 unity, then the ring of endomorphisms of M=⨁i∈IR{\displaystyle M=\bigoplus _{i\in I}R} as a right R module is isomorphic to the ring of column finite matrices CFMI(R){\displaystyle \mathbb {CFM} _{I}(R)} whose entries are indexed by I×I{\displaystyle I\times I}, and whose columns each contain only finitely many nonzero entries.  The endomorphisms of M considered as a left R module result in an analogous object, the row finite matrices RFMI(R){\displaystyle \mathbb {RFM} _{I}(R)} whose rows each only have finitely many nonzero entries.If infinite matrices are used to describe linear maps, then only those matrices can be used all of whose columns have but a finite number of nonzero entries, for the following reason. For a matrix A to describe a linear map f: V→W, bases for both spaces must have been chosen; recall that by definition this means that every vector in the space can be written uniquely as a (finite) lin