Question

设向量组a1=(1,1,1,3)^T,a2=(-1,-3,5,1)^T,a3=(3,2,-1,p+2)^T,a4=(-2

Answer 1

(a1,a2,a3,a4,a) =

1 -1 3 -2 4

1 -3 2 -6 1

1 5 -1 10 6

3 1 p+2 p 10

r3-r1-r2-r3,r3-r2,r2-r1

1 -1 3 -2 4

0 -2 -1 -4 -3

0 8 -3 16 5

0 0 p-2 p-2 -1

r3+4r2

1 -1 3 -2 4

0 -2 -1 -4 -3

0 0 -7 0 -7

0 0 p-2 p-2 -1

r3*(-1/7),r1-3r3,r2+r3,r4-(p-1)r3

1 -1 0 -2 1

0 -2 0 -4 -2

0 0 1 0 1

0 0 0 p-2 -p+1

r2*(-1/2),r1+r2

1 0 0 0 2

0 1 0 2 1

0 0 1 0 1

0 0 0 p-2 -p+1

当p=2时,r(a1,a2,a3,a4)=3,a1,a2,a3,a4线性相关,a1,a2,a3 是一个极大无关组.

当p≠2时,a1,a2,a3,a4线性无关.

此时,(a1,a2,a3,a4,a) -->...

r4*[1/(p-2)],r2-2r4

1 0 0 0 2

0 1 0 0 (3p-4)/(p-2)

0 0 1 0 1

0 0 0 1 -(p-1)/(p-2)

a=2a1+[(3p-4)/(p-2)]a2+a3-[(p-1)/(p-2)]a4.