How to prove that 2 vectors in C(A) corresponding to orthogonal vectors in row-space of A are orthgonal?

Question

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)

Answer 1

I found an explanation here and here. It explains that we choose $v_1$ such that $Av_1$ is maximum. Then we choose $v_2$ such that $v_2 \perp v_1$ and $Av_2$ is maximum. In that case, the above referred answers prove that $$Av_1 \perp Av_2$$ We choose $$u_1=Av_1/||Av_1||$$ $$u_2=Av_2/||Av_2||$$ Hence $$ u_1 \perp u_2$$

I guess Gilbert Strang omitted/missed this explanation in that lecture.

Answer 2

Recall that the SVD is given by $ A = U \Sigma V ^ T $, hence $ A ^ T A = ( U \Sigma V ^ T) (U \Sigma V ^ T)^T = U \Sigma ^ 2U ^T$. Namely, the columns of $U$ are the eigenvectors of $A ^ T A$, and as $A^TA$ is symmetric it is can be diagonilzed by an orthogonal matrix ($U$).