Consider a solution x1 of the linear system Ax=b. Justify the facts stated in parts (a) and (b): a) If xh is a solution of the system Ax=0, then x1+xh is a solution of the system Ax=b. b) If x2 is another solution of the system Ax=b, then x2−x1 is a solution of the system Ax=0

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Consider a solution x1 of the linear system Ax=b. Justify the facts stated in parts (a) and (b):  a) If xh is a solution of the system Ax=0, then x1+xh is a solution of the system Ax=b.  b) If x2 is another solution of the system Ax=b, then x2−x1 is a solution of the system Ax=0

Maciej Morrow · Accepted Answer

Consider a solution&amp;nbsp;x1&amp;nbsp;of the linear system Ax=b. Justify the information presented in parts. (a) and (b):&amp;nbsp;Thus,Axh=0&amp;nbsp;Add equation&amp;nbsp;Ax1+axh=b+0=b&amp;nbsp;A(x1+xh)=b&amp;nbsp;Therefore,(x1+xh)&amp;nbsp;is asolution of&amp;nbsp;Ax=b&amp;nbsp;since&amp;nbsp;x=x1+xh&amp;nbsp;satisfies this equation.&amp;nbsp;b) If&amp;nbsp;x2&amp;nbsp;is another solution of the system Ax=b, then&amp;nbsp;x2−x1&amp;nbsp;is asolution of the system Ax=0&amp;nbsp;Ax2=b&amp;nbsp;Ax2=Ax1=b−b=0&amp;nbsp;A(x2−x1)=0&amp;nbsp;Hence&amp;nbsp;(x2−x1)&amp;nbsp;is solution of&amp;nbsp;Ax=0&amp;nbsp;Since&amp;nbsp;x=x2−x1&amp;nbsp;satisfies the equation.&amp;nbsp;

Consider a solution x_1 of the linear system Ax=b. Justify the facts stated in parts (a) and (b): a) If x_h is a solution of the system Ax=0, then x_1+x_h is a solution of the system Ax=b. b) If x_2 is another solution of the system Ax=b, then x_2-x_1 is a solution of the system Ax=0

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