= Differentiation on an exponential-polynomial space
{title2=$D(p(t)e^{\lambda t})=e^{\lambda t}(p'(t)+\lambda p(t))$}
For distinct real or complex numbers $\lambda_j$ and positive integers $m_j$, the <vector space> spanned by $t^ke^{\lambda_jt}$, $0\leq k<m_j$, is preserved by <differentiation>. In each ordered <basis> $(e^{\lambda_jt},te^{\lambda_jt},\ldots)$, its <matrix> has $\lambda_j$ on the diagonal and $1,2,\ldots,m_j-1$ just above it. This follows from the <product rule>. Rescaling to $t^ke^{\lambda_jt}/k!$ makes each block a <Jordan block>. In particular, <differentiation> is invertible on this <vector space> exactly when every $\lambda_j\ne0$: the <determinant> is $\prod_j\lambda_j^{m_j}$.
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