if $L:V\to V'$ is a surjective linear aplication, and $W$ is a subspace of $v$, exists isomorphism $\frac{V}{W+\ker{L}}$ and $\frac{V'}{L(W)}$

Question

Let $L\colon V\to V'$ a surjective linear map. Let $W\subseteq V$ a vector subspace. I'm trying to show that exists an isomorphism $$ \frac{V}{W+\ker{L}}\simeq \frac{V'}{L(W)}. $$

I think to define the map $T'\colon V\to V'/L(W)$ such that $T'(a)=L(a)+L(W)$. But I cannot continue.

P.D. Sorry for my english

Answer 1

Observe that \begin{align} v \in \ker T' & \iff L(v)+L(W) = L(W) \\ & \iff L(v) \in L(W) \\ & \iff (\exists w \in W) \ L(v)=L(w) \\ & \iff (\exists w \in W) \ v-w \in \ker L \\ & \iff v \in W + \ker L. \end{align}