Factoring $9788111$ via Gaussian elimination over $\mathbb F_2$

Question

I am trying to follow page 142 to page 144 of An Introduction to Mathematical Cryptography by Hoffstein, Pipher & Silverman, where they give an example using Gaussian elimination over $\mathbb F_2$ as part of the process for factoring $N = 9788111$.

I am new to Gaussian Elimination done modulo $p$.

I am seeking the steps for how the results 1. and 2. below are obtained:

1. "the set of solutions turns out to be an $F_2$ vector space of dimension 8" on page 142.

2. Numbers in $v_1$ to $v_8$ on page 144.

I have attempted normal Gaussian elimination on the matrix on bottom of page 143, but cannot see how to obtain the above two results currently.

Answer 1

$$ \left( \begin{array}{ccc} 1&0 &1 \\ 1 &1 &0 \\ 1& 1& 1\\ \end{array} \right) $$

Got it, give me a minute; easy, as Henno says

$$ \left( \begin{array}{ccc} 1&0 &1 \\ 0 &1 &1 \\ 0&1 &0 \\ \end{array} \right) $$

$$ \left( \begin{array}{ccc} 1&0 &1 \\ 0&1 &1 \\ 0&0 &1 \\ \end{array} \right) $$

$$ \left( \begin{array}{ccc} & & \\ & & \\ & & \\ \end{array} \right) $$

$$ \left( \begin{array}{ccc} & & \\ & & \\ & & \\ \end{array} \right) $$