Suppose we have a real or complex finite dimensional vector space $V$ with an inner product $\langle \cdot , \cdot \rangle$ and a basis $B = \{v_1, \dots, v_n\}$. Define the matrix of the inner product in basis $B$, $| \langle \cdot , \cdot \rangle |_B$ by

$$ \left(| \langle \cdot , \cdot \rangle |_B \right)_{ij} = \langle v_i, v_j \rangle . $$

We have that $\langle v, w \rangle = (v)_B | \langle \cdot , \cdot \rangle |_B \overline{(w)_B^t}$.

Now, I thought of the following formula to get the matrix of the inner product in a different basis $B'$, and I'd like to know if it's correct. It looks like it to me, but I want to make sure I didn't overlook anything:

$$ | \langle \cdot , \cdot \rangle |_{B'} = C(B', B)^t | \langle \cdot , \cdot \rangle |_B \overline{C(B',B)} $$

Where $C(B',B)$ is the change of basis matrix from $B'$ to $B$.


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