I am studying a book on Convexity and Optimization and I found this statement.
"Let $X_0$ be any point of a linear manifold M. Then the set:
$$L := M - X_0 = \{X - X_0 | X \in M\} $$
is the unique linear subspace which runs parallel to M through the origin of $\mathbb{R}^n$ ."
I am an engineer and not a mathematician, therefore I am trying to get an intuition out of that. At the moment, I am thinking about a line of the form $y = mx + q$ whose linear subspace through the origin could be with $X_0 = q$ .
Is that a proper example? Which is the intuition behind the statement?