Proving $\{v, T(v)\}$ is linearly dependent $\iff T(v) = \pm v$ with $T \circ T = I$
Solution 1:
Your work for the first direction is correct. For the other direction, notice that $T(v) = \lambda v$ since the set is linearly dependent, and since $T \circ T = I$, we get $v = \lambda T(v) = \lambda^2 v$, therefore $\lambda^2 - 1 = 0$ if $v \neq 0$, so $\lambda \in \{-1, 1\}$.