A question on definition of generalized eigenspaces/ characteristic polynomial I am studying for quals and have a couple of questions on generalized eigenspaces/characteristic polynomials.
Suppose $T : V \to V$ is a linear map of a finite dimensional vector space over $k = \overline{k}$. Then Theorem 8.23 of Axler states:


Theorem 8.23 (Axler): Write $\lambda_1,\ldots,\lambda_m$ for the eigenvalues of $T$. There is a decomposition $$V \cong  \ker(T-\lambda_1I)^{\dim V} \oplus \ldots \oplus \ker(T -\lambda_mI)^{\dim V}.$$
Question 1: Is it true that $\ker(T-\lambda_iI)^{\dim V} = \ker(T -\lambda_iI)^{d_i},$ where  $d_i$ is the multiplicity of the linear factor $t - \lambda$ in the characteristic polynomial $\chi_T(t)$? I have a feeling this is true.
Question 2: If question 1 is true, can we see it from the structure theorem of modules over a PID?



$\ker\left ((T - \lambda I)^{\dim V}\right )$ is the generalized eigenspace of $\lambda$. $\alpha_\lambda$ is the smallest exponent such that $(T - \lambda I)^{\alpha_\lambda}$ is zero on that space, so $\alpha_\lambda \le \dim V$. But the definition of $\alpha_\lambda$ says $\ker\left ((T - \lambda I)^{\dim V}\right ) \subseteq \ker\left ((T - \lambda I)^{\alpha_\lambda}\right )$, while $\alpha_\lambda \le \dim V$ implies $\ker\left ((T - \lambda I)^{\alpha_\lambda}\right ) \subseteq \ker\left ((T - \lambda I)^{\dim V}\right )$, so the two are equal, and $\ker\left ((T - \lambda I)^{\beta_\lambda}\right )$ is caught between them.
