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向量组的线性相关性分析

向量组的线性相关性分析. 向量组的线性相关性判别 向量组的最大无关组.  . 向量组的线性相关性判别. 例 1. 讨论向量组 a 1 =(1 3 3 2), a 2 =(2 6 9 5), a 3 =(-1 -3 3 0). 的线性相关性. 解 将向量组中向量按列向量排列成矩阵并用命令 rref 化简 , 输入如下 MATLAB 命令 :. >> A=[1 3 3 2;2 6 9 5;-1 -3 3 0]; >> A=A′; >> format rat >> rref(A). 运行结果为. ans = 1 0 -5

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向量组的线性相关性分析

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  1. 向量组的线性相关性分析 向量组的线性相关性判别 向量组的最大无关组  

  2. 向量组的线性相关性判别 例1.讨论向量组a1=(1 3 3 2), a2=(2 6 9 5), a3=(-1 -3 3 0) 的线性相关性. 解 将向量组中向量按列向量排列成矩阵并用命令rref化简,输入如下MATLAB命令: >> A=[1 3 3 2;2 6 9 5;-1 -3 3 0]; >> A=A′; >> format rat >> rref(A) 运行结果为 ans = 1 0 -5 0 1 2 0 0 0 0 0 0 因其对应的阶梯形阵的秩为2小于3,故向量组a1, a2, a3线性相关.

  3. 例2.判断向量组a1=(1 2 0 1), a2=(1 3 0 -1), a3=(-1 -1 1 0) 是否线性相关,并求秩. 解 将向量组中向量按列向量排列成矩阵并用命令rref化简,输入如下MATLAB命令: >> A=[1 2 0 1;1 3 0 -1;-1 -1 1 0]; >> A=A′; >> format rat >> rref(A) 运行结果为 ans = 1 0 0 0 1 0 0 0 1 0 0 0 可见最简矩阵的秩为3,故向量组a1,a2,a3线性无关.

  4. 向量组的最大无关组 例3.求下列向量组的秩和一个最大线性无关组,并将其余向量用该最大无关组线性表示. a1=(2 -1 3 5), a2=(4 -3 1 3), a3=(3 -2 3 4), a4=(4 -1 15 17), a5=(7 -6 -7 0) 解 将向量组中向量按列向量排列成矩阵并用命令rref化简,输入如下MATLAB命令: >> A=[2 -1 3 5;4 -3 1 3;3 -2 3 4;4 -1 15 17;7 -6 -7 0]; >> A=A′; >> format rat >> rref(A) 运行结果为

  5. ans = 1 0 0 2 1 0 1 0 -3 5 0 0 1 4 -5 0 0 0 0 0 矩阵ans中有3个不全为零的行向量,所以矩阵A的秩为3.最简矩阵第一列、第二列和第三列向量线性无关,所以对应于原矩阵A的前三个行向量线性无关,即原向量组中一个最大无关组为a1, a2和a3. 矩阵ans中第四、五列中分别有三个非零元素,可将a4和a5线性表示为 a4=2a1-3a2+4a3 , a5=a1+5a2-5a3

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