The short answer is that it leads to a more consistent notation. The longer answer is that matrix multiplication is essentially 2 different operations. Conside

The short answer is that it leads to a more consistent notation. The longer answer is that matrix multiplication is essentially 2 different operations. Consider the linear funtions: f :: R3 -> R3 h :: R3 -> R3 As well as the point x :: R3 If we fix a set a basis vectors, we can represent f and h as 3x3 matrices, and x as a 3x1 matrix. The product [f][x]=[f(x)] then represents the result of a function application, while [f][h] = [f∘h] represents function composition. For the function application portion, there is no problem with your proposal. We simply represent the point as a 1x3 vector instead of a a 3x1 vector (or similarly transpose the convention for representing a function as a matrix). The problem comes with the function composition use-case. For your proposal to work, we would need to transpose only one of the two matrices, which means that the matrix representation of a function is determined both by the basis vectors, and a transposition parity bit. With multiplication as composition only making sense when the transposition parities don't match.