I have a problem that asks:
Show that if the vectors $s = \left[ \begin{array}{c} s_1\\ s_2\\ \vdots\\ s_n\\ \end{array} \right]$ and $p = \left[ \begin{array}{c} p_1\\ p_2\\ \vdots\\ p_n\\ \end{array} \right]$ are the solutions of the equation $$ a_1x_1 + a_2x_2+...+ a_nx_n = d$$
then the vector $h = s - p = \left[ \begin{array}{c} s_1-p_1\\ s_1-p_2\\ \vdots\\ s_n-p_n\\ \end{array} \right]$ is a solution of the equation $$ a_1x_1 + a_2x_2+...+ a_nx_n = 0$$
Could someone walk me through this or give me steps on what to do here? I have other problems that are similar to this one so general steps are better than just solving it.
Do you know what it means for those vectors s and p to be solutions to that equation? Can you write it out in full?
Have you tried just plugging the vector h in directly? i.e.
$a_1(s_1 - p_1) + a_2(s_2-p_2) + \dots + a_n(s_n-p_n),$
and expanding out the brackets, and seeing what you get?