Repeat Exercise 15 for points \({{\rm{v}}_1} = \left( \begin{aligned}{} - 1\\\,\,0\end{aligned} \right)\), \({{\rm{v}}_2} = \left( \begin{aligned}{}\,0\\\,3\end{aligned} \right)\), \({{\rm{v}}_3} = \left( \begin{aligned}{}\,3\\\,1\end{aligned} \right)\), \({{\rm{v}}_4} = \left( \begin{aligned}{}1\\ - \,1\end{aligned} \right)\), and \({\rm{p}} = \left( \begin{aligned}{}1\\2\end{aligned} \right)\) , given that \({\rm{p}}\, = \frac{1}{{121}}{{\rm{v}}_1} + \frac{{72}}{{121}}{{\rm{v}}_2} + \frac{{37}}{{121}}{{\rm{v}}_3} + \frac{1}{{11}}{{\rm{v}}_4}\)

and \({\rm{10}}{{\rm{v}}_1} - {\rm{6}}{{\rm{v}}_2} + 7{{\rm{v}}_3} - 11{{\rm{v}}_4} = 0\).

\(\begin{aligned}{}\frac{1}{{121}}{{\bf{v}}_1} + \frac{{72}}{{121}}{{\bf{v}}_2} + \frac{{37}}{{121}}{{\bf{v}}_3} + \frac{1}{{11}}{{\bf{v}}_4} &= \frac{1}{{121}}\left( {\begin{aligned}{{}}{ - 1}\\0\end{aligned}} \right) + \frac{{72}}{{121}}\left( {\begin{aligned}{{}}0\\3\end{aligned}} \right) + \frac{{37}}{{121}}\left( {\begin{aligned}{{}}3\\1\end{aligned}} \right) + \frac{1}{{11}}\left( {\begin{aligned}{{}}1\\{ - 1}\end{aligned}} \right)\\ &= \left( {\begin{aligned}{{}}{ - \frac{1}{{121}} + 0 + \frac{{111}}{{121}} + \frac{1}{{11}}}\\{0 + \frac{{216}}{{121}} + \frac{{37}}{{121}} - \frac{1}{{11}}}\end{aligned}} \right)\\ &= \left( {\begin{aligned}{{}}1\\2\end{aligned}} \right)\\ &= {\bf{p}}\end{aligned}\)

Substitute value in the equation \({{\bf{v}}_1} - {{\bf{v}}_2} + {{\bf{v}}_3} - {{\bf{v}}_4} = 0\).

\(\begin{aligned}{}10{{\bf{v}}_1} - 6{{\bf{v}}_2} + 7{{\bf{v}}_3} - 11{{\bf{v}}_4} &= 10\left( {\begin{aligned}{{}}{ - 1}\\0\end{aligned}} \right) - 6\left( {\begin{aligned}{{}}0\\3\end{aligned}} \right) + 7\left( {\begin{aligned}{{}}3\\1\end{aligned}} \right) - 11\left( {\begin{aligned}{{}}1\\{ - 1}\end{aligned}} \right)\\& = \left( {\begin{aligned}{{}}{ - 10 - 0 + 21 - 11}\\{0 - 18 + 7 + 11}\end{aligned}} \right)\\& = \left( {\begin{aligned}{{}}0\\0\end{aligned}} \right)\\& = {\bf{0}}\end{aligned}\)

From two steps, the sum of the coefficients is 0, so the combination is an affine dependence relationship. Both the given equations are valid.

From the equations \({\rm{p}}\, = \frac{1}{{121}}{{\rm{v}}_1} + \frac{{72}}{{121}}{{\rm{v}}_2} + \frac{{37}}{{121}}{{\rm{v}}_3} + \frac{1}{{11}}{{\rm{v}}_4}\)and\(10{{\rm{v}}_1} - 6{{\rm{v}}_2} + 7{{\rm{v}}_3} - 11{{\rm{v}}_4} = 0\), the coefficients \({{\bf{v}}_1}\) and \({{\bf{v}}_2}\) are \(\frac{1}{{1210}}\) and \(\frac{{37}}{{847}}\).

Subtract \(\frac{1}{{1210}}\) times \(10{{\rm{v}}_1} - 6{{\rm{v}}_2} + 7{{\rm{v}}_3} - 11{{\rm{v}}_4} = 0\) from \({\rm{p}}\, = \frac{1}{{121}}{{\rm{v}}_1} + \frac{{72}}{{121}}{{\rm{v}}_2} + \frac{{37}}{{121}}{{\rm{v}}_3} + \frac{1}{{11}}{{\rm{v}}_4}\).

\(\begin{aligned}{}{\rm{p}} &= \frac{1}{{121}}{{\bf{v}}_1} + \frac{{72}}{{121}}{{\bf{v}}_2} + \frac{{37}}{{121}}{{\bf{v}}_3} + \frac{1}{{11}}{{\bf{v}}_4} - \frac{1}{{1210}}\left( {10{{\bf{v}}_1} - 6{{\bf{v}}_2} + 7{{\bf{v}}_3} - 11{{\bf{v}}_4}} \right)\\ &= \left( {\frac{1}{{121}} - \frac{1}{{121}}} \right){{\bf{v}}_1} + \left( {\frac{{72}}{{121}} + \frac{6}{{1210}}} \right){{\bf{v}}_2} + \left( {\frac{{37}}{{121}} - \frac{7}{{1210}}} \right){{\bf{v}}_3} + \left( {\frac{1}{{11}} + \frac{{11}}{{1210}}} \right){{\bf{v}}_4}\\ &= \left( 0 \right){{\bf{v}}_1} + \left( {\frac{3}{5}} \right){{\bf{v}}_2} + \left( {\frac{3}{{10}}} \right){{\bf{v}}_3} + \left( {\frac{1}{{10}}} \right){{\bf{v}}_4}\\ &= \frac{3}{5}{{\bf{v}}_2} + \frac{3}{{10}}{{\bf{v}}_3} + \frac{1}{{10}}{{\bf{v}}_4}\end{aligned}\)

Similarly, add \(\frac{1}{{1210}}\) times \(10{{\rm{v}}_1} - 6{{\rm{v}}_2} + 7{{\rm{v}}_3} - 11{{\rm{v}}_4} = 0\) from \({\rm{p}}\, = \frac{1}{{121}}{{\rm{v}}_1} + \frac{{72}}{{121}}{{\rm{v}}_2} + \frac{{37}}{{121}}{{\rm{v}}_3} + \frac{1}{{11}}{{\rm{v}}_4}\).

\(\begin{aligned}{}{\rm{p}}& = \frac{1}{{121}}{{\bf{v}}_1} + \frac{{72}}{{121}}{{\bf{v}}_2} + \frac{{37}}{{121}}{{\bf{v}}_3} + \frac{1}{{11}}{{\bf{v}}_4} + \frac{1}{{1210}}\left( {10{{\bf{v}}_1} - 6{{\bf{v}}_2} + 7{{\bf{v}}_3} - 11{{\bf{v}}_4}} \right)\\ &= \left( {\frac{1}{{121}} + \frac{{10}}{{121}}} \right){{\bf{v}}_1} + \left( {\frac{{72}}{{121}} - \frac{6}{{121}}} \right){{\bf{v}}_2} + \left( {\frac{{37}}{{121}} + \frac{7}{{121}}} \right){{\bf{v}}_3} + \left( {\frac{1}{{11}} - \frac{{11}}{{121}}} \right){{\bf{v}}_4}\\ &= \frac{1}{{11}}{{\bf{v}}_1} + \frac{6}{{11}}{{\bf{v}}_2} + \frac{4}{{11}}{{\bf{v}}_3}\end{aligned}\)

So, the expressions for pare\({\rm{p}} = \frac{3}{5}{{\bf{v}}_2} + \frac{3}{{10}}{{\bf{v}}_3} + \frac{1}{{10}}{{\bf{v}}_4}\)or \({\rm{p}} = \frac{1}{{11}}{{\bf{v}}_1} + \frac{6}{{11}}{{\bf{v}}_2} + \frac{4}{{11}}{{\bf{v}}_3}\).