# Spanning Invariant Subspaces

Question: Let $T : V → V$ be a unitary linear transformation on a finite-dimensional inner product space V . Let $W ⊂ V$ be a $T$-invariant subspace. Prove that $T(W) = W$ and $T(W^⊥) = W^⊥$

Having a lot of trouble with this question and don't know where to start. Can someone lend me a hand?

You are given that $TW\subset W$; but $T$ is injective, so it preserves dimension and you get $TW=W$.
If $v\in W^\perp$ and $w\in W$, $$\langle T^*v,w\rangle=\langle v,Tw\rangle=0,$$ since $TW\subset W$. It follows that $T^*W^\perp\subset W^\perp$ and now you can deduce $T^*W^\perp=W^\perp$ since $T^*$ is injective.
• 1. Is $T$ always assumed to be injective iff $T$ is unitary? 2. Preserving dimension implies that $W ⊂ T(W)$, and thus $T(W) = W$, correct? Oct 5, 2016 at 8:05
• No. "Preserving dimension" means exactly that, that $\dim TW=\dim W$. Oct 5, 2016 at 8:23