Matrix multiplication corresponds to the composition of related linear operators. So, it goes down to the representation of a linear operator by a matrix. But w

Matrix multiplication corresponds to the composition of related linear operators. So, it goes down to the representation of a linear operator by a matrix. But we have freedom here. For example, we could choose to represent a linear operator by the transpose matrix (of the usual). But this requires changing the matrix product rule as well (to have the meaning of composition).