# Show that $\langle T(T^*T)^{-1} T^* w, Tv \rangle = \langle w, Tv \rangle$ for all $v$

The problem states that $$V$$ and $$W$$ are inner product spaces, $$V$$ is finite-dimensional, $$w \in W$$ is a fixed vector, and $$T \in \mathcal{L}(V,W)$$ is injective.

There are 2 parts to the problem:

(a) Show that $$\langle T(T^*T)^{-1} T^* w, Tv \rangle = \langle w, Tv \rangle$$ for all $$v$$

(b) Show that $$P_{\mathrm{range \,}T} = T(T^*T)^{-1}T^*$$. In particular, what can you say about $$||T(T^*T)^{-1} T^* w - w ||?$$

I don't know how to solve either of them. I'm hoping that (a) will lead me to (b). For (a), I started with $$\langle T(T^*T)^{-1} T^* w, Tv \rangle = \langle w, Tv \rangle \Longrightarrow \langle (T^*T)^{-1} T^* w, T^*Tv \rangle = \langle w, Tv \rangle$$, which I know is not very far, and I can't figure out where to go from here. I know that I can't use $$(T^*T)^{-1} = T^{-1}(T^*)^{-1}$$, because $$T$$ is not necessarily invertible.

• You're close on (a). I would move $T$ over to the left part of the inner product instead of to the right. – Cameron Williams Aug 14 at 3:22
• Ah I see it now. I don't know how I missed that. Thanks! – Jayanth Rao Aug 14 at 20:50