Given that vector group {A1, A2, A3}, {B1, B2, B3} satisfies B1 = a1 + A2, B2 = a1-2a2, B3 = a1 + a2-7a3, it is proved that vector group A is linearly independent If and only if the vector group B is linearly independent

Given that vector group {A1, A2, A3}, {B1, B2, B3} satisfies B1 = a1 + A2, B2 = a1-2a2, B3 = a1 + a2-7a3, it is proved that vector group A is linearly independent If and only if the vector group B is linearly independent

(B 1, B 2, B 3) = (a 1, a 2, a 3) P, that is, group B can be expressed linearly by group A
P =
1 1 1
1 -2 1
0 0 -7
Because | P | = - 3 * (- 7) = 21 ≠ 0
So p is reversible, that is, (B1, B2, B3) P ^ (- 1) = (A1, A2, A3)
That is, group a can be represented linearly by group B
So the two vector groups are equivalent
So r (B1, B2, B3) = R (A1, A2, A3)
So there is no linear correlation in group A
r(a1,a2,a3) = 3
r(b1,b2,b3) = 3
Vector group B is linearly independent