# Change of basis matrices

Prove that the change of basis matrix from one orthonormal basis of $\mathbb{R}_{n \times 1}$ to another is always orthogonal.
If $\{u_i\}$ and $\{v_i\}$ are bases for the same vector space, then A is the "change of basis matrix" if and only if $Au_i= v_i$ and $u_i= A^{-1}v_i$. If $\{v_i\}$ is an orthonormal basis then $ where $\delta_{ij}$ is the "Kronecker delta", equal to 1 if i= j, 0 otherwise.
$= = = \delta_{ij}$