Show that for any non-zero ploynomial $q(x)$ of a degree less than $n$ that the vector $AE_1$ is non-zero.

Question

Suppose I have the square matrix $$ \begin{pmatrix} 0 & 0 & 0 &\cdots &a_{1} \\ 1 & 0 & 0 &\cdots &a_{2} \\ 0 & 1 & 0 &\cdots &a_{3} \\ \vdots & \vdots &\vdots& \ddots&\vdots\\ 0 & 0 & 0 & \cdots&a_{n} \\ \end{pmatrix} $$ How do I show that for any non-zero ploynomial $q(x)$ of a degree less than $n$ that the vector $$ q(A) \begin{pmatrix} 1\\ 0\\ 0 \\ \vdots\\ 0 \\ \end{pmatrix} $$ is non-zero. So I have to show that the first column vector of $q(A)$ does not consist of only 0 elements. Thankful for help