For the theorem: " If v1,.....,vr are eigenvectors of a linear map T going from vector space V to V, with respect to distinct eigenvalues λ1,...,λr, then they are linearly independent eigenvectors". Are the λ-eigenspaces all dimension 1. for each λ1,......,λr.? Is the dimension of V, r? ie dim(V)=r, ie their contains r elements in a basis for V. I have another important question, is the matrix A representing the linear transformation T just the diagonal matrix (P-1AP=D, Where D contains the eigenvalues of T)? Not just in this case, but Always? This ones bugging me.