If $A$ is an affine space in $\mathbb{R}^n$, is $A-x$ a subspace of $\mathbb{R}^n$ for every $x \in A$?

Question

If $A$ is an affine space in $\mathbb{R}^n$, is $A-x$ a subspace of $\mathbb{R}^n$ for every $x \in A$?

Here $A-x=\{a-x:a \in A\}$.

I think the answer has to be yes, since it will now contain the origin, unless I'm missing some key distinction.

Answer 1

I do not remember the exact definition of affine space but I am sure that $A$ can be written as $v+L$ for a $v \in \mathbb{R}^n$ and $L$ a linear subspace. Thus a generic $x \in A$ is in the form $x=v+y$ for $y \in L$ and we easily conclude that $A -x=L-y=L$ where the last equality follows from $y \in L$.