Suppose that $T: \mathbb{R}^m \to \mathbb{R}^n$ is a linear transformation and that $v_1, v_2, \ldots, v_p$ are vectors in $\mathbb{R}^m$.
If $W = \{T(v_1), T(v_2), \ldots, T(v_p)\}$ is linearly independent in $\mathbb{R}^n$, does it follow that $S = \{v_1, v_2, \ldots, v_p\}$ is linearly independent in $\mathbb{R}^m$? Justify your answer with either a proof or a counterexample.