How to prove: Every affine set can be expressed as the solution set of a system of linear equations

Question

How can I prove that every affine set can be expressed as the solution set of a system of linear equations?

Please note that set $C \subseteq \textbf{R}^n$ is said to be affine if for any $x_1 , x_2 \in C$ and $\theta \in \textbf{R}$, we have $\theta x_1 + (1-\theta)x_2 \in C$.

Answer 1

Fix an element $x_0 \in C$.

Claim: The set $K = \{x-x_0: x \in C\}$ is a linear subspace of $\Bbb R^n$.

Claim: There exists a linear map $T$ whose kernel is precisely $K$.

Claim: $C$ is the solution to the linear system $Tx = Tx_0$ on $x$.