Spanning of Linear Mappings

Question

I have been trying to prove a fairly simple linear mapping proof although I am struggling to show the spanning part. I can show the two sets are linearly independent but I am unsure on how to prove they spam the vector spaces. The question is if L: V -> W be a linear mapping such that {L(v1), ... , L(vk)} spans W, then {v1, ... , vk} spans V

Answer 1

let $ v \in V$ then since $L(v)\in W, L(v) = a_1L(v_1)+...+a_kL(v_k)$ where each $a_i$ is in our field. By linearity, $L(v) = L(a_1v_1+...+a_kv_k)$ And hence $L(v-(a_1v_1+...+a_kv_k))=0$ (again, by linearity) and so $v=a_1v_k+...+a_kv_k$ Since $v$ was arbitrary, $({{v_1,...,v_k}})$ spans $V$