线性代数模拟试卷
[font=宋体][size=14pt]一.[/size][/font][b][font=宋体][size=14pt]填空题[/size][/font][/b][b][size=14pt]([/size][/b][b][font=宋体][size=14pt]每小题[/size][/font][/b][b][size=14pt]3[/size][/b][b][font=宋体][size=14pt]分[/size][/font][/b][b][size=14pt],[/size][/b][b][font=宋体][size=14pt]满分[/size][/font][/b][b][size=14pt]30[/size][/b][b][font=宋体][size=14pt]分[/size][/font][/b][b][size=14pt])[/size][/b][size=14pt][/size][size=12pt]1.
[/size][font=宋体][size=12pt]设[/size][/font][size=12pt][img=107,24]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image002.gif[/img]
[/size][font=宋体][size=12pt]都是[/size][/font][size=12pt]4[/size][font=宋体][size=12pt]维列向量[/size][/font][size=12pt],[/size][font=宋体][size=12pt]且[/size][/font][size=12pt]4[/size][font=宋体][size=12pt]阶行列式[/size][/font][size=12pt][img=193,27]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image004.gif[/img][/size]
[font=宋体][size=12pt]则[/size][/font][size=12pt]4[/size][font=宋体][size=12pt]阶行列式[/size][/font][size=12pt][img=128,27]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image006.gif[/img]_______________-[/size]
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[size=12pt]2.
[/size][font=宋体][size=12pt]已知[/size][/font][size=12pt][img=63,24]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image008.gif[/img][/size][font=宋体][size=12pt]线性相关[/size][/font][size=12pt],[img=19,24]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image010.gif[/img][/size][font=宋体][size=12pt]不能由[/size][/font][size=12pt][img=40,23]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image012.gif[/img][/size][font=宋体][size=12pt]线性表示则[/size][/font][size=12pt][img=40,23]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image014.gif[/img][/size][font=宋体][size=12pt]线性[/size][/font][size=12pt]__________
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[size=12pt]3.
[/size][font=宋体][size=12pt]设[/size][/font][size=12pt][img=16,17]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image016.gif[/img][/size][font=宋体][size=12pt]是[/size][/font][size=12pt][img=40,15]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image018.gif[/img][/size][font=宋体][size=12pt]阶矩阵[/size][/font][size=12pt] ,,[img=16,17]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image020.gif[/img][/size][font=宋体][size=12pt]是[/size][/font][size=12pt][img=36,15]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image022.gif[/img][/size][font=宋体][size=12pt]阶矩阵[/size][/font][size=12pt],,[img=60,23]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image024.gif[/img],[/size][font=宋体][size=12pt]且[/size][/font][size=12pt][img=51,19]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image026.gif[/img],[/size][font=宋体][size=12pt]则[/size][/font][size=12pt][img=36,23]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image028.gif[/img][/size][font=宋体][size=12pt]的取值[/size][/font][size=12pt][/size]
[font=宋体][size=12pt]范围是[/size][/font][size=12pt]________________[/size]
[size=12pt]4.[/size][font=宋体][size=12pt]设[/size][/font][size=12pt][img=16,17]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image030.gif[/img][/size][font=宋体][size=12pt]是[/size][/font][size=12pt]4[img=12,13]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image032.gif[/img]3[/size][font=宋体][size=12pt]矩阵[/size][/font][size=12pt],[/size][font=宋体][size=12pt]且[/size][/font][size=12pt][img=16,17]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image033.gif[/img][/size][font=宋体][size=12pt]的秩[/size][/font][size=12pt][img=61,23]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image035.gif[/img][/size][font=宋体][size=12pt]且[/size][/font][size=12pt][img=111,75]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image037.gif[/img][/size][font=宋体][size=12pt]则[/size][/font][size=12pt][img=60,23]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image039.gif[/img]__________-[/size]
[size=12pt]5.[/size][font=宋体][size=12pt]设[/size][/font][size=12pt]0[/size][font=宋体][size=12pt]是矩阵[/size][/font][size=12pt][img=101,75]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image041.gif[/img][/size][font=宋体][size=12pt]的特征值[/size][/font][size=12pt],[/size][font=宋体][size=12pt]则[/size][/font][size=12pt][img=27,15]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image043.gif[/img]_____________-[/size]
[size=12pt]6.[/size][font=宋体][size=12pt]设[/size][/font][size=12pt][img=312,26]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image045.gif[/img][/size][font=宋体][size=12pt]是正定二次型[/size][/font][size=12pt],[/size]
[font=宋体][size=12pt]则[/size][/font][size=12pt][img=16,22]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image047.gif[/img][/size][font=宋体][size=12pt]的取值区间为[/size][/font][u][size=12pt]
[/size][/u]
[size=12pt]7.[/size][font=宋体][size=12pt]矩阵[/size][/font][size=12pt][img=121,75]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image049.gif[/img][/size][font=宋体][size=12pt]对应的二次型是[/size][/font][size=12pt]_______________[u][/u][/size]
[size=12pt]8. [/size][font=宋体][size=12pt]设[/size][/font][size=12pt][/size]
[size=12pt][img=145,75]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image051.gif[/img][/size][font=宋体][size=12pt]相似于对角阵[/size][/font][size=12pt][img=91,75]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image053.gif[/img],[/size][font=宋体][size=12pt]则[/size][/font][size=12pt][img=27,15]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image055.gif[/img][u]
[/u][/size]
[size=12pt]9.[/size][font=宋体][size=12pt]设[/size][/font][size=12pt][img=16,17]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image056.gif[/img][/size][font=宋体][size=12pt]为[/size][/font][size=12pt]3[/size][font=宋体][size=12pt]阶方阵[/size][/font][size=12pt],[img=20,20]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image058.gif[/img][/size][font=宋体][size=12pt]为伴随矩阵[/size][/font][size=12pt],[img=48,41]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image060.gif[/img],[/size][font=宋体][size=12pt]则[/size][/font][size=12pt][img=100,56]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image062.gif[/img]=___________[u][/u][/size]
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10.[/size][font=宋体][size=12pt]设[/size][/font][size=12pt][img=127,75]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image064.gif[/img][/size][font=宋体][size=12pt]是不可逆矩阵[/size][/font][size=12pt],[/size][font=宋体][size=12pt]则[/size][/font][size=12pt][img=27,15]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image066.gif[/img]____________[/size]
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[font=宋体][size=12pt]二.
[/size][/font][size=12pt](8[/size][font=宋体][size=12pt]分[/size][/font][size=12pt])[/size][font=宋体][size=12pt]计算行列式[/size][/font][size=12pt][/size]
[size=12pt][img=168,96]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image068.gif[/img][/size]
[font=宋体][size=12pt]三[/size][/font][size=12pt].(8[/size][font=宋体][size=12pt]分[/size][/font][size=12pt]) [/size][font=宋体][size=12pt]三阶方阵[/size][/font][size=12pt][img=32,21]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image070.gif[/img][/size][font=宋体][size=12pt]满足关系式[/size][/font][size=12pt]:[img=117,20]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image072.gif[/img],[/size][font=宋体][size=12pt]且[/size][/font][size=12pt]
[img=101,75]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image074.gif[/img],[/size][font=宋体][size=12pt]求[/size][/font][size=12pt][img=16,17]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image076.gif[/img][/size]
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[font=宋体][size=12pt]四[/size][/font][size=12pt].(10[/size][font=宋体][size=12pt]分[/size][/font][size=12pt])[/size][font=宋体][size=12pt]设[/size][/font][size=12pt][/size]
[size=12pt][img=395,53]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image078.gif[/img]
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[font=宋体][size=12pt]求向量组的秩及其一个极大无关组[/size][/font][size=12pt].[/size]
[size=12pt] [/size] [font=宋体][size=12pt]五[/size][/font][size=12pt]. (12[/size][font=宋体][size=12pt]分[/size][/font][size=12pt])[/size][font=宋体][size=12pt]问常数[/size][/font][size=12pt][img=13,19]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image080.gif[/img][/size][font=宋体][size=12pt]取何值时[/size][/font][size=12pt], [/size][font=宋体][size=12pt]方程组[/size][/font][size=12pt][/size]
[size=12pt][img=220,75]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image082.gif[/img][/size]
[font=宋体][size=12pt]无解[/size][/font][size=12pt],[/size][font=宋体][size=12pt]有唯一解[/size][/font][size=12pt],[/size][font=宋体][size=12pt]或有无穷多解[/size][/font][size=12pt],[/size][font=宋体][size=12pt]并在有无穷多解时写出其一般解[/size][/font][size=12pt].[/size]
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[font=宋体][size=12pt]六[/size][/font][size=12pt]. (16[/size][font=宋体][size=12pt]分[/size][/font][size=12pt])[/size][font=宋体][size=12pt]求正交变换[/size][/font][size=12pt][img=61,16]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image084.gif[/img],[/size][font=宋体][size=12pt]将二次型[/size][/font][size=12pt][/size]
[size=12pt][img=386,29]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image086.gif[/img][/size][font=宋体][size=12pt]化为[/size][/font][size=12pt][/size]
[font=宋体][size=12pt]标准形[/size][/font][size=12pt],[/size][font=宋体][size=12pt]并写出其标准形[/size][/font][size=12pt].[/size]
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[font=宋体][size=12pt]七[/size][/font][size=12pt]. (8[/size][font=宋体][size=12pt]分[/size][/font][size=12pt])[/size][font=宋体][size=12pt]设[/size][/font][size=12pt][img=32,21]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image088.gif[/img] [/size][font=宋体][size=12pt]都是[/size][/font][size=12pt][img=13,15]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image090.gif[/img][/size][font=宋体][size=12pt]阶矩阵[/size][/font][size=12pt],[/size][font=宋体][size=12pt]且[/size][/font][size=12pt][img=16,17]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image092.gif[/img][/size][font=宋体][size=12pt]可逆[/size][/font][size=12pt],[/size][font=宋体][size=12pt]证明[/size][/font][size=12pt][img=27,17]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image094.gif[/img][/size][font=宋体][size=12pt]与[/size][/font][size=12pt][img=25,17]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image096.gif[/img][/size][font=宋体][size=12pt]有相同的特征值[/size][/font][size=12pt][/size]
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[font=宋体]八[/font]. (8[font=宋体]分[/font])[font=宋体]设向量组[/font][img=21,19]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image098.gif[/img][img=87,24]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image100.gif[/img][font=宋体]线性无关[/font],[font=宋体]向量[/font][img=19,23]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image102.gif[/img][font=宋体]可由向量组[/font][img=16,17]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image104.gif[/img][font=宋体]线性表示[/font],[font=宋体]而向量[/font][img=20,23]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image106.gif[/img][font=宋体]不能由[/font]
[font=宋体]向量组[/font][img=16,17]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image107.gif[/img][font=宋体]线性表示[/font].[font=宋体]证明[/font]:[img=39,19]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image109.gif[/img][font=宋体]个向量[/font][img=147,24]http://dis.njtu.edu.cn/bjtu/wlkj/10006yp/course/simtest/st1.files/image111.gif[/img][font=宋体]必线性无关[/font].
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