Suppose for contradiction that the input vectors v_n are linearly dependent. Then let V be the matrix made up of these v_n's.
As the vectors are dependent V has a nonzero null space.
So for any x in that null space Vx = 0.
TVx = 0
so the null space of TV is at least the null space of V. It could be higher dimensional if T isn't invertible.
Thus the vectors of TV are not linearly independent.
So the vectors v_n must be linearly independent.