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.