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Well, Does orthogonal linear map preserves the inner product?

I know $T$ is orthogonal iff $||T(x)||^2=||x||^2$, and I know that two vector may not be orthogonal if we change the inner product, I have not example right now though.

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up vote 5 down vote accepted

For an inner product space over $\mathbb{R}$, we can use the identity \begin{equation} \langle x, y \rangle = \frac14 \left( \| x + y \|^2 - \|x - y\|^2 \right). \end{equation}

So assuming $T$ preserves norms, \begin{align*} \langle Tx, Ty \rangle &= \frac14 \left( \| Tx + Ty \|^2 - \| Tx - Ty \|^2 \right) \\ &= \frac14 \left( \| T(x + y) \|^2 - \| T(x - y) \|^2 \right) \\ &= \frac14 \left( \| x + y \|^2 - \| x - y \|^2 \right) \\ &= \langle x, y \rangle. \end{align*}

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