If $B = (e_1,e_2,\ldots, e_n)$ is a basis for an inner product space $V$ and $B' = (f_1,f_2,\ldots,f_n)$ is an orthonormal basis of $V$.
Is the change of basis matrix $P$ necessarily orthogonal?
[Math] Change of basis matrix for inner product space
inner-productslinear algebra
Best Answer
The change basis matrix $P$ is $$P=\left(f_1 f_2\cdots f_n\right)$$ where $f_i$ is a column vector and since the basis $B'$ is orthonormal we have $$f_i \ . f_j=\delta_{ij}$$ where $\delta$ is the Kroneker symbol and hence $P$ is orthogonal