Components of a linear operator with respect to a basis?

As I understand, a matrix represents the action of a linear operator by taking components of vectors with respect to one stated basis and outputting the components of the vector, transformed by the linear operator, with respect to another basis which may or may not be the same as the input basis.

Now I have seen several questions regarding the meaning of 'components of a linear operator with respect to a basis', and I think i understand the basic concept pretty well. My question is more about the details. Specifically, when one refers to 'the components of a linear operator with respect to a basis', is this the basis with respect to which the INPUT vector components are given, or the OUTPUT vector components? These need not, after all, be the same. Heck, I think you could decompose any matrix that has different input and output bases into a basis transformation matrix(either first or second) and the transformation matrix (note- I am not sure about this. Certainly it would be true for a linear operator mapping a vector space to itself. Would have to think about the other cases, but that's not the point here!)

So my question in a nutshell: are the components of a linear operator with respect to a basis the basis for the input vector, output vector, or does this phrase assume the bases for input and output are the same?