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In an n-dimensional Euclidean space V two bases $\{\alpha_1,\alpha_2,…,\alpha_n\}$ and $\{\beta_1,\beta_2,…,\beta_n\}$ are given so that $(\alpha_i │\alpha_j)=(\beta_i│\beta_j)$ for all i and j. Prove that there exists an orthogonal linear operator T on V such that $T\alpha_i=\beta_i$.

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Since the the sets $\{\alpha_i\mid 1\leq i\leq n\}$ and $\{\beta_i\mid 1\leq i\leq n\}$ are bases, you can certainly define a linear operator such that $T\alpha_i=\beta_i$. The question is whether it is orthogonal.

We need to show that $(Tu,Tv)=(u,v)$ for any $u,v\in V$ which, by bilinearity, reduces to showing that $(T \alpha_i,T\alpha_j)=(\alpha_i,\alpha_j)$ for all $i$ and $j$. But this follows because $$(T\alpha_i,T\alpha_j)=(\beta_i,\beta_j)=(\alpha_i,\alpha_j).$$

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