Suppose T is a linear map and dim(Im(T))=k

[mivanova]

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

 

[HallsofIvy]

If T maps an n dimensional space into an m dimensional space, then the kernel of T must be of dimension n- k. And, of course, every vector in the kernel of T is an eigenvector with eigenvalue 0. Now, since each eigenvalue has a corresponding "eigen"space of dimension at least 1, how many other eigenvalues can there be?