From Today's Algebra Class

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 (Λ₂), Λ₁ matrix.

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