Chapter 1.3, Problem 48E

Consider a solution ${\stackrel{\to }{x}}_1$ of the linear system $A\stackrel{\to }{x}=\stackrel{\to }{b}$ .Justify the facts stated in parts (a) and (b):
a. If ${\stackrel{\to }{x}}_h$ , is a solution of the system $A\stackrel{\to }{x}=\stackrel{\to }{0}$ , then ${\stackrel{\to }{x}}_1+{\stackrel{\to }{x}}_h$ is a solution of the system $A\stackrel{\to }{x}=\stackrel{\to }{b}$ .
b. If ${\stackrel{\to }{x}}_2$ is another solution of the system $A\stackrel{\to }{x}=\stackrel{\to }{b}$ , then ${\stackrel{\to }{x}}_2-{\stackrel{\to }{x}}_1$ is a solution of the system $A\stackrel{\to }{x}=\stackrel{\to }{0}$ .
c. Now suppose A is a $2\times 2$ matrix. A solution vector ${\stackrel{\to }{x}}_1$ of the system $A\stackrel{\to }{x}=\stackrel{\to }{b}$ is shown in the accompanying figure. We are told that the solutions of the system $A\stackrel{\to }{x}=\stackrel{\to }{0}$ form the line shown in the sketch.Draw the line consisting of all solutions of the system $A\stackrel{\to }{x}=\stackrel{\to }{b}$ .

If you are puzzled by the generality of this problem,think about an example first:
  $A=\left[\begin{array}{cc}1& 2\\ 3& 6\end{array}\right],\stackrel{\to }{b}=\left[\begin{array}{l}3\\ 9\end{array}\right],$ and ${\stackrel{\to }{x}}_1=\left[\begin{array}{l}1\\ 1\end{array}\right]$ .