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Here's what I think.

Because S is an element of L(V,W), S:V-->W means that S has a basis of {v1...vn}, and two vector spaces that form a bijective linear map (which S and T do because they have the same basis) are isomorphic. Moreover, because T(vi)=S(vi) then by their isomorphism, T and S must be equal.

This is the last question on my practice midterm and I'm unsure if this is how to proceed with the proof. Any comments, especially on how I could do this more "formally"?