Theorem 7.3. Let $T$ be a linear operator on a finite-dimensional vector space $V$ such that the characteristic polynomial of $T$ splits, and lett $\lambda_1,\lambda_2,...,\lambda_k$ be the distinct eigenvalues of $T$. Then, for every $x\in V$, there exist vectors $v_i \in K_{\lambda_i}$, $1\le i \le k$, such that
$x = v_1 + v_2 + \cdots +v_k$.
Proof. The proof is by mathematical induction on the number $k$ of distinct eigenvalues of $T$. First suppose that $k = 1$, and let $m$ be the multiplicity of $\lambda_1$. Then $(\lambda_1 - t)^m$is the characteristic polynomial of $T$, and hence $(\lambda_1 I - T)^m = T_0$ by the Cayley-Hamilton theorem(p.317). Thus $V=K_{\lambda_i}$, and the result follows.
where $K_{\lambda_i}$ = $\{v \in V: (T-\lambda_1 I)^p(v) = 0$ for some positive integer $p$}.
I wanted to know why it follows that $V=K_{\lambda_i}$?
