The class I took was very careful to be precise about this matter. Let V,W be finite-dimensional vector spaces and f: V --> W a linear map between them. Then f is completely determined by how it transforms some basis for V. Say B = (v_1,v_2,...,v_n) is a basis for V, and C = (w_1,w_2,...,w_m) is a basis for W. Then let M be the m-by-n matrix where the i-th column is f(v_i) written with respect to the basis C in W. Then we can say that f(v) = Mv for all vectors v in V - when v is written in the basis B and f(v) is written in the basis C. In particular, we would frequently write f = _{B}[M]_{C} to emphasize that the matrix transforms from the basis B to the basis C.

One other minor thing here is that R^{3} is special in that is it literally the set of ordered triples of real numbers, and so there are naturally "coordinates". But if we took, say, R[ x ]_{<3}, the vector space of real-valued polynomials in x with degree less than three, there aren't really "coordinates". (1,x,x^2) is a simple basis for the space, but the vectors themselves do not have coordinates. 4x^2 + 2x + 7 is a vector in this space, and with respect to that basis it is (7,2,4). Then if you apply a linear transformation to this vector, you can instead use a matrix with respect to this basis and apply it to (7,2,4); it doesn't make any sense to multiply a matrix by 4x^2 + 2x + 7. In fact, R[ x ]_{<3} is isomorphic to R^{3}, but here it's more clear that the vectors need to be written with respect to a basis.