Let M {\displaystyle {}M} be an m × n {\displaystyle {}m\times n} -matrix over the field K {\displaystyle {}K} of rank r {\displaystyle {}r} . Show that there exists an r × n {\displaystyle {}r\times n} -matrix A {\displaystyle {}A} , and an m × r {\displaystyle {}m\times r} -matrix B {\displaystyle {}B} , both of rank r {\displaystyle {}r} , such that M = B ∘ A {\displaystyle {}M=B\circ A} holds.
