From Introduction to Functional Analysis with Applications by Kreszig:
Fix $n$.
Let $$\sum_{k=0}^n \alpha_k^{(n)}(t_k^{(n)})^j = \frac{1}{j+1}(b^{j+1} - a^{j+1}), \text{ for $j = 0,1, \dots, n$}$$ where $\alpha_k^{(n)}$ are unknowns and $t_k^{(n)}$ are fixed elements of the interval $[a,b]$ such that $t_k^{(n)}$ is a partition of $[a,b]$.
Then a unique solution exists if the corresponding homogenous system $$\sum_{k=0}^n (t_k^{(n)})^j \gamma_k= 0, \text{ where $j = 0, 1, \dots, n$}$$ has only the trivial solution $\gamma_k=0$ for all $k$ or, equally well, $\sum_{j=0}^n(t_k^{(n)})^j\gamma_j = 0, \text{ for $k = 0, \dots, n$}$
$(1.)$ How is the corresponding homogenous system formed here? I thought $\alpha$ were the unknowns?
$(2.)$ If a solution exists in the second to last equation, why is that equivalent to a solution existing in the last equation?