Chapter 8.2, Problem 9E

In Exercises 9 and 10, mark each statement True or False. Justify each answer.

9.

1. a. If v₁,....,v_p are in ℝⁿ and if the set {v₁ − v₂, v₃ − v₂,....,v_p − v₂} is linearly dependent, then {v₁,....,v_p} is affinely dependent. (Read this carefully.)
2. b. If v₁,....,v_p are in ℝⁿ and if the set of homogeneous forms { ${\tilde{v}}_1,\mathrm{....},{\tilde{v}}_p$} in ℝ^n+l is linearly independent, then {v₁,....,v_p} is affinely dependent.
3. c. A finite set of points {v₁,....,v_k} is affinely dependent if there exist real numbers c₁,.....,c_k , not all zero, such that c₁ + ....+ c_k = 1 and c₁v₁ + .... + c_kv_k = 0
4. d. If S = {v₁,....,v_p} is an affinely independent set in ℝⁿ and if p in ℝⁿ has a negative barycentric coordinate determined by S, then p is not in aff S.
5. e. If v₁, v₂, v₃, a, and b are in ℝ³ and if a ray a + tb for t ≥ 0 intersects the triangle with vertices v₁, v₂, and v₃, then the barycentric coordinates of the intersection point are all nonnegative.