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Let $S$ be an operator on $\mathbb{R}^4$ having eigenvectors $$((1 ,1, 1, 1)^T, (1 ,1, -1, -1)^T, (1 ,-1, 1, -1)^T, (1, -1, -1, 1)^T)$$ with corresponding eigenvalues $2$, $3$, $4$, $5$. Let $T$ be the operator on $\mathbb{R}^4$ having eigenvectors $$((1, 1, 1, 1)^T ,(1, 1, -1, -1)^T ,(1, -1, 1, -1)^T, (1, 0 ,0 ,0)^T)$$ with corresponding eigenvalues $2$, $3$, $4$, $5$. Prove that there exists an invertible operator $āˆ$ such that $āˆ^{-1}Sāˆ=T$, but it is not possible for $āˆ$ to be an isometry.

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