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?