给定向量组a1=(1,0,1,0)T,a2=(2,1,-3,7)T,a3=(3,1,0,3)T,a4=(4,1,3,-1)T求向量组的秩及一个极大无关组，并

Question

如题所述

Answer 1

解: (a1,a2,a3,a4)=
1 2 3 4
0 1 1 1
1 -3 0 3
0 7 3 -1

r3-r1
1 2 3 4
0 1 1 1
0 -5 -3 -1
0 7 3 -1

r1-2r2,r3+3r2,r4-7r2
1 0 1 2
0 1 1 1
0 0 2 4
0 0 -4 -8

r4+2r3
1 0 1 2
0 1 1 1
0 0 2 4
0 0 0 0

r3*(1/2),r1-r3,r2-r3
1 0 0 0
0 1 0 -1
0 0 1 2
0 0 0 0

向量组的秩为3, a1,a2,a3 是一个极大无关组, 且 a4=-a2+2a3.