# Similarity and change of basis

1. Jul 21, 2007

### teleport

1. The problem statement, all variables and given/known data

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 :

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

2. Relevant equations

3. 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.

2. Jul 21, 2007