Orthogonality vs Orthonormality

Question

Let $v_{1}, ..., v_{n}$ be orthonormal vectors in $\mathbb{R}^{n}$. How can we show that $Av_{1}, ..., Av_{n}$ are also orthonormal if and only if $A \in \mathbb{R}^{n \times n}$ is orthogonal?

Answer 1

Notice that $\left\langle Av_i,Av_j\right\rangle=\left\langle A^TAv_i,v_j\right\rangle$. So clearly if $A$ is orthogonal, $Av_1,\dots ,Av_n$ are orthogonal as well.

Conversely, suppose that $Av_1,\dots ,Av_n$ are orthogonal. Then $\left\langle Av_i,Av_j\right\rangle=\left\langle A^TAv_i,v_j\right\rangle=\delta_{i,j}$. What does this tell you about $A^TA$?