Show that the orthogonal projection onto $Range T$ is equal to $T(T^\ast T)^{-1}T^\ast$ given that $T: V \to W$ is injective

by allonsy676   Last Updated August 14, 2019 09:20 AM - source

Given $V$ and $W$ as finite-dimensional inner product spaces and an injective linear map $T: V \to W$, how can we show that $$P_{Range T} = T(T^\ast T)^{-1}T^* \in \mathcal{L}(W)$$ where $P_{Range T}$ is the orthogonal projection onto $Range T$ and $T^\ast$ is the adjoint of $T$? Supposedly this result can be used to derive a useful matrix formula for orthogonal projection, but I'm not sure how to figure out how we can show the equality above. Does $(T^\ast T)^{-1}$ even make sense? How do we show that? And what is the intuition of the above formulation?

Among my ideas are trying to show that the right hand side is in the range of $T$ (which it seems to be) and trying to utilize the fact that $P_{Range T}^2 = P_{Range T}$. But I'm not sure how to proceed from there.

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