Let K {\displaystyle {}K} be a field, and let V {\displaystyle {}V} and W {\displaystyle {}W} be vector spaces over K {\displaystyle {}K} of dimensions n {\displaystyle {}n} and m {\displaystyle {}m} . Let
be a linear map, described by the matrix M ∈ Mat m × n ( K ) {\displaystyle {}M\in \operatorname {Mat} _{m\times n}(K)} with respect to two bases. Prove that