Question

Let \(\mathcal{P}_{k}\) be the space of polynomials of degree at most \(k\) and define the linear map \(L: \mathcal{P}_{k} \rightarrow \mathcal{P}_{k+1}\) by \(L p:=p^{\prime \prime}(x)+x p(x) .\)

 

a) Show that the polynomial \(q(x)=1\) is not in the image of \(L\). [SUGGESTION: Try the case \(k=2\) first.]
b) Let \(V=\left\{q(x) \in \mathcal{P}_{k+1} \mid q(0)=0\right\}\). Show that the map \(L: \mathcal{P}_{k} \rightarrow V\) is invertible. [Again, try \(k=2\) first.]