Suppose there are 2 orthogonal vectors $v_1$ and $v_2$ so $v_1^Tv_2=0$. Suppose we take linear transformations of the two vectors. Let $A$ be the transformation matrix. Let $u_1=Av_1; u_2=Av_2;$ Are $u_1$ are $u_2$ orthogonal? i.e. is $u_1^Tu_2=0$? How to prove it. I tried the following way, but I'm stuck.
I'm stuck here, unable to proceed further.
If the above equation is not true for any random vectors $v_1$ and $v_2$, is it true for any specific vectors? What if $v_1$ and $v_2$ are in $Row$ $space$ $of$ $A$?