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Consider the two linear transformations on the vector space $V=\mathbf{R}^{n}$ :
$R=$ right shift: $\left(x_{1}, \ldots, x_{n}\right) \rightarrow\left(0, x_{1}, \ldots, x_{n-1}\right)$
$L=$ left shift: $\left(x_{1}, \ldots, x_{n}\right) \rightarrow\left(x_{2}, \ldots, x_{n}, 0\right)$
Let $A \subset$ End $(V)$ be the real algebra generated by $\mathrm{R}$ and $\mathrm{L}$. Find the dimension of $A$ considered as a real vector space.
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