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Let $L: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ be a linear map with the property that $L \mathbf{v} \perp \mathbf{v}$ for every $\mathbf{v} \in \mathbb{R}^{3}$. Prove that $L$ cannot be invertible.
Is a similar assertion true for a linear map $L: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ ?
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