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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.]
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