# determine a orthogonal basis $(v_1,v_2,v_3)$ of $\mathbb R^3$, such that $Av_1,Av_2,Av_3$ are pairwise orthogonal

Let $A= \begin{bmatrix} 1& 0& -4\\ 2& 1& 1\\ 2& -1& 1 \end{bmatrix}$

$A^TA=\begin{bmatrix} 9& 0& 0\\ 0& 2& 0\\ 0& 0& 18 \end{bmatrix}$

I am asked to determine a orthogonal basis $(v_1,v_2,v_3)$ of $\mathbb R^3$, such that $Av_1,Av_2,Av_3$ are pairwise orthogonal.

If we take the columns of $A^TA$ as the basis $(v_1,v_2,v_3)$, they are obviously an orthogonal basis of $\mathbb R^3$ and we can calculate

$A\begin{bmatrix} 9\\ 0 \\ 0 \end{bmatrix}=$$\begin{bmatrix} 9\\ 18 \\ 18 \end{bmatrix}. A\begin{bmatrix} 0\\ 0 \\ 18 \end{bmatrix}=$$\begin{bmatrix} -72\\ 18 \\ 18 \end{bmatrix}$