Theorem: Let $V$ be a vector space over $\mathbb{C}$, and let $A: V \to V$ be a linear map. Let $P(t) $ be a polynomial such that P(A) = O, and let
$$P(t) = (t - \alpha_1)^{m_1}....(t-\alpha_n)^{m_r}$$ be its factorzation, the $\alpha_1, \alpha_2, .... , \alpha_r$ being the distinct roots. Let $W_i$ be the kernel of $(A - \alpha_i I)^{m_i}$. Then $V$ is the direct sum of the subspaces $W_1, ... , W_r$.
The author of my textbook (Serge Lang, Linear Algebra UTM) states that the following claim follows from the above theorem.
Claim: Let $V \neq \lbrace O \rbrace $ be a finite-dimensional complex vector space. Let $A: V \to V$ be a linear map. Then there exists a number $\alpha$ and an integer $r\geq1 $ such that $(A - \alpha I)^r = O$.
My problem is that I don't see how.
I just realized that Qiaochu is correct. Consider the matrix $A = \begin{bmatrix}1 & 0\\0 & 0\end{bmatrix}$. Assume that $(A - \alpha I)^r = 0$. Then: $${\begin{bmatrix}1-\alpha & 0\\0 & -\alpha\end{bmatrix}}^r=\begin{bmatrix}(1-\alpha)^r & 0\\0 & (-1)^r \alpha\end{bmatrix}=0$$, necessitating that $\alpha = 0$. However $(1-(0))^r = 1^r = 1$, contradicting our assumption that $(A - \alpha I)^r = 0$. Therefore no such $\alpha$ and $r$ exist for this matrix, invalidating the claim.