So say I have some linear transformation $T(\vec{x}) = A\vec{x}$
So we say $$ \left[ \begin{array}{ccc} a_1 & \dots & a_n\\ a_1 & \ddots & \vdots\\ a_1 & \dots & a_n\\ \end{array} \right] \times \left[ \begin{array}{c} x_1 \\ \vdots \\ x_n \\ \end{array} \right] = \left[ \begin{array}{c} b_1 \\ \vdots \\ b_n \\ \end{array} \right] $$
So my question is, if we take the following augmented matrix: $$ \left[ \begin{array}{ccc|c} a_1 & \dots & a_n & b_1\\ a_1 & \ddots & \vdots & \vdots\\ a_1 & \dots & a_n & b_n\\ \end{array} \right] $$ If we were to put the above matrix into reduced row echelon form, (assuming we are given some numbers to plug into matrix A), while leaving the b vector in terms of $b_1, b_2, ... b_n$, the right side of the matrix would end up being just a bunch of combinations of the terms $b_1, b_2, ... b_n$.
My question is, what does this resulting vector tell us about the range of this function.
I am interested in any usefulness that this new vector has, or any information we can glean from it.