求向量a1=(4,-1,3,-2)a2=(8,-2,6,-4)a3=(3,-1,4,-2)a4=(6,-2,8,-4)的所有极大线性无关向量组

Question

求向量a1=(4,-1,3,-2) a2=(8,-2,6,-4) a3=(3,-1,4,-2) a4=(6,-2,8,-4)的所有极大线性无关向量组

Answer 1

(a1^T,a2^T,a3^T,a4^T) =
4 8 3 6
-1 -2 -1 -2
3 6 4 8
-2 -4 -2 -4
r1+4r2,r3+3r2,r4-2r2
0 0 -1 -2
-1 -2 -1 -2
0 0 1 2
0 0 0 0
r2-r1,r3+r1
0 0 -1 -2
-1 -2 0 0
0 0 0 0
0 0 0 0
r1*(-1),r2*(-1),r1r2
1 2 0 0
0 0 1 2
0 0 0 0
0 0 0 0
a1,a2,a3,a4 的极大无关组有:a1,a3; a1,a4; a2,a3; a2,a4.
不过,一般不会让求所有的极大无关组.