91Ó°ÊÓ

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 \({c_1},{c_2},...,{c_p}\) not all zero, and the sum must be zero \({c_1} + {c_2} + ... + {c_p} = 0\), and \({c_1}{{\bf{v}}_1} + {c_2}{{\bf{v}}_2} + ... + {c_p}{{\bf{v}}_p} = 0\).

Let \({{\mathop{\rm v}\nolimits} _1} = \left( {\begin{aligned}{{}}3\\{ - 3}\end{aligned}} \right),{{\mathop{\rm v}\nolimits} _2} = \left( {\begin{aligned}{{}}0\\6\end{aligned}} \right),{{\mathop{\rm v}\nolimits} _3} = \left( {\begin{aligned}{{}}2\\0\end{aligned}} \right)\).

Compute translated points \({{\mathop{\rm v}\nolimits} _2} - {{\mathop{\rm v}\nolimits} _1}\) and \({{\mathop{\rm v}\nolimits} _3} - {{\mathop{\rm v}\nolimits} _1}\)as shown below:

\({{\mathop{\rm v}\nolimits} _2} - {{\mathop{\rm v}\nolimits} _1} = \left( {\begin{aligned}{{}}{ - 3}\\9\end{aligned}} \right)\),

\({{\mathop{\rm v}\nolimits} _3} - {{\mathop{\rm v}\nolimits} _1} = \left( {\begin{aligned}{{}}{ - 1}\\3\end{aligned}} \right)\)

It is observed that two points are multiples of each other.

Theorem 5states that an indexed set \(S = \left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}} \right\}\) in \({\mathbb{R}^n}\), with \(p \ge 2\), the following statement is equivalent. This means that either all the statements are true or all the statements are false.

1. The set \(S\) isaffinely dependent.
2. Each of the points in \(S\)is an affine combination of the other points in \(S\).
3. In \({\mathbb{R}^n}\), the set \(\left\{ {{{\mathop{\rm v}\nolimits} _2} - {{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p} - {{\mathop{\rm v}\nolimits} _1}} \right\}\)is linearly dependent.
4. The set \(\left\{ {{{\bar v}_1},...,{{\bar v}_p}} \right\}\) of homogeneous forms in \({\mathbb{R}^{n + 1}}\) is linearly dependent.

Since two points are multiples of each other hence form a linearly dependent set.

Therefore, all statements in theorem 5 are true and thus \(S\) are affinely dependent.

\(\begin{aligned}{}{{\mathop{\rm v}\nolimits} _2} - {{\mathop{\rm v}\nolimits} _1} = 3\left( {{{\mathop{\rm v}\nolimits} _3} - {{\mathop{\rm v}\nolimits} _1}} \right)\\{{\mathop{\rm v}\nolimits} _2} - {{\mathop{\rm v}\nolimits} _1} = 3{{\mathop{\rm v}\nolimits} _3} - 3{{\mathop{\rm v}\nolimits} _1}\\2{{\mathop{\rm v}\nolimits} _1} + {{\mathop{\rm v}\nolimits} _2} - 3{{\mathop{\rm v}\nolimits} _3} = 0\end{aligned}\)

Thus, the set of points is affinely dependent and \(2{{\mathop{\rm v}\nolimits} _1} + {{\mathop{\rm v}\nolimits} _2} - 3{{\mathop{\rm v}\nolimits} _3} = 0\).