Question

How do I show that if the Transition Matrix P B1-> B2 is diagonal, then the vectors of B1 must be a scalar multiple of some vector in B2. B1 and B2 are bases for Rn.

Answer 1

I'll try..
I'm not entirely sure about my proof because I don't understand the concept of P 100%.

B1 = {a1,a2,..an} (ordered basis)
B2 = {c1,c2,...cn}

coordinate vector of a1 respect to B1 = [1,0,....0]
coordinate vector of a1 respect to B2 = P * [1,0....0] = [d1,0,0..0] (P is diagonal)

Hence,
1 * a1 = d1 * c1