Answer is false.
If {v₁, w₁} is a linearly dependent set of vectors and if {v₂, w₂} is a set of linearly independent vectors, then you can never find such an A.
If {v₁, w₁} is linearly dependent, then v₁ = cw₁ for some scalar c, hence
v₂ = A(v₁) = A(cw₁) = cA(w₁) = cw₂
That implies {v₂, w₂} to be linearly independent, but it is taken to be a linearly independent set.