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Given a unit vector $\mathbf{w} \in \mathbb{R}^{n}$, let $W=\operatorname{span}\{\mathbf{w}\}$ and consider the linear map $T: \mathbb{R}^{n} \rightarrow$ $\mathbb{R}^{n}$ defined by
$T(\mathbf{x})=2 \operatorname{Proj}_{W}(\mathbf{x})-\mathbf{x},$
where $\operatorname{Proj}_{W}(\mathbf{x})$ is the orthogonal projection onto $W$. Show that $T$ is one-to-one.
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