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I have two problems here; one I think I almost have but I'm stuck, and the other I'm pretty much stumped on.

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).

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?

Suppose n is a positive integer and T in L(F

Not really an equation, but the notation "F" represents either R or C.

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!

## Homework Statement

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).

## Homework Equations

## 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?

## Homework Statement

Suppose n is a positive integer and T in L(F

^{n}) is defined by T(x_{1}, ... , x_{n}) = (x_{1}+ ... + x_{n}, ... , x_{1}+ ... + x_{n}). Find the characteristic polynomial of T.## Homework Equations

Not really an equation, but the notation "F" represents either R or C.

## 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!

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