You can define a natural isomorphism from Hom_{R}(R^{(Λ1)}, R^{(Λ2)}) to (R^{(Λ2)})^{Λ1}. (Where Hom_{R}(R^{(Λ1)}, R^{(Λ2)}) is the R-linear homomorphisms from R^{(Λ1)} to R^{(Λ2)}.)

Hom_{R}(R^{(Λ1)}, R^{(Λ2)}) ---> R^{(Λ2) , Λ1}

If *f* is an element of Hom_{R}(R^{(Λ1)}, R^{(Λ2)}) then that map sends *f* to [*f*], a (Λ_{2}) , Λ_{1} matrix.

Isn't that the most beautiful thing you've ever seen?