$T(v_1), \ldots,T(v_k)$ are independent if and only if $\operatorname{span}(v_1,\ldots,v_k)\cap \ker(T)=\{0\}$ I need help with this:

If $T:V\rightarrow W$ is a linear transformation and $\{v_1,v_2,\ldots,v_k\}$ is a linearly independent set in $V$, prove that $T(v_1), T(v_2),\ldots,T(v_k)$  are independent in $W$ if and only if 
  $$\operatorname{span}(v_1,\ldots,v_k)\cap \ker(T)=\{0\}.$$

I know how to prove that a independent set transforms to a independent set, but how do I show that this intersection equals the zero vector?

Just consider the subspace $S$ spanned by your vectors $v_1,\dots,v_k$. Then, if the condition holds, you have an isomorphism $T_S$ from $S \to T(S) \subseteq W$. For an isomorphism you have $\ker T_S = \{0\}$, but  
$$\{0\} = \ker T_S = S \cap \ker T = \langle v_1,\dots,v_k \rangle \cap \ker T,$$
so this gives the result.
