Editing Module homomorphism (section)

== A matrix representation ==
The relationship between matrices and linear transformations in [[linear algebra]] generalizes in a natural way to module homomorphisms between free modules. Precisely, given a right ''R''-module ''U'', there is the [[canonical isomorphism]] of the abelian groups
:<math>\operatorname{Hom}_R(U^{\oplus n}, U^{\oplus m}) \overset{f \mapsto [f_{ij}]}\underset{\sim}\to M_{m, n}(\operatorname{End}_R(U))</math>
obtained by viewing <math>U^{\oplus n}</math> consisting of column vectors and then writing ''f'' as an ''m'' × ''n'' matrix. In particular, viewing ''R'' as a right ''R''-module and using <math>\operatorname{End}_R(R) \simeq R</math>, one has
:<math>\operatorname{End}_R(R^n) \simeq M_n(R)</math>,
which turns out to be a ring isomorphism (as a composition corresponds to a [[matrix multiplication]]).

Note the above isomorphism is canonical; no choice is involved. On the other hand, if one is given a module homomorphism between finite-rank [[free module]]s, then a choice of an ordered basis corresponds to a choice of an isomorphism <math>F \simeq R^n</math>. The above procedure then gives the matrix representation with respect to such choices of the bases. For more general modules, matrix representations may either lack uniqueness or not exist.