Let $φ: \mathbb{R^2} \to \mathbb{R^2}$ be a linear transformation and $A = \begin{bmatrix}1&-2\\2&2 \end{bmatrix}$ be its representation with respect to the standard basis. Find all of the $φ-$invariant subspaces.
{$0$} and $\mathbb{R^2}$ are trivial invariant subspaces of dimensions $0$ and $2$.
For Dimension $1$, I've shown that there's no invariant subspace of dimension $1$. So, the only invariant subspaces are {$0$} and $\mathbb{R^2}$. Are these really the only ones? Do I need to show there are no other ones?