Let V be a vector space and T elementof L(V). (a) Prove that if v is an eigenvector of T with eigenvalue lambda, and lambda notequalto 0, then v elementof Ran(T). Give a counterexample to show that this statement is false if you don't assume lambda notequalto 0. (b) Suppose that dim Ran(T) = k. Prove that T has at most k+1 distinct eigenvalues.
Show transcribed image textLet V be a vector space and T elementof L(V). (a) Prove that if v is an eigenvector of T with eigenvalue lambda, and lambda notequalto 0, then v elementof Ran(T). Give a counterexample to show that this statement is false if you don't assume lambda notequalto 0. (b) Suppose that dim Ran(T) = k. Prove that T has at most k+1 distinct eigenvalues.