Suppose that\(\left\{ {{p_1},{p_2},{p_3}} \right\}\)is an affinely independent set in\({\mathbb{R}^{\bf{n}}}\)and q is an arbitrary point in\({\mathbb{R}^{\bf{n}}}\). Show that the translated set\(\left\{ {{p_1} + q,{p_2} + q,{p_3} + {\bf{q}}} \right\}\)is also affinely independent.

The set is said to be affinely dependent if the set \(\left\{ {{{\bf{v}}_{\bf{1}}},{{\bf{v}}_{\bf{2}}},...,{{\bf{v}}_p}} \right\}\) in the dimension \({\mathbb{R}^n}\) exists such that for non-zero scalars \({c_1},{c_2},...,{c_p}\), the sum of scalars is zero i.e. \({c_1} + {c_2} + ... + {c_p} = 0\), and \({c_1}{{\bf{v}}_1} + {c_2}{{\bf{v}}_2} + ... + {c_p}{{\bf{v}}_p} = 0\).

The set is affinely dependent if the set\(\left\{ {{p_1} + q,{p_2} + q,{p_3} + {\bf{q}}} \right\}\)in\({\mathbb{R}^n}\)exists such that for non-zero scalars\({c_1},{c_2}\)and\({c_3}\)the is zero i.e.\({c_1} + {c_2} + {c_3} = 0\).

So,\({c_1}\left( {{p_1} + q} \right) + {c_2}\left( {{p_2} + q} \right) + {c_3}\left( {{p_3} + q} \right) = 0\).

Simplify\({c_1}\left( {{p_1} + q} \right) + {c_2}\left( {{p_2} + q} \right) + {c_3}\left( {{p_3} + q} \right) = 0\)as shown below:

\(\begin{array}{c}{c_1}{p_1} + {c_1}q + {c_2}{p_2} + {c_2}q + {c_3}{p_3} + {c_3}q = 0\\{c_1}{p_1} + {c_2}{p_2} + {c_3}{p_3} + \left( {{c_1} + {c_2} + {c_3}} \right){\bf{q}} = 0\end{array}\)

As\({c_1} + {c_2} + {c_3} = 0\),\({c_1}{p_1} + {c_2}{p_2} + {c_3}{p_3} = 0\).

Thus,\({c_1}{{\bf{p}}_1} + {c_2}{{\bf{p}}_2} + {c_3}{{\bf{p}}_3} = 0\)and\({c_1} + {c_2} + {c_3} = 0\)show that\(S = \left\{ {{p_1},{p_2},{p_3}} \right\}\) is an affinely dependent set of points in \({\mathbb{R}^n}\).

But\(\left\{ {{p_1},{p_2},{p_3}} \right\}\)is affinely independent. Thus, the translated set is also affinely independent.