Consider a linear transformation T from V to V, where V is an n-dimensional linear space. Let B be a basis of V. Then there exists one (and only one) nxn matrix B that transforms the coordinates of f to the coordinates of T(f). The columns of B are the B-coordiante vectors of the transformation of the basic elements f1, f2, ..., fn of V. (Compare with Section 3.4 where we identified the matrix of a linear transformation for vectors v1, v2, ..., vm.)
Suppose that U and B are two bases of an n-dimensional linear space V. Then there exists one (and only one) nxn matrix S such that the [f]U = S [f]B , for all f in V. That is, the matrix S takes you from one coordinate system to another coordinate system. This invertible matrix S is called the change of basis matrix from B to U.
Let V be a linear space with two given bases U and B. Consider a linear transformation T from V to V, and let A and B be the U- and B-matrix of T, respectively. Let S be the change of basis matrix from B to U. Then A is similar to B, and AS = SB or A = SBS^(-1) or B = S^(-1)AS. (See diagram on p. 179.)
Section 4.3 - 1, 3, 5, 13, 21, 23, 25, 46, 49