Originally Posted by
SlipEternal Here are the bases for the $\displaystyle \phi$-invariant subspaces: $\displaystyle \emptyset, \{x^0\}, \{x^0,x^1\}, \ldots, \{x^0,x^1,\ldots, x^{n-1}, x^n\}$. So, there are $\displaystyle n+2$ total. (The empty set is the basis for the trivial subspace.)