLVp/oOKeNSoffI10AOkEa0jQR0Sk1/2JuCguWCSiM67YC79taamY8PjN3I1rScQOSco0WBJIGUIDmvpo+ts8ETL/TIn4RQBk1bdQDNJEFegbxH3vOJLHZWL9/GyZKOFlMfI11S+K0Lk0d32oOrHZj1XlxzZy8dK74XuskscaQjV8Dm/1TP93ze03bKqriJj4BMPY+MIq2wmVik3FaYWdaJE9dqAZjv3tPy3pK1vlnDP0EqdMWEi18dHxm7sK679NTsY0tqxqsu5C38UsWDtTO/EtCabyK25bYuo9U9LumXctqu6H5Y4=[/tex]得 [tex=11.0x6.357]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[/tex]即 [tex=5.214x1.0]uh36dzsJfggBlWeE2D47K5OTAySr97q/rOeyvmSNz3UMluRSO/VXWEegTrLjLtU8/hGQFzmamzNHMhLEym7GiA2NXFBJewHe2BaQP7Y1pNZPcVwn3MV9cyC5wVNYNBP31eMXJpbddrlthu8TjUFMOg==[/tex]都能由 [tex=5.857x1.0]qulE2au0sCsC2RUF6/a3Jzvc72/Ojlqjke3MkB3mi/ocVm1zbQVoiZ3n4s0Tg+DcUYb3pwbmP3EXwTF2glwNookvPqLjZ8WM2JwV9JFjusoGV3okOYIXPKl6YqwjmmTB[/tex] 线性表示，因为任意 [tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex] 维向量能由单位向量线性表 示 故任意 [tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]维向量都可以由 [tex=5.857x1.0]qulE2au0sCsC2RUF6/a3Jzvc72/Ojlqjke3MkB3mi/ocVm1zbQVoiZ3n4s0Tg+DcUYb3pwbmP3EXwTF2glwNookvPqLjZ8WM2JwV9JFjusoGV3okOYIXPKl6YqwjmmTB[/tex] 线性表示. 再证充分性 已知任意 [tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex] 维向量都可由 [tex=5.857x1.0]qulE2au0sCsC2RUF6/a3Jzvc72/Ojlqjke3MkB3mi/ocVm1zbQVoiZ3n4s0Tg+DcUYb3pwbmP3EXwTF2glwNookvPqLjZ8WM2JwV9JFjusoGV3okOYIXPKl6YqwjmmTB[/tex] 线性表示 则单位向量组 [tex=5.214x1.0]uh36dzsJfggBlWeE2D47K5OTAySr97q/rOeyvmSNz3UMluRSO/VXWEegTrLjLtU8/hGQFzmamzNHMhLEym7GiA2NXFBJewHe2BaQP7Y1pNZPcVwn3MV9cyC5wVNYNBP31eMXJpbddrlthu8TjUFMOg==[/tex] 可由 [tex=5.857x1.0]qulE2au0sCsC2RUF6/a3Jzvc72/Ojlqjke3MkB3mi/ocVm1zbQVoiZ3n4s0Tg+DcUYb3pwbmP3EXwTF2glwNookvPqLjZ8WM2JwV9JFjusoGV3okOYIXPKl6YqwjmmTB[/tex] 线性表示 已由上一题知 [tex=5.857x1.0]qulE2au0sCsC2RUF6/a3Jzvc72/Ojlqjke3MkB3mi/ocVm1zbQVoiZ3n4s0Tg+DcUYb3pwbmP3EXwTF2glwNookvPqLjZ8WM2JwV9JFjusoGV3okOYIXPKl6YqwjmmTB[/tex] 线性无关
举一反三
- 设 [tex=6.357x1.0]qulE2au0sCsC2RUF6/a3Jzvc72/Ojlqjke3MkB3mi/ocVm1zbQVoiZ3n4s0Tg+DcUYb3pwbmP3EXwTF2glwNookvPqLjZ8WM2JwV9JFjusoGV3okOYIXPKl6YqwjmmTB[/tex]是一组 [tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex] 维向量,证明: [tex=6.357x1.0]qulE2au0sCsC2RUF6/a3Jzvc72/Ojlqjke3MkB3mi/ocVm1zbQVoiZ3n4s0Tg+DcUYb3pwbmP3EXwTF2glwNookvPqLjZ8WM2JwV9JFjusoGV3okOYIXPKl6YqwjmmTB[/tex] 线性无关的充分必要条件是任一 [tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex] 维向量都可被它们线性表出.
- 证明：在[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]维向量空间[tex=1.429x1.0]id8CqLD3sKgZOEL0mYn1xA==[/tex]中，任一线性无关的向量组所含向量的个数不超过[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]。
- 设[tex=1.0x1.0]uNeXgLsfwRCqSCLKXiHNMw==[/tex]，[tex=1.0x1.0]i+smJv8fQxqX02gk5G5Gig==[/tex]，[tex=1.286x0.786]lRSLJav0cvc1uYdx/9plcw==[/tex]，[tex=1.071x1.0]T16p9QhVJeZysiw1Vc9goA==[/tex] 是一组[tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex]维向量，证明它们线性无关的充分必要条件是: 任一 [tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex]维向量都可由它们线性表示.
- 证明：每一个[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]维线性空间都可以表示成[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]个一维子空间的直和。
- 证明：在[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]维向量空间[tex=1.429x1.0]id8CqLD3sKgZOEL0mYn1xA==[/tex]中，[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]个向量[tex=4.786x1.0]qulE2au0sCsC2RUF6/a3Jz+bwfwPeEraqg452x6rdusBeuEbPyHyGf2YX5cxbX/RrQfAZgEO5fckVGjHK61J8A==[/tex]线性无关当且仅当[tex=1.429x1.0]id8CqLD3sKgZOEL0mYn1xA==[/tex]中任一向量都可以由[tex=4.786x1.0]qulE2au0sCsC2RUF6/a3Jz+bwfwPeEraqg452x6rdusBeuEbPyHyGf2YX5cxbX/RrQfAZgEO5fckVGjHK61J8A==[/tex]线性表出。
内容
- 0
设[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]是正整数。证明:在任意一组[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]个连续的正整数中恰好有1个被[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]整除。
- 1
设n维向量[tex=5.929x1.0]t2ggWTTv2XPLQkBXnhzUm5zvgFR1nlU7rFaEFtE3qjhJV81uDcStjXLY0p3pcOuvFFxrmN66jfvz5qcD3RdxPMYdcSh9+n4PGarPZ3aTZ9E=[/tex].证明:它们线性无关的充分必要条件是任一n维向量都可由它们线性表示.
- 2
需要用多少字节来编码[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]位的数据，其中[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]等于7
- 3
设[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是一个[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]级矩阵，证明： １） [tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是反对称矩阵当且仅当对任一[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]维向量[tex=0.857x1.0]N7iCrOsS+NNEUUlnsYCi1g==[/tex]，有[tex=4.0x1.143]rLVONmXxLnhl8YaM4UacI9oY4xHCd5UxvQ2cXFY3Iyc=[/tex]； ２） 如果[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是对称矩阵，且对任一个[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]维向量[tex=0.857x1.0]N7iCrOsS+NNEUUlnsYCi1g==[/tex] ，有[tex=4.0x1.143]rLVONmXxLnhl8YaM4UacI9oY4xHCd5UxvQ2cXFY3Iyc=[/tex]，那么[tex=2.071x1.0]P1sZi5Sh6qXV+PX80otJJg==[/tex]．
- 4
设[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex] 是一个 [tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex] 级矩阵,证明 如果 [tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex] 是对称矩阵,且对任一个[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex] 维向量 [tex=0.857x1.0]N7iCrOsS+NNEUUlnsYCi1g==[/tex] 有[tex=4.0x1.143]rLVONmXxLnhl8YaM4UacI9oY4xHCd5UxvQ2cXFY3Iyc=[/tex] 那么[tex=3.429x1.0]gDaSCeRv2nAY2ZKE6tr+4g==[/tex]