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The part that I don't understand is how the author gets to the equality in 10.6. Lemma 10.3 simply states that a system of $m$ homogeneous linear equations in $n$ indeterminates has a solution in which not all the variables are zero if $n>m$. Since this lemma is used to derive the equation 10.6 my guess is that there had to be some system of linear equations the author used to arrive at 10.6, but what is such system?
I would really appreciate any help/thoughts.
(10.6) is literally a system of homogeneous linear equations. Namely, the variables are $y_1,\dots,y_n$, and for each $j=1,\dots,m$, we have the homogeneous equation $$y_1g_1(x_j)+\dots+y_ng_n(x_j)=0$$ in these variables (with the scalars $g_1(x_j),\dots,g_n(x_j)$ as the coefficients). This is a system of $m$ linear equations in $n$ variables, so since $n>m$, it has a nonzero solution.

(10.6) is literally a system of homogeneous linear equations. Namely, the variables are $y_1,\dots,y_n$, and for each $j=1,\dots,m$, we have the homogeneous equation $$y_1g_1(x_j)+\dots+y_ng_n(x_j)=0$$ in these variables (with the scalars $g_1(x_j),\dots,g_n(x_j)$ as the coefficients). This is a system of $m$ linear equations in $n$ variables, so since $n>m$, it has a nonzero solution.
20171214 06:37:34