All affine sets are affine subspaces (and vice-versa)

Question

Show that a set $∅ \neq A ⊂ \mathbb{R}^n$ is affine if and only if it is an affine subspace.

Definitions I know:

Affine set: If a set $A \subset\mathbb{R}^n$ is affine, then for all $x,y\in A$, we have $tx + (1-t)y \in A$ where $t\in\mathbb{R}$.
Affine subspace: A set $A$ is an affine subspace of $\mathbb{R}^n$ if every element $a\in A$ can be written as $p + V$, where $p\in\mathbb{R}^n$ is a fixed vector, and $V$ is a subspace of $\mathbb{R}^n$.

My work: $[\Leftarrow]$ seems easier. Consider two elements $p+v_1$ and $p+v_2$ in $A = p+V$ as defined above. Then $t(p+v_1) + (1-t)(p + v_2) = p + tv_1 + (1-t)v_2 \in p+V$ because $V$ is a subspace. So all affine subspaces are affine sets.

How do I show the other direction, namely $[\Rightarrow]$?

I started with a non-empty affine set $A\subset\mathbb{R}^n$, and I'm trying to construct $p$ and $V$ so that $p+V = A$, but I don't know how to! I'd appreciate any help!

Answer 1

Hint: Fix $x \in A$ and put $V:= A-x$. Check that $V$ is a vector space.

Answer 2

Let $p$ be any point in $A$. Let $V=A-p$ (i.e. $\{a-p: a \in A\}$). Let $x \in V$ so $x=a-p$ with $a \in A$ and take $c \in \mathbb R$. Let us show that $cx \in V$. $cx=ca-cp=(ca-cp+p) -p$ and $ca-cp+p=ca+(1-c)p \in A$. So $cx \in V$.

Now consider $(a_1-p)+(a_2 -p)$ where $a_1.a_2 \in A$. Write this as $ 2(\frac {a_1+a_2} 2 -p)$. can you finish? .