Let $T:\mathbb{R}^2\to\mathbb{R}^2$ denote the linear transformation that is the projection onto the line $y=2x$. List all $F[X]$-submodules of $\mathbb{R}^2$ (where the $F[X]$-module structure here is determined by $T$.)
So if $f(x) = a_{0}+\cdots+a_{n}x^{n}$, we define scalar multiplication as follows: $$ f(x) \cdot \begin{bmatrix} x\\ y\\ \end{bmatrix} = a_{0} \begin{bmatrix} x\\ y\\ \end{bmatrix} + a_{1}T\begin{bmatrix} x\\ y\\ \end{bmatrix} + \cdots + a_{n}T^{n}\begin{bmatrix} x\\ y\\ \end{bmatrix} $$
Using orthogonal projection we have the following: $$ f(x) \cdot \begin{bmatrix} x\\ y\\ \end{bmatrix} = a_{0}\begin{bmatrix} x\\ y\\ \end{bmatrix} + a_{1}\begin{bmatrix} \frac{x+2y}{5}\\ \frac{2x+4y}{5}\\ \end{bmatrix} + \cdots + a_{n}\begin{bmatrix} \frac{x+2y}{5}\\ \frac{2x+4y}{5}\\ \end{bmatrix} $$
So this is the scalar multiplication of $\mathbb{R}^2$ as an $F[x]$-module. But now the question is what do all submodules look like? I know that submodules are closed under addition and scalar multiplication, where the scalar multiplication looks as I've described, but I'm having a hard time listing out the specifics of the spaces. Any hints/suggestions for tackling this are welcome. Thank you!