设向量组：a1(2,0,1,1)a2（-1,-1,0,1）a3(1,-1,0,0)a4（0,-2,-1,-1）求向量组呵一个最大无关组,并用无关组将其余向量线性表示

Question

设向量组：a1(2,0,1,1)a2（-1,-1,0,1）a3(1,-1,0,0)a4（0,-2,-1,-1）求向量组呵一个最大无关组,并用无关组将其余向量线性表示

Answer 1

(a1,a2,a3,a4)=
2 -1 1 0
0 -1 -1 -2
1 0 0 -1
1 1 0 -1
r1-2r3,r4-r3
0 -1 1 2
0 -1 -1 -2
1 0 0 -1
0 1 0 0
r1+r4,r2+r4
0 0 1 2
0 0 -1 -2
1 0 0 -1
0 1 0 0
r2+r1
0 0 1 2
0 0 0 0
1 0 0 -1
0 1 0 0
交换行
1 0 0 -1
0 1 0 0
0 0 1 2
0 0 0 0
所以a1,a2,a3是一个极大无关组,且a4=-a1+2a3.