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Given an inner product product $\langle \cdot, \cdot\rangle$ on a finite dimensional vector space $V$ over $F$, $F=\mathbb{R}$ or $F=\mathbb{C}$.

My Questions

  1. Is there a basis $\beta$ for $V$ such that $\langle \mathbf{v}, \mathbf{w}\rangle=\langle [\mathbf{v}]_{\beta}, [\mathbf{w}]_{\beta}\rangle$ for every $\mathbf{v}, \mathbf{w}\in V$, where the second inner product is the standard inner product on $F^n$.

  2. Which books I can find this theorem? (I believe it is true.)

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    $\begingroup$ Yes, provided that by inner product you mean Hermitian and positive definite. This should be in any linear algebra book that deals with Hermitian forms. The keyword is Gram-Schmidt orthonormalization. $\endgroup$ – user239203 Aug 3 '20 at 7:46
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You need $\beta$ to be an orthonormal basis. You can always obtain such basis by using the Gram-Schmidt process.

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  • $\begingroup$ Aha! I see. Thanks. $\endgroup$ – bfhaha Aug 3 '20 at 7:58

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