Question

Let\(\left\{ {{p_1},{p_2},{p_3}} \right\}\)be an affinely dependent set of points in\({\mathbb{R}^{\bf{n}}}\)and let\(f:{\mathbb{R}^{\bf{n}}} \to {\mathbb{R}^{\bf{m}}}\)be a linear transformation. Show that\(\left\{ {f\left( {{{\bf{p}}_1}} \right),f\left( {{{\bf{p}}_2}} \right),f\left( {{{\bf{p}}_3}} \right)} \right\}\)is affinely dependent in\({\mathbb{R}^{\bf{m}}}\).

Short answer

\(\left\{ {f\left( {{{\bf{p}}_1}} \right),f\left( {{{\bf{p}}_2}} \right),f\left( {{{\bf{p}}_3}} \right)} \right\}\)is affinely dependent in\({\mathbb{R}^{\bf{m}}}\).

Step-by-step solution

State the condition for affinely dependence

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

Show affinely dependence

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

Apply the linear transformation of functions as shown below:

\(\begin{array}{c}{c_1}f\left( {{{\bf{p}}_1}} \right) + {c_2}f\left( {{{\bf{p}}_2}} \right) + {c_3}f\left( {{{\bf{p}}_3}} \right) = f\left( 0 \right)\\f\left( {{c_1}{{\bf{p}}_1} + {c_2}{{\bf{p}}_2} + {c_3}{{\bf{p}}_3}} \right) = f\left( 0 \right)\\{c_1}{{\bf{p}}_1} + {c_2}{{\bf{p}}_2} + {c_3}{{\bf{p}}_3} = 0\end{array}\)

Here, \({c_1}{{\bf{p}}_1} + {c_2}{{\bf{p}}_2} + {c_3}{{\bf{p}}_3} = 0\) shows that \({c_1} + {c_2} + ... + {c_p} = 0\). Thus, \(\left\{ {f\left( {{{\bf{p}}_1}} \right),f\left( {{{\bf{p}}_2}} \right),f\left( {{{\bf{p}}_3}} \right)} \right\}\) is affinely dependent in \({\mathbb{R}^{\bf{m}}}\).