Note Set 02a: Solving $\mathbf{A}\vec{x}=\vec{b}$ with magnficient row operations

Okay, let's solve some linear systems. We describe how to use row operations in a formalized way that will help us in the future (we have been warned by the words of dead monks). Row swaps appear as well, and we finish the job with back substitution. We begin to see how row operations, the way of Gauss, creates a triangular system $\mathbf{U}\vec{x}=\vec{c}$ with the same solution(s) as $\mathbf{A}\vec{x}=\vec{b}$. We show how augmented matrices are the way to go, and we uncover pivots and multipliers, very important matrix features.
Best consumed by 2016/08/31