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Let $V$, $W$ be finite-dimensional vector spaces over $k$. What is the explicit coordinate-free form of the canonical isomorphism $\mathrm{\mathop{Hom}}(\mathrm{\mathop{Hom}}(V, W), k) = \mathrm{\mathop{Hom}}(W, V)?$

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If $A:V\to W$ then $A$ gives a linear map $a:Hom(W,V)\to k$ via $a(B)=Tr(AB)$. That's it.