Existence of non-zero scalars s.t. $\sum_{i=0}^{n^2}\alpha_i A^i=\mathcal{O}_n$

Question

I have the following problem to which I have not been able to find an answer given the theory so far covered in my linear algebra course:
Given a matrix $A\in$Mat$_{n,n}(\mathbb{F})$ (square matrix, $\mathbb{F}$ is any field), show that there exist $\alpha_o,\alpha_1,\dots,\alpha_{n^2}\in\mathbb{F}$ not all zero such that $$ \sum_{i=0}^{n^2}\alpha_i A^i=\mathcal{O}_n $$ where $\mathcal{O}_n$ is the zero matrix and $A^0$ is the identity matrix $\mathcal{I}_n$.
The theory covered thus far is: inverse matrices, linear operators, linear (in)dependence, bases and dimension, but no dimension formulae have been proved yet.