Finding the dimension of a subspace in $\mathbb R^n$ Given $S_{u}^\bot:=\{v\in \mathbb R^n:v \cdot u=0\}$, I've proved that $S_{u}^\bot$ is a subspace of $\mathbb R^n$. The next part of the question asks me to prove that $S_{u}^\bot$ has dimension of $n-1$.
I assume that it has something to do with the number of elements in the basis of the condition inside $S_{u}^\bot$ but do not know how to tackle. I also solved some other parts of the same question but I don't think that they're related to this part particularly.

If you expand $v\cdot u=0$ in coordinates, you get a single homogeneous linear equation satisfied by the elements of $S_u^\perp$. Now apply the rank-nullity theorem to this system of one homogeneous equation.
