Chapter 8.3, Problem 15E

Let v₁ = $\left[\begin{array}{c}1\\ 0\end{array}\right]$, v₂ = $\left[\begin{array}{c}1\\ 2\end{array}\right]$, v₃ = $\left[\begin{array}{c}4\\ 2\end{array}\right]$, v₄ = $\left[\begin{array}{c}4\\ 0\end{array}\right]$, and p = $\left[\begin{array}{c}2\\ 1\end{array}\right]$. Confirm that

$p=\frac{1}{3}{v}_1+\frac{1}{3}{v}_2+\frac{1}{6}{v}_3+\frac{1}{6}{v}_4$ and v₁ − v₂ + v₃ − v₄ = 0.

Use the procedure in the proof of Caratheodory’s Theorem to express p as a convex combination of three of the v_i’s. Do this in two ways.