How can I show the following statements are equivalent?

Question

Let $C ⊂ \mathbb{R}^n$ Prove that the following statements are equivalent.

(i) $C$ is an affine set

(ii) For every $x_0 ∈ C$ , the set $C − x_0 := \{ z − x_0: z ∈ C \}$ is a subspace.

(iii) There exists $x_0 ∈ C$ such that the set $ C − x_0$ is a subspace.

For $C$ to be an affine set contains the line through any two points $x_1, x_2$ in the set.

so (i) is defined as: $x_1\lambda + (1-\lambda)x_2 ∈ C$ for $\lambda ∈ \mathbb{R}$

then for (ii) be a subspace, we must show it contains the 0 space and that its closed under multiplication and addition. Since by definition an affine set is closed for affine combinations whenever $x_0,x_1, \lambda ∈\mathbb{R}$ then is this sufficient to show the equivalence of (i) and (ii) ?

Answer 1

Hints:

The following lemmas are not quite difficult to prove and can be helpful.

Let $A\subseteq\mathbb R^n$. Then:

• $A$ is a subspace $\iff$ $A$ is affine and contains $\mathbf0$.
• $A$ is affine $\Rightarrow$ $\mathbf{u}+A$ is affine for every $\mathbf{u}\in\mathbb R^n$.