I could use some help with the following $2$ questions.
$1)$ I have to prove that for any non-trivial rotation $\rho \in O(V)$ there exists a unique line A (the axis) such that $\rho(a) = a$ $\forall a \in A$. with $dim(V) = 3$
I tried:
Suppose $A \in O(V)$ and then I have to show $Av = v$. So we are asked to find out whether A has an eigenvalue $\lambda = 1$.
I proved the above, but I am not sure if this is the right way to prove the question.
$2)$ Show that the restriction of $\rho$ to $A^⊥$ is a rotation in $O^+(A^⊥)$, where we write $O^+$ for the orientation preserving elements of $O(V)$.
Thanks in advance.