Exercise Let $T: \mathbb{R}^3 \to \mathbb{R}^3$ be the linear operator given by
$$T\begin{pmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \end{pmatrix} = \begin{bmatrix} x_3 \\ x_1 \\ x_2 \end{bmatrix}$$
Determine all $T$-invariant subspaces of $\mathbb{R}^3$.
We attempt to find all $T$-invariant subspaces is to find all $W \leqslant \mathbb{R}^3$ satisfying
$$\forall w \in W, T(w) \in W$$
First, $W = \begin{Bmatrix} \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \end{Bmatrix}$ and $W = \mathbb{R}^3$ are $T$-invariant.
Second, if we let $W = Span \begin{Bmatrix} \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \end{Bmatrix}$, we see that for all $w \in W, T(w) \in W$ since $x_1 = x_2 = x_3$
Third, range$(T)$ and null$(T)$ are both $T$-invariant.
Now, consider that $T\begin{pmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \end{pmatrix} = \begin{bmatrix} x_3 \\ x_1 \\ x_2 \end{bmatrix}$. Then the matrix corresponding to $T$, call it $M_T$, is
$$M_T = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ \end{bmatrix}$$
We verify this by seeing that
$$M_T = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} x_3 \\ x_1 \\ x_2 \end{bmatrix}$$
Where do I go from here? Is there something to do with eigenvalues or eigenvectors at this point?
Or have I already found all of the $T$-invariant subspaces of $\mathbb{R}^3$?
Clearly, you do have to consider eigenvectors, for it is the very reason why this is a paramount concept in linear algebra. (Note that $range(T)$ is not invariant in general. Since the matrix is non-singular, $range(T)=\mathbb{R}^3$ and $null(T)={0}$ are obviously invariant as you had pointed out)
You can check that $M^3=I_3$, meaning the eigenvalues are the third roots of unity, and the only real one (1) corresponds to the eigenvector that you found.
This is not however the only invariant subset: since this linear map permutates the basis vectors, it is actually a rotation of $+120°$ around the span you described, meaning all planes orthogonal to $(1,1,1)$ are also invariant.
You can learn more here: https://en.wikipedia.org/wiki/Rotation_matrix#General_rotations