Find the dimension of \(A\) considered as a real vector space.

Question

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.