Letting $U, V$ be vector spaces over $\mathbb{F}$ with $W\subseteq V$ a subspace. I want to show that if $B = \{T\in Hom_\mathbb{F}(U,V) | im(T)\subseteq W\}$ that $B\approx Hom_{\mathbb{F}}(U,W)$$Hom_\mathbb{F}(U,V)/B\approx Hom_\mathbb{F}(U,V/W) $
For the first one, I'd like to say that I can just use the identity map, but I feel as though I have to take into consideration the codomain of $T\in B$ somehow to be precise. I'm not entirely sure what the best approach is to do this.
The second one I'm pretty lost. I feel as though utilizing the universal mapping properties of the quotient space and then invoking the first isomorphism theorem is relevant, but again I can't quite write it down precisely...
Any and all help is much appreciated!