Question

求矩阵A的列向量组的一个最大无关组，并把其余列向量用最大无关组线性表示

A=
2 3 1 -3 -7
1 2 0 -2 -4
3 -2 8 3 0
2 -3 7 4 3

请给出详细方法步骤，谢谢您啦！

Answer 1

1 1 2 2 1
0 2 1 5 -1
2 0 3 -1 3
1 1 0 4 -1
r3-2r1, r4-r1
1 1 2 2 1
0 2 1 5 -1
0 -2 -1 -5 1
0 0 -2 2 -2
r3+r2, r4*(-1/2)
1 1 2 2 1
0 2 1 5 -1
0 0 0 0 0
0 0 1 -1 1
r1-2r4, r2-r4
1 1 0 4 -1
0 2 0 6 -2
0 0 0 0 0
0 0 1 -1 1
r2*(1/2)
1 1 0 4 -1
0 1 0 3 -1
0 0 0 0 0
0 0 1 -1 1
r1-r2, r3<->r4
1 0 0 1 0
0 1 0 3 -1
0 0 1 -1 1
0 0 0 0 0

极大无关组: a1,a2,a3
a4 = a1+3a2-a3
a5 = -a2 +a3
这样可以么？

追问

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Answer 2

A =
[2 3 1 -3 -7]
[1 2 0 -2 -4]
[3 -2 8 3 0]
[2 -3 7 4 3]
初等行变换为
[1 2 0 -2 -4]
[0 -1 1 1 1]
[0 -8 8 9 12]
[0 -7 7 8 11]
初等行变换为
[1 0 2 0 -2]
[0 1 -1 -1 -1]
[0 0 0 1 4]
[0 0 0 1 4]
初等行变换为
[1 0 2 0 -2]
[0 1 -1 0 3]
[0 0 0 1 4]
[0 0 0 0 0]
则 a1, a2, a4 为一个最大无关组，
a3 = 2a1-a2
a5 = -2a1+3a2+4a4.