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Answer on Abstract Algebra Question for jeremy

Question #23571
Let R be a k-algebra where k is a field, and M,N be left Rmodules, with dimkM <∞. It is known that, for any field extension K ⊇ k, the natural map θ : (HomR(M,N))K → HomRK(MK,NK) is an isomorphism of K-vector spaces. Replacing the hypothesis dimkM < ∞ by “M is a finitely presented R-module,” give a basis-free proof for the fact that θ is a K-isomorphism.
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