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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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Generally speaking, one way to prove that two things are isomorphic is first to try to see whether you can construct an obvious map in one direction, for example using universal properties. It is not always entirely obvious which direction the obvious map goes, but this is a good place to start. – Qiaochu Yuan Sep 24 '12 at 6:04
@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! – AsinglePANCAKE Sep 24 '12 at 6:46
up vote 2 down vote accepted

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 $Hom_{\mathbb{F}}(U, V)/B \approx Hom_{\mathbb{F}}(U, V/W)$.

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