TITLE: The subspace associated with the affine set $C$ is the nullspace of $A$.
QUESTION [7 upvotes]: This is an example from my book:
Example 2.1: The solution set of a system of linear
equations, $C = \{ x | Ax = b\}$,  Where  $A \in R^{m\times n}$ and $b \in R^m$, is an affine set. To show this suppose $x_1, x_2 \in C$ then for any $\theta$ we have:
$
A(\theta x_1 + (1-\theta)x_2) = \theta Ax_1 + (1-\theta)Ax_2 = \theta b + (1-\theta)b = b
$
Which shows that the affine combination $\theta x_1 +(1-\theta)x_2 $ is also in C.  The subspace associated with the affine set $C$ is the nullspace of $A$.
I understand the proof but I cant understand the last sentence. From what I know if $x$ is a solution for $Ax = b$ Then every linear combination of x is a also a solution, And that is the definition of the range space. What am I missing here?

REPLY [7 votes]: The nullspace of a matrix $A$ is the set of vectors $v$ for which $Av=0$. Basically, what it’s saying is that the general solution to the system is the sum of a particular solution $x_0$ and the solution to the homogeneous system $Ax=0$.