Fix vectors $x,y\in\mathbb{R}^d$ and a smooth function $\phi:\mathbb{R}\rightarrow \mathbb{R}$. Define $\phi^d: \mathbb{R}^d \rightarrow \mathbb{R}^d$ as applying $\phi$ entrywise (i.e. $\phi^d(x_1, x_2, \dots) = (\phi(x_1),\phi(x_2), \dots)$). Under what conditions do we have the equality

$$\mathbb{R}^d=\text{span}(\{\phi^d(\alpha x+y)~\big |~\alpha\in \mathbb{R}\})$$

Clearly if $\phi$ is the identity operator $\phi(x)=x$ or if any two entries of $x$ are equal, this equality does not hold. Main claim is equality holds for generic (i.e. almost all) $x,\phi$ choices.