1. The problem statement, all variables and given/known data Let V be a finite dimensional vector space over the field F and let S and T be linear operators on V. We ask: When do there exist ordered bases B and B' for V such that B = [T]B'? Prove that such bases exist only if there is an invertible linear operator U on V such that T = USU-1. 3. The attempt at a solution I have a hand-waiving argument and am not too sure of what to do. Assume that T = USU-1. Multiplying both sides from the left by U-1. You have U-1T = SU-1. Now since S, U and T are functions, lets "plug in" a vector whose co-ordinates are from B', say Y. U-1T (Y) = SU-1 (Y) T will "act on" Y since it is from B' using its respective matrix and the U-1 will take it from B' to B after it has been transformed. On the right hand side, the vector Y comes in, gets converted to a vector in B by being "acted on" by U. Then it will be transformed by S. Now the two vectors on either side are equal to each other, so effectively, their co-ordinate matrices from B and B' will be equal and so [T]B' = B. Apart from this vague line of thinking, I don't know where to start on a formal proof. Please and thank you.