How to construct basis of an affine hyperplane $\mathbb{H} = \{x \in \mathbb{R}^n | \sum_{i = 1}^n x_i = b, b \geq 0\}$?

Question

Consider an affine hyperplane of the form,

$\mathbb{H} = \{x \in \mathbb{R}^n | \sum_{i = 1}^n x_i = b, b \geq 0\}$

Is there a method for constructing a basis for $\mathbb{H}$?

That is, is there a method of finding a set of (linearly independent or orthonormal) vectors $v_1, \ldots, v_n$, such tha any vector in $\mathbb{H}$ can be expressed as $c_1 v_1 + \ldots + c_n v_n \in \mathbb{H}$ for some $c_1, \ldots, c_n \in \mathbb{R}$?

Answer 1

Let $v_1 = \frac{1}{\sqrt{n}}(1, \ldots, 1)$ be the normalized all-ones vector, and extend it to an orthonormal basis $v_1, \ldots, v_n$. Check that the elements of $\mathbb{H}$ are of the form $c_1 v_1 + \cdots + c_n v_n$ with $c_1 = \sqrt{n} b$ and any scalars $c_2, \ldots, c_n$.

Answer 2

If $b \neq 0$ then the linearly independent elements $be_k \in H$, $k=1,...n$ form a basis for $\mathbb{R}^n$.

If $b=0$ then the linearly independent elements $e_k-e_{k-1} \in H$, $k=2,...,n$ form a basis for the $n-1$ dimensional subspace $H$.

Regardless of $b$, the elements $e_k-e_{k-1}+{1 \over n} be$, $k=2,...,n$, along with $e_1-e_n+{1 \over n} be$ (where $e=(1,...,1)$) are an affinely independent subset of $H$ whose affine span is $H$. The dimension of $H$ as an affine set is the dimension of the parallel subspace which is seen to be $n-1$.