## Acalytica

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Let $\mathcal{P}_{1}$ be the linear space of real polynomials of degree at most one, so a typical element is $p(x):=a+b x$, where $a$ and $b$ are real numbers. The derivative, $D: \mathcal{P}_{1} \rightarrow \mathcal{P}_{1}$ is, as you should expect, the map $D P(x)=b=b+0 x$. Using the basis $e_{1}(x):=1$, $e_{2}(x):=x$ for $\mathcal{P}_{1}$, we have $p(x)=a e_{1}(x)+b e_{2}(x)$ so $D p=b e_{1}$.
Using this basis, find the $2 \times 2$ matrix $M$ for $D$. Note the obvious property $D^{2} p=0$ for any polynomial $p$ of degree at most 1 . Does $M$ also satisfy $M^{2}=0$ ? Why should you have expected this?
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