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

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.)

• 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. – user239203 Aug 3 '20 at 7:46

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