Similarity and change of basis

[teleport]

Homework Statement

Consider the 2-dimensional complex vector space V of functions spanned by sin x and cos x. For a fixed real number α, define a linear operator T ≡ Tα on V by putting T(f(x)) = f(x + α). Find the matrices [T]B and [T]E of T relative to the bases B = {cos x, sin x} and E = {cos x + i sin x, cos x − i sin x}. Find an invertible matrix P implementing the similarity between [T ]B and [T]E :

$$[T]_B = P[T]_EP^{-1}$$

Homework Equations

The Attempt at a Solution

I found T_B and T_E and I think I'm probably right because their tr and det are equal. Sorry if I don't show all that work because it is long. My problem is to get P. I am not sure how to get this. Thanks for any suggestions.

 

[radou]

Well, what do you know about transformation matrices between different bases?

 

[teleport]

OK. Got it now.