Consider the ordered bases B = {1, x, x2} and C = {1, (x − 1), (x − 1)2} for P2.
(a) Find the transition matrix from C to B.
b) Find the transition matrix from B to C.
(c) Write p(x) = a + bx + cx2 as a linear combination of the polynomials in C.
Now consider the "variable substitution" map
T : P2 → P2, defined by T(p(x)) = p(2x − 1).
In other words, T : p(x) → p(2x − 1).
(d) Show that T is a linear transformation.
(e) Find the matrix representation [T]B of T with respect to the ordered basis B,
(f) Find the matrix representation [T]C of T with respect to the ordered basis C directly, using the definition of [T]C.
(g) Find the matrix representation [T ]C of T again, using [T ]B and the change of basis formula.
(h) What can you say about the eigenvectors and eigenvalues of T? Give a brief explanation.