Show that the solution-set of a system of m linear equations in n unknowns is an affine subspace of $F^n$. What is its corresponding subspace?
The definition I have of an affine subspace for a subset A of a vector space V is that if A is a translate of a subspace W of V, $A = v_0 + W = (v_0 + w, w \in W)$ and then W is the corresponding subspace to A. How would I find $v_0$ and W here? Or I do not need to find them to show the result, I am stuck on how to solve this problem so any general directions to get me started would be great.
Hint: You (should) know that the solution set of a system of linear equations where the right-hand side is 0 is a vector space.
It could help to consider a practical case, say $$\left\{ \begin{array}{l} 2x-3y+z=4\\ x+2y+z=5 \end{array} \right.$$ compared to $$\left\{ \begin{array}{l} 2x-3y+z=0\\ x+2y+z=0 \end{array} \right.$$ The solutions of the second system form a line through $(0,0,0)$ that is a vector subspace. What about the first?