I'm doubting myself a little bit on my logic when answering this question:
Suppose $T$ is a linear map, and $v_1,...,v_k$ are vectors. If $T(v_1), ..., T(v_k)$ are dependent, need $v_1, ..., v_k$ be dependent?
My reasoning is that yes, they do need to be dependent since: If $T(v_1), ..., T(v_k)$ are dependent, then there are some not-all-zero real numbers $c_1,...,c_k$ for which the equation $$c_1T(v_1) + ... + c_kT(v_k) = 0$$ holds. Suppose $F$ is a linear map that is the reverse of $T$. Then: $$c_1F(T(v_1)) + ... + c_kF(T(v_k)) = 0$$ which simplifies to: $$c_1v_1 + ... + c_kv_k = 0$$ Thus, $v_1,...,v_k$ are also linearly dependent.