Correspondence between set of subspaces W and set of subspaces of V/K

Question

I have a Linear Algebra question on Vector Spaces that I'm a bit stuck on.

Let K be a subspace of a vector space and $W \subseteq V$. Show that the assignment $W \mapsto W/K$ ($W/K \subseteq V/K$) defines 1-1 correspondence between the set of subspaces W of V with $K \subseteq W \subseteq V$ and the set of subspaces of V/K.

I was hoping someone would be able to give me some guiding help to get me started on it.

Thanks in advance.

Answer 1

For it to be one-to-one, it will have to be injective and surjective.

Injective: The only way for it to fail to be injective is if two subspaces $W_1$, $W_2$ get sent to the same $W'/K\subseteq V/K$. When would it be possible for this to happen? Why does that mean it's not possible in this case?

Surjective: Imagine we have some subset $U\subseteq V/K$. Can you think of a way to "go backwards" to get a subset $U'\subseteq V$ such that $U'/K = U$?