No, that's not it. You are trying to prove that the columns are linearly independent.Thank you very much AKG. So the problem was that the hint was vaguely written. Had I known what they meant exactly, I could have found that eventually.
The columns of the nxn matrix in your last line are linearly independent and so the matrix is invertible. Left multiplying both sides of the equation by the inverse matrix results in the ai all being zero.