(a1,a2,a3,a4)=
1 3 2 5
2 2 3 5
3 1 1 2
-1 -1 1 0
r1+r4,r2+2r4,r3+3r4,r4*(-1)
0 2 3 5
0 0 5 5
0 -2 4 2
1 1 -1 0
r2*(1/5),r1-3r2,r3-4r2,r4+r2
0 2 0 2
0 0 1 1
0 -2 0 -2
1 1 0 1
r3+r1,r1*(1/2),r4-r1
0 1 0 1
0 0 1 1
0 0 0 0
1 0 0 0
所以 a1,a2,a3 是一个极大无关组
且 a4 = a2+a3.