Affine subspaces and parallel linear subspaces Let $\mathcal{X}$ be a real vector space and $C\subset \mathcal{X}$ an affine subspace of $\mathcal{X}$, i.e. $C\neq\emptyset$ and $C=\lambda C + (1-\lambda)C$ for all $\lambda\in\mathbb{R}$. In the text I am reading, they have defined the linear subspace parallel to $C$ to be $V=C-C=\{a-b : a\in C, b\in C\}$. What is the significance of subtracting the entire set $C$ compared to just one vector from $C$? Why not just take $V=C-c$ for some $c\in C$?
The example I was looking at was a line in $\mathbb{R}^2$, $C = \{(3,y) : y\in \mathbb{R}\}$. Doesn't $C-(3,0) = \{c-(3,0) : c\in C\}$ produce the same set as $C-C$ or am I missing some subtlety?

I am not sure if you believe me or not! but the only reason that people like to write parallel subspace in the way $V=C-C$ is that it looks  NICER.
But what you said is correct both representations are equivalent. it is a simple exercise .
