4. (15 points.) Letbe eigenvectors corresponding to I: distinct cigenvalues of a lincar operator T on a vector space V. Prove that the T-cyclic subspace generated by has dimension k.

Problem 6 :Let A and B be elements in (C). Suppose that AB - BA = c⋅(A - B) for some non-zero c ∈ C. Prove that there exists an invertible matrix P ∈ (C) such that AP and BP are upper-triangular matrices with the same diagonal entries.