Suppose M is the matrix that describes T in a given basis B, so that the columns of M represent the images of the members of B expressed as linear combinations of the members of B, and M is the matrix similarly describing T with respect to basis B.
What is the relation of M to M?
Let J be the (Jacobian) matrix whose columns are the basis vectors of B expressed in terms of those of B.
Then MJ has columns which are the images of the basis vectors of B expressed in terms of those of B.
To reexpress these in terms of the basis vectors of B you must multiply on the left by the matrix which expresses the members of B as linear combinations of those of B.
This is the inverse Jacobian, J^-1. We therefore have M = J^-1 MJ, and, by our last result, as claimed:

M=J^-1MJ = J^-1JM=IM=M.