I have two problems here; one I think I almost have but I'm stuck, and the other I'm pretty much stumped on. 1. The problem statement, all variables and given/known data Suppose V is a complex vector space and T is in L(V). Prove that T has an invariant subspace of dimension j for each j = 1, ... dim(V). 2. Relevant equations 3. The attempt at a solution I know that every operator on a finite-dimensional, nonzero complex vector space has an eigenvalue. I think we also know that V has subspaces of dimension j for each j = 1, ... dim(V). Each of these subspaces is complex, so each of them has at least one eigenvalue, so T(v) = [tex]\lambda[/tex]v for each subpace, which makes T invariant on that subspace. Am I totally off the wall here, or am I close? 1. The problem statement, all variables and given/known data Suppose n is a positive integer and T in L(Fn) is defined by T(x1, ... , xn) = (x1 + ... + xn, ... , x1 + ... + xn). Find the characteristic polynomial of T. 2. Relevant equations Not really an equation, but the notation "F" represents either R or C. 3. The attempt at a solution With respect to the standard basis B, the matrix M(T, B) is an n x n matrix consisting of all 1s. To find the characteristic polynomial, I need to find det(I*x - M(T, B)), but I am having trouble doing that. Any tips on how to calculate this determinant? Or is there perhaps a simpler approach? I know the eigenvalues are 0 and n. Thanks for your help, guys!