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 $\mathfrak { R } ^ { n }$ ?
I: $\left\{ \mathbf { v } _ { 2 } - \mathbf { v } _ { 1 } , \ldots , \mathbf { v } _ { \mathrm { k } } - \mathbf { v } _ { 1 } \right\}$ is linearly dependent if and only if $S$ is affinely dependent.
II: If $S$ is affinely independent and a point $p$ in $R ^ { n }$ has all positive barycentric coordinates determined by $S$, then $p$ is not in aff $S$.
A) I
B) II
C) Neither I or II
D) Both I and II

Question

Quizplus · Accepted Answer

I: The statement is true because affine dependence of a set $S$ means that there exists a non-trivial affine combination of the points in $S$ that equals zero. This directly translates to the linear dependence of the set $\left\{ \mathbf { v } _ { 2 } - \mathbf { v } _ { 1 } , \ldots , \mathbf { v } _ { \mathrm { k } } - \mathbf { v } _ { 1 } \right\}$, as the subtraction of one point from the others essentially checks for linear combinations that sum to the zero vector, indicating linear dependence.II: This statement is false because if $p$ has all positive barycentric coordinates with respect to $S$, it actually implies that $p$ lies inside the affine hull of $S$ (aff $S$). Barycentric coordinates are used to express a point as a combination of points that define a simplex, and having all positive coordinates means the point is strictly within the simplex formed by $S$, not outside of it.

Find the Barycentric Coordinates of P with Respect to the Affinely