Let $0<s\leq t\leq u\leq v$ and $W_x$ be a Brownian motion. Show that $aW_s+bW_t$ and $\frac{1}{v}W_v-\frac{1}{u}W_u$ are independent for $a,b$ satisfying $as+bt=0$.
The question seems easy but somehow I can't rearrange the terms to show the independence. I know that increments of Brownian motion are independent, so I tried to add, subtract, multiply, divide but I didn't manage to make these increments appear. For example $aW_s+bW_t=aW_s+bW_t -(as+bt)W_s$ or $aW_s+bW_t=aW_s+bW_t -(as+bt)W_t$ don't lead me anywhere.
Thank you
Independence is equivalent to zero covariance when normals are jointly normal. So just compute the expectation of the product and see if it's zero.