Linear Algebra - Polynomials Suppose that    $z_1 , . . . , z_{m+1}$    are distinct    elements    of  $\mathbb{R}$    and that $w_1, . . . , w_{m+1}$ ∈ $\mathbb{R}$. Prove that there exists a unique polynomial $p$ with degree $m$ such that
$p(z_j) = w_j$

Write $p(x) = c_mx^m + c_{m-1}x^{m-1} + \cdots + c_1x + c_0$.  The conditions $p(z_j) = w_j$ give you a system of $m + 1$ equations in the $m + 1$ variables $c_m, \ldots, c_0$.  To get that the solution exists and is unique you will use the fact that the Vandermonde matrix is non-singular.
