If V has finite dimension n, show that two projections have the same diagonal form if and only if their kernels have the same dimension. (A projection is defined to be a linear transformation P:V-->V for which P^2=P; V is a vector space). For the forward direction, I was thinking that if T and P are projections that have the same diagonal form, their matrices of transformation are similar. I think I need to use the formula dimV= dim ker(T)+ dim Im(T) Knowing that dim Im(T)= rank of the matrix of transformation of T and dim ker(T)= n-dim Im(T) So the result follows. But I am not sure about the backward direction.