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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


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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.
in Mathematics by ♦ Platinum (93,241 points) | 207 views

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