Invariant space of linear transformation

Question

Let $V$ be a vector space of a finite nonzero dimension $n$ over some field. Let $T$ be a linear transformation of $V$, such that $T$ is nonzero and not one-to one.
(a)Give a $T$-invariant linear subspace $V_1$ of $V$ such that $0<dim V_1 <n$.
(b)Suppose that for some vector $x\in V$ the set of all natural numbers $m$ such that $T^m x=0$ is nonempty. Show that this set must be of the form $\{k,k+1,\ldots\}$ for some natural number $k$ that is no greater than $n$.

Answer 1

Hint: (a) What can you say about $\dim\ker T$?
(b) Let $M = \{m \in \mathbb N \mid T^m x \}$. As $M \ne \emptyset$, we can set $k = \min M$. Show that than $M = \{k, k+1, \ldots\}$, as from $m \in M$ you have $T^{m+1}x = T(T^m x) = \cdots$.