Is there a basis $\beta$ for $V$ such that $\langle \mathbf{v}, \mathbf{w}\rangle=\langle [\mathbf{v}]_{\beta}, [\mathbf{w}]_{\beta}\rangle$.

Question

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.)

Answer 1

You need $\beta$ to be an orthonormal basis. You can always obtain such basis by using the Gram-Schmidt process.