Multiplication of a vector by an orthogonal matrix

Question

I have a question, consider $V$ an orthogonal matrix, and $u$ and $z$ are vectors, and W is a matrix does :

$V'u = W V'z \implies u = W z$ ?

I want to get rid of the orthogonal matrix $V'$, my intuition says that I can, but I don't know which property of the orthogonal matrices will help me to do say. Thank you in advance.

Answer 1

Assuming $V'u = WV'z$, we have

$$V'(u - Wz) = (WV' - V'W) z.$$

For the left-hand side to be zero for arbitrary $z$, $W$ and $V'$ have to commute, so your statement is not true in general.