Orthogonal complement of vector of all 1's

Question

In the vector space of $n$ dimensional real valued vectors, and considering the vector of all $1$'s, $\mathbf{x}=\mathbf{1},$ do we know the orthogonal complement of $\mathbf{x},$ i.e. $\mathbf{1}^\perp?$ in other words, the set of all vectors $\mathbf{b}$ such that $\langle \mathbf{b},\mathbf{1}\rangle=0.$

I ask because I have a matrix whose kernel is $\mathbb{R}\mathbf{1}$ and I'm trying to figure out the orthogonal complement of this kernel and do not know how to express it.

Answer 1

The $n-1$ dimensional subspace spanned by $$\{(-1,1,0,0,...,0), (0,-1,1,0,...,0),$$

$$(0,0,-1,1,0,...,0),...,(0,0,0,...,-1,1)\}$$ is the orthogonal complement of $(1,1,1,....,1)$