Question

Use only the definition of affine dependence to show that anindexed set \(\left\{ {{v_1},{v_2}} \right\}\) in \({\mathbb{R}^{\bf{n}}}\) is affinely dependent if and only if \({v_1} = {v_2}\).

Short answer

An indexed set \(\left\{ {{v_1},{v_2}} \right\}\) in \({\mathbb{R}^n}\) is affinely dependent if and only if \({{\bf{v}}_1} = {{\bf{v}}_2}\).

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

An indexed set \(\left\{ {{{\bf{v}}_{\bf{1}}},{{\bf{v}}_{\bf{2}}}} \right\}\)in\({\mathbb{R}^n}\)is affinely dependent if and only if\({{\bf{v}}_{\bf{1}}} = {{\bf{v}}_{\bf{2}}}\).

The set\(\left\{ {{{\bf{v}}_{\bf{1}}},{{\bf{v}}_{\bf{2}}}} \right\}\)in\({\mathbb{R}^n}\)exists if for non-zero scalars\({c_1}\)and\({c_2}\), the sum is zero i.e.\({c_1} + {c_2} = 0\)and\({c_1}{{\bf{v}}_1} + {c_2}{{\bf{v}}_2} = 0\).

Here,\({c_1} = - {c_2} \ne 0\). So,\({c_1}{{\bf{v}}_1} + {c_2}{{\bf{v}}_2} = 0\)can be written as shown below:

\(\begin{aligned}{}\left( { - {c_2}} \right){{\bf{v}}_1} + {c_2}{{\bf{v}}_2} = 0\\ - {c_2}{{\bf{v}}_1} + {c_2}{{\bf{v}}_2} = 0\\ - {c_2}\left( {{{\bf{v}}_1} - {{\bf{v}}_2}} \right) = 0\\{{\bf{v}}_1} - {{\bf{v}}_2} = 0\end{aligned}\)

So,\({{\bf{v}}_1} = {{\bf{v}}_2}\).

Thus, an indexed set \(\left\{ {{v_1},{v_2}} \right\}\) in \({\mathbb{R}^n}\) is affinely dependent if and only if \({{\bf{v}}_1} = {{\bf{v}}_2}\).