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13第三章节与第四章节习题课.ppt

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习题课 * * * * Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. 第四章习题三选讲 一.4. 方程组 有解,则常数 a1, a2, a3, a4应满足: a1+ a2+ a3+ a4 = 0 Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. 二.选择题 1.对n元方程组,下列结论正确的是( ) 若Ax = 0只有零解, 则Ax = b有唯一解. Ax = 0有非零解的充要条件是|A| = 0 Ax = b有唯一解的充要条件是R(A) = n 若Ax = b有两个不同的解, 则Ax = 0有无穷多解. 反例: (矛盾方程) Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. 2. 设?1, ?2, ?3是Ax = 0基础解系,则该方程的基础解系还可以表示成( ) ?1, ?2, ?3的一个等价向量组 ?1, ?2, ?3的一个等秩向量组 ?1 +?2, ?2 + ?3, ?3 + ?1 ?1 - ?2, ?2 - ?3, ?3 - ?1 未必线性无关 未必是方程的解 Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. 3. 设?1, ?2, ?3是方程组Ax = b的三个解向量, 并且R(A) = 3, ?1 = (1, 2, 3, 4)T, ?1 + ?2 = (0, 1, 2, 3)T, k表示任意常数,则线性方程组Ax = b的通解为( ) (1, 2, 3, 4)T + k(1, 1, 1, 1)T (1, 2, 3, 4)T + k(0, 1, 2, 3)T (1, 2, 3, 4)T + k(2, 3, 4, 5)T (1, 2, 3, 4)T + k(3, 4, 5, 6)T Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. 四. 求一个齐次线性方程组,使它的基础解系为?1 = (0, 1, 2, 3)T, ?2 = (3, 2, 1, 0)T. 解: 设所求的方程为: 则(a1,1, a1,2, a1,3, a1,4)T和(a2,1, a2,2, a2,3, a2,4)T皆是方程组 的解. 即: ,令y2=3c1, y3 = 3c2 Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. 特别选取: 所求的方程为: Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. 五. 设n阶方阵A满足A2 = A, E为n阶单位方阵, 证明: R(A) + R(A – E) = n. 证明: 记R(A) = r, R(A – E) = m 由A(A – E) = O, 即 得到: 皆是齐次方程Ax = 0的解, 而该方程的解空间的维数为n – r, 所以有: 故R(A) + R(A – E) ? r + n – r = n …………(1) Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. 记 由: -E = (A – E) – A 得到: …………….(2) 综合(1)、(2)得到:R(A) +
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