This can't be expressed as a linear transformation A.

Your original vectors v1+v2+v3+v4 = 0

However, your new vectors v1'+v2'+v3'+v4' = (-2,-2,-2)

If this was a linear transformation:

Av1 + Av2 + Av3 + Av4 = A(v1+v2+v3+v4) = 0

If you wanted this transformation, you will at least have to use affine transformations.