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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.

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

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