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How do I find the basis $B$ of $U$ in $\mathbb{R}^3$ where $U=\{X \in \mathbb{R}^3 | \exists b\in\mathbb{R}^3 : ax + b \cdot y=z \forall x,y,z \in \mathbb{R}^3\}$

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The set $U$ has the standard basis,
$$\mathbb{R}^3 =\{(x,y,z)| x,y,z \in\mathbb{R}\} \;\text{ (assume } 0\; \text{to reflect the standard basis)}$$
where the standard basis has the property that $aX+bY+cZ=0$, for $a,b,c \in \mathbb{R}$ and $X=(x,y,z),Y=(y,z,0),Z=(z,0,0)$, and the property
$$\forall x,y,z \in \mathbb{R}\qquad (aX+bY+cZ=0\;\text{ implies } a

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