# Math Help - eigenvalues

1. ## eigenvalues

Hi,
Plese help!
Suppose T is a linear map and dim(Im(T))=k. Prove that T has at most k+1 distinct eigenvalues.
Thank you!

2. 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 $dim(Im(T)) \ge dim(span(T(A))) = dim(span(A)) = |A|$
so there are at most dim(Im(T)) non zero eigenvalues, or dim(Im(T))+1 eigenvalues