= Nullity of a cyclic linear operator
{title2=$\dim\ker\alpha=\mathbf1_{\{d(0)=0\}}$}
For a nonzero finite-dimensional cyclic <linear operator>, identify its <module> with $K[X]/(d)$, where $d$ is the monic <minimal polynomial>. Its <kernel> consists of classes of $h$ with $d\mid Xh$. If $d(0)\ne0$, coprimality gives $d\mid h$, so the <kernel> is zero. If $d=Xe$, it consists precisely of the multiples of $e$ modulo $Xe$, a one-dimensional space. Thus the multiplicity of zero as a root of $d$ does not increase the <nullity>. The same companion block reads off invertibility from the constant term and gives the characteristic and minimal <polynomials> directly, while explicitly listing <eigenvalues> may require factoring $d$.
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