$V$ is Euclidean space, $W$ is it's subspace.
$T$ is an orthogonal transformation T: V $\rightarrow$ V
$W$ is $T$ invariant.
Is W equal to the range of $T(w)$, $w \in W$?
I thought about it visually:
Orthogonal transformation can be viewed as a (rigid or improper) rotation.
Given a unique vector the result of the transformation will also be unique, so the range of $T(w)$ must be the same size than that of W.
Given that W is $T$ invariant: $T(w) \in W$, $w \in W$, then they must be equal.
Is this correct?