Invariant subspaces and dimentions

Question

Let V is a finite dimensional vector space over C and T be a linear operator on V . How to prove T has an invariant subspace of dimension k for each k = 1,2, . . . ,dimV .

Answer 1

It can be solved by Jordan form. I just give a hint.

Let $\{e_i\}$ is a basis of $V$. Put $J=\lambda I+N$, where $I$ is the identity map, and $N$ is the nilpotent linear map such that sent $e_i$ to $e_{i+1}$(if $i+1>\dim V$, then 0). Every subspace of the form $<a_{k+1}e_{k+1}+,...,+a_ne_n>$ is an invariant subspace of $\dim n-k$.