for every distinct eigenvalue you have at least one eigenvector. let the set A contain one eigenvector for every non zero eigenvalue. eigenvectors for distinct eigenvalues are independent. because they are eigenvectors for nonzero eigenvalues you have

span(A) = span(T(A)) - so

so there are at most dim(Im(T)) non zero eigenvalues, or dim(Im(T))+1 eigenvalues