How can I create an isomorphism between the solution space and a parallel vector space. I'm not sure how to define the vector space and the isomorphism.
$$ \begin{bmatrix} -2 & 4 \\ 4 & -8\\ \end{bmatrix} * \begin{bmatrix} x \\ y \\ \end{bmatrix} = \begin{bmatrix} -8 \\ 16 \\ \end{bmatrix} $$
The solution space is the line $y=x/2-2$. Since it does not pass through the origin $(0,0)$, it is not a vector subspace of $\mathbb R^2$. Rather, it is an affine subspace, which by definition is a set of the form $V+u$ where $V$ is a vector subspace and $u$ is a fixed vector. This $V$ is the parallel vector space that you are looking for. The map $x\mapsto x+u$ is a desired isomorphism. The space $V$ is unique for a given affine subspace, but $u$ generally is not.
Since we are talking about lines on a plane, things are not very complicated. Two lines are parallel if they have the slope. So, $y=x/2$ is the parallel vector space. An example of isomorphism between them is $(x,x/2)\mapsto (x,x/2-2)$.
In general, the solution space of the system $Ax=b$ has parallel vector space described by the system $Ax=0$.