Let $\varphi: \Bbb R^3 \to \Bbb R^3$ be the rotation around the line through the origin and the point $(1, 1, 1)$ by 120 degrees. Decompose $\Bbb R^3$ as a direct sum of two subspaces that are each stable under $\varphi$.
This is the exercise. So I though that this answer is good??? But I am not sure though. I am really doubthful about it.
My answer:
Let $\varphi: \Bbb R^3\to \Bbb R^3$ be a rotation around the line through the origin and the point $(1,1,0)$ by 60 degrees. Also let $\lambda : \Bbb R^3\to \Bbb R^3$ be the rotation through the points $(1,1,0)$ and $(1,1,1)$ by 60 degrees. Then their sum equals the rotation that is described above?
The question asks for a decomposition $R^3 = U \oplus V$ such that $\phi(U) \subset U$ and $\phi(V) \subset V$.
Hint: What happens to the axis of rotation (a one-dimensional subspace) under $\phi$? What is a complementary subspace of the axis of rotation that is also invariant under $\phi$?