You usually think of the product of a matrix with a vector as the matrix acting on the vector, i.e. every row of the matrix tells you what the corresponding com

You usually think of the product of a matrix with a vector as the matrix acting on the vector, i.e. every row of the matrix tells you what the corresponding component of the resulting vector is. This turns the thing on its head and describes the result as the vector acting on the matrix. Because you usually use matrices to describe operations, e.g. transformations like translations or rotations in (computer) geometry or observables in quantum physics, it seem more natural to think of the matrix as acting on the vector, i.e. a point of the geometry or a physical state, not the other way round.