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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!

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    @QiaochuYuan: This tends to be my approach, but I'm still a bit of a noob. And forgive this mathematically irrelevant digression, but I must confess I'm a tad starstruck!2012-09-24

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Let $\iota\colon W \rightarrow V$ be the canonical injection. Let $\phi\colon Hom_{\mathbb{F}}(U, W) \rightarrow Hom_{\mathbb{F}}(U, V)$ be the map defined by $\phi(f) = \iota\circ f$. $\phi$ is clearly injective. It is clear that Im$(\phi) = B$. Hence $Hom_{\mathbb{F}}(U, W) \approx B$.

Let $\pi\colon V \rightarrow V/W$ be the canonical map. Let $\psi\colon Hom_{\mathbb{F}}(U, V) \rightarrow Hom_{\mathbb{F}}(U, V/W)$ be the map defined by $\psi(f) = \pi\circ f$. Let $g \in Hom_{\mathbb{F}}(U, V/W)$. Since $\pi\colon V \rightarrow V/W$ is surjective, there exists $f \in Hom_{\mathbb{F}}(U, V)$ such that $g = \pi\circ f$. Hence $\psi$ is sujective. Clearly $Ker(\psi) = B$. Hence $Hom_{\mathbb{F}}(U, V)/B \approx Hom_{\mathbb{F}}(U, V/W)$.