1. Show that if ?1 and ?2 are different eigenvalues of A and u1and u2 are associated eigenvectors, then u1 and u2 are independent.More generally, show that if ?1, ..., ?k are distinct eigenvaluesof A and ui is an eigenvector associated to ?i for i=1, ..., k,then u1, ..., uk are independent.
2. Show that for each eigenvalue ?, the set E(?) = {u LaTeX:in?Rn: u is an eigenvector associated to ?} is a subspace ofRn.
