Chapter 8.2, Problem 25E

To determine

To find: If $S=\left\{v_1,v_2,\dots v_k\right\}$ in ${ℝ}^n$ is an affinely independent set then there exist real numbers $c_1,c_2,\dots ,c_k$ such that $p=c_1{v}_1+\cdots +c_k{v}_k$ where $c_1+c_2+\cdots +c_k=1$

The expression $p=c_1{v}_1+\cdots +c_k{v}_k$ can be written as,

  $\left[\begin{array}{c}p\\ 1\end{array}\right]=c_1\left[\begin{array}{c}{v}_1\\ 1\end{array}\right]+\cdots +c_k\left[\begin{array}{c}{v}_k\\ 1\end{array}\right]$

The objective is to prove that it will have a unique solution.

That is, $p=c_1{v}_1+\cdots +c_k{v}_k$ where $c_1+c_2+\cdots +c_k=1$ must have an unique solution.