As someone who can go through algebra and linear math this was wayyyy too heavy me. It was very hard to follow and map it to how to code such an algorithm.
You really need to give a number of examples. Remember human brains are fast example based learners. We see patterns and make abstract concepts out of it. Here the abstract concepts is the crazy math.
Another thing - the linear algebra presented in the article is all at high school or early-university level: systems of linear equations, matrices, row reduction, unique solutions, solution space. It might be heavy in terms of quantity, but I would not consider it to be terribly advanced math.
I think the only part that most students have not seen is the finite field arithmetic, because that knowledge is pretty specialized for people who actually design math for computers to execute.
Sorry about not meeting your expectations. I was planning to make an interactive JavaScript demo in the future to show all the steps of the computation with actual numbers as examples.
For the time being, try running the Python code and adding print statements to the places of interest.
You really need to give a number of examples. Remember human brains are fast example based learners. We see patterns and make abstract concepts out of it. Here the abstract concepts is the crazy math.