Can it be true that $\|v^t x\|>1$?
Let $\|v\|=1$ and $v\in R^n$. Let $x = (1,-1,1,1,-1,....,1)\in R^n$ ($x$ is just a vector of $1$'s and $-1$'s. Can it be true that $\|v^t x\|>1$? I'm looking for examples or a proof that it can't.
Solution 1:
Sure, let $v=\frac{x}{||x||}$ then
$$|v^tx| = \frac{||x||^2}{||x||} = \sqrt{n} \geq 1$$