= Solution
Write $K_i=\ker T^i$, with $K_0=\{0\}$. The <minimal polynomial> $x^m$ means $T^m=0$ and $T^{m-1}\ne0$. Choose $w$ with $T^{m-1}w\ne0$. For each $0\leq i<m$, set $w_i=T^{m-1-i}w$. Then
$$
T^{i+1}w_i=0,\qquad T^iw_i=T^{m-1}w\ne0.
$$
Thus $w_i\in K_{i+1}\setminus K_i$. Also $K_i\subseteq K_{i+1}$ because applying $T$ to zero still gives zero. Therefore the <kernel dimensions of powers of a nilpotent map> increase strictly until the <nilpotency index>:
$$
\boxed{\{0\}=K_0\subsetneq K_1\subsetneq\cdots\subsetneq K_m=V}.
$$
The same reasoning covers $m=1$, when the <nilpotent linear map> is zero on a nonzero <vector space>.
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