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

I was watching Singular Value Decomposition Lecture by Gilbert Strang. He takes two orthonormal vectors $v_1$, $v_2$. Let $\sigma_1$$u_1=Av_1 and \sigma_2$$u_2$=A$v_2$. He takes $u_1$ is orthogonal to $u_2$. How? How to prove this?

• You're assuming $\sigma_1\ne \sigma_2$ here. Consider $\langle Mv_1,u_2\rangle$, and use $Mv_1=\sigma_1 u_1$ and $M^*u_2 = \sigma_2 v_2$. – Ted Shifrin May 12 '18 at 5:01