Intuition behind pre-multiplication of a vector with a matrix

Question

Given a matrix $\mathbf{A}$ and a vector $\mathbf{x}$, the MVP $\mathbf{Ax}$ gives the geometric interpretation of transforming $\mathbf{x}$ into the coordinate system with basis vectors defined by the columns of $\mathbf{A}$. Does such an interpretation exist when a vector $\mathbf{y}$ is pre-multiplied to a matrix, i.e., $\mathbf{yA}$? One interpretation is in terms of the transpose of $\mathbf{A}$, but I'm looking for an explanation in terms of $\mathbf{A}$ itself.