I have an exercise which is:
Let $u = (1,2,3)$. If $H=\{u\in\mathbb{R}^3 \mid u\cdot v\}=0$, show that $H$ is a subspace of $\mathbb{R}^3$.
I know if $u\cdot v=0$, they are linearly independent so they should generate a subspace and it would be $\mathbb{R}^2$ but I don't know how to demonstrate it, I never did something like this.
Thank you for your help :)