We can mechanize the above transformation by defining a function \phi_B, called the standard representation of V with respect to B, that takes every vector to its coordinate representation: \phi_B(v)=[v]_B.
Alternatively, we could have defined \phi_B^{-1} to be the above function from the beginning, realized that \phi_B^{-1} is an isomorphism, and defined \phi_B to be its inverse.