给定向量组:a1=(1,-1,-1,-3),a2=(-1,1,-3,-1),a3=(-1,-3,1,-1),a4=(-3,-1,-1,1)求1.向量组a1,a2,a3,a4秩2.求该向量组的极大无关组,将其余向量用极大无关组线性表示.

Question

给定向量组:a1=(1,-1,-1,-3),a2=(-1,1,-3,-1),a3=(-1,-3,1,-1),a4=(-3,-1,-1,1)求1.向量组a1,a2,a3,a4秩
2.求该向量组的极大无关组,将其余向量用极大无关组线性表示.

Answer 1

(a1,a2,a3,a4) =
1 -1 -1 -3
-1 1 -3 -1
-1 -3 1 -1
-3 -1 -1 1
r2+r1,r3+r1,r4+3r1
1 -1 -1 -3
0 0 -4 -4
0 -4 0 -4
0 -4 -4 -8
r4-r3,r3*(-1/4),r1+r3
1 0 -1 -2
0 0 -4 -4
0 1 0 1
0 0 -4 -4
r4-r2,r2*(-1/4),r1+r2
1 0 0 -1
0 0 1 1
0 1 0 1
0 0 0 0
r2r3
1 0 0 -1
0 1 0 1
0 0 1 1
0 0 0 0
a1,a2,a3,a4的秩为3,a1,a2,a3是其一个极大无关组,且 a4=-a1+a2+a3.