I was watching Singular Value Decomposition Lecture by Gilbert Strang. He takes two orthonormal vectors $v_1$, $v_2$. Let $\sigma_1$$u_1$=A$v_1$ and $\sigma_2$$u_2$=A$v_2$. He takes $u_1$ is orthogonal to $u_2$. How? How to prove this?
Edit1: Adding link to video lecture https://www.youtube.com/watch?v=Nx0lRBaXoz4 (Start from 3:30 mins)