Find the barycentric coordinates of p with respect to the affinely independent set of points that precedes it.
-Which of the following statements are true for the set $S = \left\{ \mathbf { v } _ { 1 } , \ldots , \mathbf { v } _ { \mathrm { k } } \right\}$ in $\mathfrak { R } \mathrm { n } ^ { \text {? } }$
I: If $S$ is affinely independent, then a point $p$ in $R ^ { n }$ cannot have any barycentric coordinates determined by $\mathrm { S }$ that are equal to 0 .
II: If $S$ is affinely independent, then a point $p$ in aff $S$ has a unique representation as an affine combination of $\mathbf { v } _ { 1 } , \ldots , \mathbf { v } _ { \mathrm { k } }$.
A) II
B) Both I and II
C) I
D) Neither I or II

Question

Quizplus · Accepted Answer

II is true because if $S$ is affinely independent, then any point $p$ in the affine hull of $S$ (denoted as aff $S$) can indeed be expressed uniquely as an affine combination of the points in $S$. This means there is exactly one way to write $p$ as a sum of the points in $S$, scaled by coefficients that sum to 1. Statement I is false because a point $p$ can have barycentric coordinates equal to 0 with respect to $S$, indicating that $p$ lies on the affine hull of the subset of $S$ excluding the points corresponding to the zero barycentric coordinates.

Find the Barycentric Coordinates of P with Respect to the Affinely