Determine If the Set of Points Is Affinely Dependent A) the Set Is Affinely Dependent

Multiple Choice

Determine if the set of points is affinely dependent. If so, construct an affine dependence relation for the points.
- $\left[ \begin{array} { r } 4 \\ 8 \\ - 1 \end{array} \right] , \left[ \begin{array} { r } - 8 \\ - 16 \\ 4 \end{array} \right] , \left[ \begin{array} { r } 0 \\ 1 \\ 12 \end{array} \right] , \left[ \begin{array} { r } - 16 \\ - 31 \\ 22 \end{array} \right]$

A) The set is affinely dependent. $2 \left[ \begin{array} { r } 4 \\ 8 \\ - 1 \end{array} \right] - 3 \left[ \begin{array} { r } - 8 \\ - 16 \\ 4 \end{array} \right] - \left[ \begin{array} { r } 0 \\ 1 \\ 12 \end{array} \right] + \left[ \begin{array} { r } - 16 \\ - 31 \\ 22 \end{array} \right] = \left[ \begin{array} { l } 0 \\ 0 \\ 0 \end{array} \right]$
B) The set is affinely dependent. $5 \left[ \begin{array} { r } 4 \\ 8 \\ - 1 \end{array} \right] - 3 \left[ \begin{array} { r } - 8 \\ - 16 \\ 4 \end{array} \right] - \left[ \begin{array} { r } 0 \\ 1 \\ 12 \end{array} \right] - \left[ \begin{array} { r } - 16 \\ - 31 \\ 22 \end{array} \right] = \left[ \begin{array} { l } 0 \\ 0 \\ 0 \end{array} \right]$
C) The set is affinely dependent. $2 \left[ \begin{array} { r } 4 \\ 8 \\ - 1 \end{array} \right] + 3 \left[ \begin{array} { r } - 8 \\ - 16 \\ 4 \end{array} \right] + \left[ \begin{array} { r } 0 \\ 1 \\ 12 \end{array} \right] - \left[ \begin{array} { r } - 16 \\ - 31 \\ 22 \end{array} \right] = \left[ \begin{array} { l } 0 \\ 0 \\ 0 \end{array} \right]$
D) The set is affinely independent.