How to prove that Linear Transformation cannot preserve independence

Question

I know how to prove linear transformation can preserve dependence, but how to prove the converse without using a counter example?

Answer 1

Because some of the vectors may in the null space of the domain, e.g. $\textsf{V}$, of a linear transformation, e.g. $\textsf{T}:\textsf{V}\to\textsf{W}$. You can't promise that any independent subset of $\textsf{V}$ won't contain a element $v'$ such that $\textsf{T}(v')=0_\textsf{W}$

Answer 2

How about to use $\det(AB)=\det(A)\det(B)$? Determinant is zero equivalent to linear dependence of columns.