# Thread: Change of basis matrices

1. ## Change of basis matrices

Prove that the change of basis matrix from one orthonormal basis of $\displaystyle \mathbb{R}_{n \times 1}$ to another is always orthogonal.

2. If $\displaystyle \{u_i\}$ and $\displaystyle \{v_i\}$ are bases for the same vector space, then A is the "change of basis matrix" if and only if $\displaystyle Au_i= v_i$ and $\displaystyle u_i= A^{-1}v_i$. If $\displaystyle \{v_i\}$ is an orthonormal basis then $\displaystyle <v_i, v_j}= \delta_{ij}$ where $\displaystyle \delta_{ij}$ is the "Kronecker delta", equal to 1 if i= j, 0 otherwise.

$\displaystyle <v_i, v_j>= <Au_i, Au_j>= <u_i, A^TAu_j>= \delta_{ij}$