Let $S$ be the vector space of functions $f: \mathbf{R} \rightarrow \mathbf{R}$ such that the $n$th derivative of $f$ is defined and continuous for every $n \geqslant 0$. Define linear maps $A, B: S \rightarrow S$ by $A(f)=d f / d x$ and $B(f)(x)=x f(x)$. Show that

$$[A, B]=1_{S},$$

where in this question $[A, B]$ means $A B-B A$ and $1_{S}$ is the identity map on $S$.

Now let $V$ be any real vector space with linear maps $A, B: V \rightarrow V$ such that $[A, B]=1_{V}$. Suppose that there is a nonzero element $y \in V$ with $A y=0$. Let $W$ be the subspace of $V$ spanned by $y, B y, B^{2} y$, and so on. Show that $A(B y)$ is in $W$ and give a formula for it. More generally, show that $A\left(B^{i} y\right)$ is in $W$ for each $i \geqslant 0$, and give a formula for it.

Show, using your formula or otherwise, that $\left\{y, B y, B^{2} y, \ldots\right\}$ are linearly independent. (Or, equivalently: show that $y, B y, B^{2} y, \ldots, B^{n} y$ are linearly independent for every $n \geqslant 0$.)