Let V be a finite dimensional vector space over a field F. Let T be linear transformation on V satisfying T^2 = T. Show that V admits basis of eigenvectors for T. Solution Given that T^2=T T^2-T=0 T(T-1)=0 that means characteristic equatuon wl be x(x-1)=0 I.e two distinxt eigen values and which implies that T is daigonazble then eigen vectors corresponding to distinct eigen values are mutualy linearly independent and hence form basis for V ( since T has distinct (dimV ) eigen values .).