Show that a set\(\left\{ {{{\bf{v}}_{\bf{1}}},...,{{\bf{v}}_p}} \right\}\)in\({\mathbb{R}^{\bf{n}}}\)is affinely dependent when \(p \ge n + 2\).

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 \({c_1}{{\bf{v}}_1} + {c_2}{{\bf{v}}_2} + ... + {c_p}{{\bf{v}}_p} = 0\).

Consider the set of points \(\left\{ {{{\bf{v}}_{\bf{1}}},{{\bf{v}}_{\bf{2}}},...,{{\bf{v}}_p}} \right\}\)is in \({\mathbb{R}^n}\).

It is given that\(p \ge n + 2\)or \(p - 1 \ge n + 1\).

Let \(\left\{ {{{\bf{v}}_2} - {{\bf{v}}_1},{{\bf{v}}_3} - {{\bf{v}}_1},...,{{\bf{v}}_p} - {{\bf{v}}_1}} \right\}\) be the set of vectors in \({\mathbb{R}^n}\). This new set of points must be linearly dependent when a set of vectors is translated by eliminating the first point or any other point.

Here,\(p \ge n + 2\) shows that the set contains more vectors or points than the number of entries. Thus, the original set of points \(\left\{ {{{\bf{v}}_{\bf{1}}},{{\bf{v}}_{\bf{2}}},...,{{\bf{v}}_p}} \right\}\) is affinely dependent.

Therefore, the set \(\left\{ {{{\bf{v}}_{\bf{1}}},{{\bf{v}}_{\bf{2}}},...,{{\bf{v}}_p}} \right\}\) in\({\mathbb{R}^n}\) is affinely dependent.