I'm trying to find all subspaces of $\Bbb F^n$ (where $\Bbb F$ can be $\Bbb R$ or $\Bbb C$) invariant under $\Phi$ where $\Phi:\Bbb F^n\to \Bbb F^n$ is given by
$$\Phi(x_1,\dots, x_n) = (x_1,2x_2, \dots, nx_n)$$
I've already figured out that eigenspaces have the form $\operatorname{span}(0,\dots, 0,1,0,\dots, 0)$ where the $1$ is in the $i$th position for all $i\in \{1,\dots, n\}$. So these are all the $1$-dimensional invariant subspaces. Beyond these and the two trivial invariant subspaces, every sum of the $1$-dimensional subspaces should be an invariant subspace as well. For instance, the subspace $$\operatorname{span}(1,0,\dots, 0) + \operatorname{span}(0,1,0,\dots, 0) = \text{the $xy$-plane}$$ should be invariant as well.
So can I conclude that every subspace of $\Bbb F^n$ is an invariant subspace under $\Phi$? Or are there subspaces of $\Bbb F^n$ that can't be written as a sum of the spans of the standard basis vectors? And if I am forgetting some (as I suspect I am), how can be either be sure I've gotten all of the invariant subspaces or find the ones I've missed?
Thanks.
Edit: As I learned from Mark in the comments, clearly I can't conclude that every subspace of $\Bbb F^n$ is an invariant subspace under $\Phi$. So how can I either rule out any subspaces that aren't a sum of spans of the basis vectors or find the subspaces that I'm missing so far?
Edit 2: I have only just gotten to eigenspaces/ invariant subspaces. I don't know what a minimal polynomial is or what a diagonalisable map is.
Suppose $U\leqslant \mathbb{F}^n$ is $\Phi$-invariant, and consider the restriction of $\Phi$ to $U$, denoted say by $\Psi$. Then the minimal polynomial of $\Psi$ divides the minimal polynomial of $\Phi$; hence is also a product of distinct linear factors, so that $\Psi$ is diagonalisable. That is, $U$ is the direct sum of eigenspaces of $\Psi$; but each eigenspace of $\Psi$ is an eigenspace of $\Phi$. Hence $U$ is one of the $2^n$ possible sums of the $n$ eigenspaces of $\Phi$.