Exercise 2.A.17

Proof. Suppose by contradiction that $p_0,p_1,\dots ,p_m$ is linearly indepedent. For every polynomial $p\in \mathrm{span}<!--\; FUNCTION\; APPLICATION\; -->(p_0,p_1,\dots <!--\; FUNCTION\; APPLICATION\; -->,p_m)$ we have $p(2)=0$. Therefore $1\notin \mathrm{span}<!--\; FUNCTION\; APPLICATION\; -->(p_0,p_1,\dots <!--\; FUNCTION\; APPLICATION\; -->,p_m)$ and, by the Linear Dependence Lemma, the list $p_0,p_1,\dots ,p_m,1$ is linearly independent. However, this list has length $m+2$ and the list $1,x,\dots ,x^m$, which spans $P_m(R)$, has length $m+1$. This is a contradiction by 2.23. □