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Let $V$ be a finite dimension $k$-vector space, and $W \subset V$ be the subspace, prove the natrual map $$\text{Hom}(V,V) \to \text{Hom}(W,V/W)$$ is surjective. where it maps $\alpha \in \text{Hom}(V,V)$ by first restriting the domain to $W$ then compose it with the projection $V\to V/W$.

My attempt:

Since for vector space, we can always take zero extension we have the surjection $\text{Hom}(V,V) \to \text{Hom}(W,V)$. Or alternatively $\text{Hom}(-,V)$ is a exact contravariant functor it will sent injective morphism into surjective morphism.

only needs to prove that canonical map $\text{Hom}(W,V) \to \text{Hom}(W,V/W)$ induced from $V\to V/W$ is surjective.By noticing that in the category of vector space $\text{Hom}(W,-)$ is an exact covariant functor therefore it preserves the surjective map. Is my proof correct?

yi li
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    Yes, I would say this is a nice and correct proof. Note that this argument does not work in the context of modules, since then $\operatorname{Hom}$ might not be exact in either argument. But for vector spaces it is true. – Jonas Linssen Apr 13 '23 at 14:05
  • Thank you @Jonas Linssen , I also notice that it depends on the special property of the vector space. – yi li Apr 13 '23 at 14:11
  • I found this question when computing the tangent space of the Grassmannian : https://mathoverflow.net/a/135913/97016 – yi li Apr 13 '23 at 14:16

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