Given a differential operator like the regular derivative, or grad or curl or div etc, it can act on a function to its right to yield a new function. Because it is linear, it is effectively like a an infinite matrix acting on an infinite column vector (roughly, obviously the space of smooth functions in uncountable, +other differences).
But matrices can be applied to the other side on a row vector. What would be analogous for differential operators?
There is no natural "action on a function to its left" that can be assigned to a differential operator. For a linear operator, what matters is the result of its action; what does not matter is how we write it. The fact that we say "a matrix applied to other side on a row vector" is just our usage of the language. It does not create a new kind of action of a linear operator represented by the matrix.