Let $F_m[z]$ denote the vector space of all polynomials with degree less than or equal to $m\in Z_+$ and having coefficient over $F$, and suppose that $p_0, p_1, \cdots p_m \in F_m[z]$ satisfy $p_j(2) = 0$. Prove that $(p_0, p_1, \cdots p_m)$ is a linearly dependent list of vectors in $F_m[z]$.
I don't know how to prove it. We should find some $c_i$ (not all zero) such that $\sum c_ip_i=0$. It must be equal to zero for all $z$. How is it possible?
Consider the matrix $P$ whose rows are the coefficient sequences of the $p_i.$ This is an $(m+1)\times (m+1)$ matrix, which has the property that $P(v_2)= 0,$ where $v_2 = (1, 2, 4, \dotsc 2^m)$ The rows of this matrix are thus linearly dependent.