For each $m,n\in \mathbb{N}$ think ${\mathbb{R}}^{m}$ as the space of real column vectors of size $m$ and ${\mathbb{R}}^{m\times n}$ as the space of matrices of size $m\times n$.

Let $d\in \mathbb{N}$

Let $a:\{1,\dots ,{2}^{d}\}\to \{1\}\times \{0,1{\}}^{d}$ be an enumeration (injective and surjective map).

Let $A\in {\mathbb{R}}^{(d+1)\times {2}^{d}}$ be the matrix whose columns are $a(1),\dots ,a({2}^{d})$

Is it true that for each $b\in \{1\}\times [0,1{]}^{d}$ there exists $x\in [0,+\mathrm{\infty}{)}^{{2}^{d}}$ such that $Ax=b?$?

Let $d\in \mathbb{N}$

Let $a:\{1,\dots ,{2}^{d}\}\to \{1\}\times \{0,1{\}}^{d}$ be an enumeration (injective and surjective map).

Let $A\in {\mathbb{R}}^{(d+1)\times {2}^{d}}$ be the matrix whose columns are $a(1),\dots ,a({2}^{d})$

Is it true that for each $b\in \{1\}\times [0,1{]}^{d}$ there exists $x\in [0,+\mathrm{\infty}{)}^{{2}^{d}}$ such that $Ax=b?$?